Fractal and Fractional Latest open access articles published in Fractal Fract. at https://www.mdpi.com/journal/fractalfract https://www.mdpi.com/journal/fractalfract MDPI en Creative Commons Attribution (CC-BY) MDPI support@mdpi.com Fractal Fract, Vol. 5, Pages 76: A Nonlocal Fractional Peridynamic Diffusion Model https://www.mdpi.com/2504-3110/5/3/76 This paper proposes a nonlocal fractional peridynamic (FPD) model to characterize the nonlocality of physical processes or systems, based on analysis with the fractional derivative model (FDM) and the peridynamic (PD) model. The main idea is to use the fractional Euler–Lagrange formula to establish a peridynamic anomalous diffusion model, in which the classical exponential kernel function is replaced by using a power-law kernel function. Fractional Taylor series expansion was used to construct a fractional peridynamic differential operator method to complete the above model. To explore the properties of the FPD model, the FDM, the PD model and the FPD model are dissected via numerical analysis on a diffusion process in complex media. The FPD model provides a generalized model connecting a local model and a nonlocal model for physical systems. The fractional peridynamic differential operator (FPDDO) method provides a simple and efficient numerical method for solving fractional derivative equations. Fractal Fract, Vol. 5, Pages 76: A Nonlocal Fractional Peridynamic Diffusion Model

Fractal and Fractional doi: 10.3390/fractalfract5030076

Authors: Yuanyuan Wang HongGuang Sun Siyuan Fan Yan Gu Xiangnan Yu

This paper proposes a nonlocal fractional peridynamic (FPD) model to characterize the nonlocality of physical processes or systems, based on analysis with the fractional derivative model (FDM) and the peridynamic (PD) model. The main idea is to use the fractional Euler–Lagrange formula to establish a peridynamic anomalous diffusion model, in which the classical exponential kernel function is replaced by using a power-law kernel function. Fractional Taylor series expansion was used to construct a fractional peridynamic differential operator method to complete the above model. To explore the properties of the FPD model, the FDM, the PD model and the FPD model are dissected via numerical analysis on a diffusion process in complex media. The FPD model provides a generalized model connecting a local model and a nonlocal model for physical systems. The fractional peridynamic differential operator (FPDDO) method provides a simple and efficient numerical method for solving fractional derivative equations.

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A Nonlocal Fractional Peridynamic Diffusion Model Yuanyuan Wang HongGuang Sun Siyuan Fan Yan Gu Xiangnan Yu doi: 10.3390/fractalfract5030076 Fractal and Fractional 2021-07-23 Fractal and Fractional 2021-07-23 5 3 Article 76 10.3390/fractalfract5030076 https://www.mdpi.com/2504-3110/5/3/76
Fractal Fract, Vol. 5, Pages 75: Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions https://www.mdpi.com/2504-3110/5/3/75 The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment. Fractal Fract, Vol. 5, Pages 75: Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract5030075

Authors: Gábor Maros Ferenc Izsák

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

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Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions Gábor Maros Ferenc Izsák doi: 10.3390/fractalfract5030075 Fractal and Fractional 2021-07-21 Fractal and Fractional 2021-07-21 5 3 Article 75 10.3390/fractalfract5030075 https://www.mdpi.com/2504-3110/5/3/75
Fractal Fract, Vol. 5, Pages 74: Modeling and Application of Fractional-Order Economic Growth Model with Time Delay https://www.mdpi.com/2504-3110/5/3/74 This paper proposes a fractional-order economic growth model with time delay based on the Solow model to describe the economic growth path and explore the underlying growth factors. It effectively captures memory characteristics in economic operations by adding a time lag to the capital stock. The proposed model is presented in the form of a fractional differential equations system, and the sufficient conditions for the local stability are obtained. In the simulation, the theoretical results are verified and the sensitivity analysis is performed on individual parameters. Based on the proposed model, we predict China’s GDP in the next thirty years through optimization and find medium-to-high-speed growth in the short term. Furthermore, the application results indicate that China is facing the disappearance of demographic dividend and the deceleration of capital accumulation. Therefore, it is urgent for China to increase the total factor productivity (TFP) and transform its economic growth into a trajectory dependent on TFP growth. Fractal Fract, Vol. 5, Pages 74: Modeling and Application of Fractional-Order Economic Growth Model with Time Delay

Fractal and Fractional doi: 10.3390/fractalfract5030074

Authors: Ziyi Lin Hu Wang

This paper proposes a fractional-order economic growth model with time delay based on the Solow model to describe the economic growth path and explore the underlying growth factors. It effectively captures memory characteristics in economic operations by adding a time lag to the capital stock. The proposed model is presented in the form of a fractional differential equations system, and the sufficient conditions for the local stability are obtained. In the simulation, the theoretical results are verified and the sensitivity analysis is performed on individual parameters. Based on the proposed model, we predict China’s GDP in the next thirty years through optimization and find medium-to-high-speed growth in the short term. Furthermore, the application results indicate that China is facing the disappearance of demographic dividend and the deceleration of capital accumulation. Therefore, it is urgent for China to increase the total factor productivity (TFP) and transform its economic growth into a trajectory dependent on TFP growth.

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Modeling and Application of Fractional-Order Economic Growth Model with Time Delay Ziyi Lin Hu Wang doi: 10.3390/fractalfract5030074 Fractal and Fractional 2021-07-21 Fractal and Fractional 2021-07-21 5 3 Article 74 10.3390/fractalfract5030074 https://www.mdpi.com/2504-3110/5/3/74
Fractal Fract, Vol. 5, Pages 73: Visualization of Mandelbrot and Julia Sets of Möbius Transformations https://www.mdpi.com/2504-3110/5/3/73 This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented. Fractal Fract, Vol. 5, Pages 73: Visualization of Mandelbrot and Julia Sets of Möbius Transformations

Fractal and Fractional doi: 10.3390/fractalfract5030073

Authors: Leah K. Mork Darin J. Ulness

This work reports on a study of the Mandelbrot set and Julia set for a generalization of the well-explored function η(z)=z2+λ. The generalization consists of composing with a fixed Möbius transformation at each iteration step. In particular, affine and inverse Möbius transformations are explored. This work offers a new way of visualizing the Mandelbrot and filled-in Julia sets. An interesting and unexpected appearance of hyperbolic triangles occurs in the structure of the Mandelbrot sets for the case of inverse Möbius transforms. Several lemmas and theorems associated with these types of fractal sets are presented.

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Visualization of Mandelbrot and Julia Sets of Möbius Transformations Leah K. Mork Darin J. Ulness doi: 10.3390/fractalfract5030073 Fractal and Fractional 2021-07-17 Fractal and Fractional 2021-07-17 5 3 Article 73 10.3390/fractalfract5030073 https://www.mdpi.com/2504-3110/5/3/73
Fractal Fract, Vol. 5, Pages 72: Lévy Processes Linked to the Lower-Incomplete Gamma Function https://www.mdpi.com/2504-3110/5/3/72 We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior. Fractal Fract, Vol. 5, Pages 72: Lévy Processes Linked to the Lower-Incomplete Gamma Function

Fractal and Fractional doi: 10.3390/fractalfract5030072

Authors: Luisa Beghin Costantino Ricciuti

We start by defining a subordinator by means of the lower-incomplete gamma function. This can be considered as an approximation of the stable subordinator, easier to be handled in view of its finite activity. A tempered version is also considered in order to overcome the drawback of infinite moments. Then, we study Lévy processes that are time-changed by these subordinators with particular attention to the Brownian case. An approximation of the fractional derivative (as well as of the fractional power of operators) arises from the analysis of governing equations. Finally, we show that time-changing the fractional Brownian motion produces a model of anomalous diffusion, which exhibits a sub-diffusive behavior.

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Lévy Processes Linked to the Lower-Incomplete Gamma Function Luisa Beghin Costantino Ricciuti doi: 10.3390/fractalfract5030072 Fractal and Fractional 2021-07-17 Fractal and Fractional 2021-07-17 5 3 Article 72 10.3390/fractalfract5030072 https://www.mdpi.com/2504-3110/5/3/72
Fractal Fract, Vol. 5, Pages 71: Approximation of Space-Time Fractional Equations https://www.mdpi.com/2504-3110/5/3/71 The aim of this paper is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes. Fractal Fract, Vol. 5, Pages 71: Approximation of Space-Time Fractional Equations

Fractal and Fractional doi: 10.3390/fractalfract5030071

Authors: Raffaela Capitanelli Mirko D’Ovidio

The aim of this paper is to provide approximation results for space-time non-local equations with general non-local (and fractional) operators in space and time. We consider a general Markov process time changed with general subordinators or inverses to general subordinators. Our analysis is based on Bernstein symbols and Dirichlet forms, where the symbols characterize the time changes, and the Dirichlet forms characterize the Markov processes.

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Approximation of Space-Time Fractional Equations Raffaela Capitanelli Mirko D’Ovidio doi: 10.3390/fractalfract5030071 Fractal and Fractional 2021-07-17 Fractal and Fractional 2021-07-17 5 3 Article 71 10.3390/fractalfract5030071 https://www.mdpi.com/2504-3110/5/3/71
Fractal Fract, Vol. 5, Pages 70: Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method https://www.mdpi.com/2504-3110/5/3/70 In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example. Fractal Fract, Vol. 5, Pages 70: Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method

Fractal and Fractional doi: 10.3390/fractalfract5030070

In this paper, by means of the second Chebyshev wavelet and its operational matrix, we solve a system of fractional-order Volterra–Fredholm integro-differential equations with weakly singular kernels. We estimate the functions by using the wavelet basis and then obtain the approximate solutions from the algebraic system corresponding to the main system. Moreover, the implementation of our scheme is presented, and the error bounds of approximations are analyzed. Finally, we evaluate the efficiency of the method through a numerical example.

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Solving a System of Fractional-Order Volterra-Fredholm Integro-Differential Equations with Weakly Singular Kernels via the Second Chebyshev Wavelets Method Esmail Bargamadi Leila Torkzadeh Kazem Nouri Amin Jajarmi doi: 10.3390/fractalfract5030070 Fractal and Fractional 2021-07-14 Fractal and Fractional 2021-07-14 5 3 Article 70 10.3390/fractalfract5030070 https://www.mdpi.com/2504-3110/5/3/70
Fractal Fract, Vol. 5, Pages 69: Iterated Functions Systems Composed of Generalized θ-Contractions https://www.mdpi.com/2504-3110/5/3/69 The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space X×⋯×X into X, where (X,d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS. Fractal Fract, Vol. 5, Pages 69: Iterated Functions Systems Composed of Generalized θ-Contractions

Fractal and Fractional doi: 10.3390/fractalfract5030069

Authors: Pasupathi Rajan María A. Navascués Arya Kumar Bedabrata Chand

The theory of iterated function systems (IFSs) has been an active area of research on fractals and various types of self-similarity in nature. The basic theoretical work on IFSs has been proposed by Hutchinson. In this paper, we introduce a new generalization of Hutchinson IFS, namely generalized θ-contraction IFS, which is a finite collection of generalized θ-contraction functions T1,…,TN from finite Cartesian product space X×⋯×X into X, where (X,d) is a complete metric space. We prove the existence of attractor for this generalized IFS. We show that the Hutchinson operators for countable and multivalued θ-contraction IFSs are Picard. Finally, when the map θ is continuous, we show the relation between the code space and the attractor of θ-contraction IFS.

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Iterated Functions Systems Composed of Generalized θ-Contractions Pasupathi Rajan María A. Navascués Arya Kumar Bedabrata Chand doi: 10.3390/fractalfract5030069 Fractal and Fractional 2021-07-14 Fractal and Fractional 2021-07-14 5 3 Article 69 10.3390/fractalfract5030069 https://www.mdpi.com/2504-3110/5/3/69
Fractal Fract, Vol. 5, Pages 68: Some New Results on Hermite–Hadamard–Mercer-Type Inequalities Using a General Family of Fractional Integral Operators https://www.mdpi.com/2504-3110/5/3/68 The aim of this article is to obtain new Hermite–Hadamard–Mercer-type inequalities using Raina’s fractional integral operators. We present some distinct and novel fractional Hermite–Hadamard–Mercer-type inequalities for the functions whose absolute value of derivatives are convex. Our main findings are generalizations and extensions of some results that existed in the literature. Fractal Fract, Vol. 5, Pages 68: Some New Results on Hermite–Hadamard–Mercer-Type Inequalities Using a General Family of Fractional Integral Operators

Fractal and Fractional doi: 10.3390/fractalfract5030068

Authors: Erhan Set Barış Çelik M. Emin Özdemir Mücahit Aslan

The aim of this article is to obtain new Hermite–Hadamard–Mercer-type inequalities using Raina’s fractional integral operators. We present some distinct and novel fractional Hermite–Hadamard–Mercer-type inequalities for the functions whose absolute value of derivatives are convex. Our main findings are generalizations and extensions of some results that existed in the literature.

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Some New Results on Hermite–Hadamard–Mercer-Type Inequalities Using a General Family of Fractional Integral Operators Erhan Set Barış Çelik M. Emin Özdemir Mücahit Aslan doi: 10.3390/fractalfract5030068 Fractal and Fractional 2021-07-13 Fractal and Fractional 2021-07-13 5 3 Article 68 10.3390/fractalfract5030068 https://www.mdpi.com/2504-3110/5/3/68
Fractal Fract, Vol. 5, Pages 67: Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia https://www.mdpi.com/2504-3110/5/3/67 We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency-phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system. Fractal Fract, Vol. 5, Pages 67: Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia

Fractal and Fractional doi: 10.3390/fractalfract5030067

Authors: Jun-Sheng Duan Di-Chen Hu

We considered forced harmonic vibration systems with the Liouville–Weyl fractional derivative where the order is between 1 and 2 and with a distributed-order derivative where the Liouville–Weyl fractional derivatives are integrated on the interval [1, 2] with respect to the order. Both types of derivatives enhance the viscosity and inertia of the system and contribute to damping and mass, respectively. Hence, such types of derivatives characterize the viscoinertia and represent an “inerter-pot” element. For such vibration systems, we derived the equivalent damping and equivalent mass and gave the equivalent integer-order vibration systems. Particularly, for the distributed-order vibration model where the weight function was taken as an exponential function that involved a parameter, we gave detailed analyses for the weight function, the damping contribution, and the mass contribution. Frequency–amplitude curves and frequency-phase curves were plotted for various coefficients and parameters for the comparison of the two types of vibration models. In the distributed-order vibration system, the weight function of the order enables us to simultaneously involve different orders, whilst the fractional-order model has a single order. Thus, the distributed-order vibration model is more general and flexible than the fractional vibration system.

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Vibration Systems with Fractional-Order and Distributed-Order Derivatives Characterizing Viscoinertia Jun-Sheng Duan Di-Chen Hu doi: 10.3390/fractalfract5030067 Fractal and Fractional 2021-07-12 Fractal and Fractional 2021-07-12 5 3 Article 67 10.3390/fractalfract5030067 https://www.mdpi.com/2504-3110/5/3/67
Fractal Fract, Vol. 5, Pages 66: Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation https://www.mdpi.com/2504-3110/5/3/66 This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations. Fractal Fract, Vol. 5, Pages 66: Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation

Fractal and Fractional doi: 10.3390/fractalfract5030066

Authors: Azmat Ullah Khan Niazi Jiawei He Ramsha Shafqat Bilal Ahmed

This paper concerns with the existence and uniqueness of the Cauchy problem for a system of fuzzy fractional differential equation with Caputo derivative of order q∈(1,2], 0cD0+qu(t)=λu(t)⊕f(t,u(t))⊕B(t)C(t),t∈[0,T] with initial conditions u(0)=u0,u′(0)=u1. Moreover, by using direct analytic methods, the Eq–Ulam-type results are also presented. In addition, several examples are given which show the applicability of fuzzy fractional differential equations.

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Existence, Uniqueness, and Eq–Ulam-Type Stability of Fuzzy Fractional Differential Equation Azmat Ullah Khan Niazi Jiawei He Ramsha Shafqat Bilal Ahmed doi: 10.3390/fractalfract5030066 Fractal and Fractional 2021-07-11 Fractal and Fractional 2021-07-11 5 3 Article 66 10.3390/fractalfract5030066 https://www.mdpi.com/2504-3110/5/3/66
Fractal Fract, Vol. 5, Pages 65: Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior https://www.mdpi.com/2504-3110/5/3/65 This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1&amp;lt;d&amp;lt;2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t−1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model. Fractal Fract, Vol. 5, Pages 65: Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior

Fractal and Fractional doi: 10.3390/fractalfract5030065

Authors: Vincent Tartaglione Jocelyn Sabatier Christophe Farges

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1&amp;lt;d&amp;lt;2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t−1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model.

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Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior Vincent Tartaglione Jocelyn Sabatier Christophe Farges doi: 10.3390/fractalfract5030065 Fractal and Fractional 2021-07-10 Fractal and Fractional 2021-07-10 5 3 Article 65 10.3390/fractalfract5030065 https://www.mdpi.com/2504-3110/5/3/65
Fractal Fract, Vol. 5, Pages 64: Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator https://www.mdpi.com/2504-3110/5/3/64 In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator. Fractal Fract, Vol. 5, Pages 64: Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator

Fractal and Fractional doi: 10.3390/fractalfract5030064

Authors: Igor V. Malyk Mykola Gorbatenko Arun Chaudhary Shivani Sharma Ravi Shanker Dubey

In this manuscript, the time-fractional diffusion equation in the framework of the Yang–Abdel–Cattani derivative operator is taken into account. A detailed proof for the existence, as well as the uniqueness of the solution of the time-fractional diffusion equation, in the sense of YAC derivative operator, is explained, and, using the method of α-HATM, we find the analytical solution of the time-fractional diffusion equation. Three cases are considered to exhibit the convergence and fidelity of the aforementioned α-HATM. The analytical solutions obtained for the diffusion equation using the Yang–Abdel–Cattani derivative operator are compared with the analytical solutions obtained using the Riemann–Liouville (RL) derivative operator for the fractional order γ=0.99 (nearby 1) and with the exact solution at different values of t to verify the efficiency of the YAC derivative operator.

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Numerical Solution of Nonlinear Fractional Diffusion Equation in Framework of the Yang–Abdel–Cattani Derivative Operator Igor V. Malyk Mykola Gorbatenko Arun Chaudhary Shivani Sharma Ravi Shanker Dubey doi: 10.3390/fractalfract5030064 Fractal and Fractional 2021-07-02 Fractal and Fractional 2021-07-02 5 3 Article 64 10.3390/fractalfract5030064 https://www.mdpi.com/2504-3110/5/3/64
Fractal Fract, Vol. 5, Pages 63: An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions https://www.mdpi.com/2504-3110/5/3/63 An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established. Fractal Fract, Vol. 5, Pages 63: An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions

Fractal and Fractional doi: 10.3390/fractalfract5030063

Authors: Emilia Bazhlekova

An initial-boundary-value problem is considered for the one-dimensional diffusion equation with a general convolutional derivative in time and nonclassical boundary conditions. We are concerned with the inverse source problem of recovery of a space-dependent source term from given final time data. Generalized eigenfunction expansions are used with respect to a biorthogonal pair of bases. Existence, uniqueness and stability estimates in Sobolev spaces are established.

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An Inverse Source Problem for the Generalized Subdiffusion Equation with Nonclassical Boundary Conditions Emilia Bazhlekova doi: 10.3390/fractalfract5030063 Fractal and Fractional 2021-06-30 Fractal and Fractional 2021-06-30 5 3 Article 63 10.3390/fractalfract5030063 https://www.mdpi.com/2504-3110/5/3/63
Fractal Fract, Vol. 5, Pages 62: Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games https://www.mdpi.com/2504-3110/5/3/62 This paper investigates the distributed computation issue of generalized Nash equilibrium (GNE) in a multi-player game with shared coupling constraints. Two kinds of relatively fast distributed algorithms are constructed with alternating inertia and overrelaxation in the partial-decision information setting. We prove their convergence to GNE with fixed step-sizes by resorting to the operator splitting technique under the assumptions of Lipschitz continuity of the extended pseudo-gradient mappings. Finally, one numerical simulation is given to illustrate the efficiency and performance of the algorithm. Fractal Fract, Vol. 5, Pages 62: Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games

Fractal and Fractional doi: 10.3390/fractalfract5030062

Authors: Zhangcheng Feng Wenying Xu Jinde Cao

This paper investigates the distributed computation issue of generalized Nash equilibrium (GNE) in a multi-player game with shared coupling constraints. Two kinds of relatively fast distributed algorithms are constructed with alternating inertia and overrelaxation in the partial-decision information setting. We prove their convergence to GNE with fixed step-sizes by resorting to the operator splitting technique under the assumptions of Lipschitz continuity of the extended pseudo-gradient mappings. Finally, one numerical simulation is given to illustrate the efficiency and performance of the algorithm.

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Alternating Inertial and Overrelaxed Algorithms for Distributed Generalized Nash Equilibrium Seeking in Multi-Player Games Zhangcheng Feng Wenying Xu Jinde Cao doi: 10.3390/fractalfract5030062 Fractal and Fractional 2021-06-28 Fractal and Fractional 2021-06-28 5 3 Article 62 10.3390/fractalfract5030062 https://www.mdpi.com/2504-3110/5/3/62
Fractal Fract, Vol. 5, Pages 61: Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations https://www.mdpi.com/2504-3110/5/3/61 Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved. Fractal Fract, Vol. 5, Pages 61: Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations

Fractal and Fractional doi: 10.3390/fractalfract5030061

Authors: Stanislav Harizanov Nikola Kosturski Ivan Lirkov Svetozar Margenov Yavor Vutov

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, Ω⊂Rd, d∈{1,2,3}. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aαu=f, α∈(0,1). Although matrix A∈RN×N is sparse, symmetric and positive definite (SPD), matrix Aα is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation rα,k(t) of degree k of tα in the interval [0,1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(Nlog2N) can be improved.

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Reduced Multiplicative (BURA-MR) and Additive (BURA-AR) Best Uniform Rational Approximation Methods and Algorithms for Fractional Elliptic Equations Stanislav Harizanov Nikola Kosturski Ivan Lirkov Svetozar Margenov Yavor Vutov doi: 10.3390/fractalfract5030061 Fractal and Fractional 2021-06-28 Fractal and Fractional 2021-06-28 5 3 Article 61 10.3390/fractalfract5030061 https://www.mdpi.com/2504-3110/5/3/61
Fractal Fract, Vol. 5, Pages 60: On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus https://www.mdpi.com/2504-3110/5/3/60 Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods. Fractal Fract, Vol. 5, Pages 60: On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus

Fractal and Fractional doi: 10.3390/fractalfract5030060

Authors: Gul Sana Pshtiwan Othman Mohammed Dong Yun Shin Muhmmad Aslam Noor Mohammad Salem Oudat

Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods.

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On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus Gul Sana Pshtiwan Othman Mohammed Dong Yun Shin Muhmmad Aslam Noor Mohammad Salem Oudat doi: 10.3390/fractalfract5030060 Fractal and Fractional 2021-06-25 Fractal and Fractional 2021-06-25 5 3 Article 60 10.3390/fractalfract5030060 https://www.mdpi.com/2504-3110/5/3/60
Fractal Fract, Vol. 5, Pages 59: Degenerate Derangement Polynomials and Numbers https://www.mdpi.com/2504-3110/5/3/59 In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al. Fractal Fract, Vol. 5, Pages 59: Degenerate Derangement Polynomials and Numbers

Fractal and Fractional doi: 10.3390/fractalfract5030059

Authors: Minyoung Ma Dongkyu Lim

In this paper, we consider a new type of degenerate derangement polynomial and number, which shall be called the degenerate derangement polynomials and numbers of the second kind. These concepts are motivated by Kim et al.’s work on degenerate derangement polynomials and numbers. We investigate some properties of these new degenerate derangement polynomials and numbers and explore their connections with the degenerate gamma distributions for the case λ∈(−1,0). In more detail, we derive their explicit expressions, recurrence relations, and some identities involving our degenerate derangement polynomials and numbers and other special polynomials and numbers, which include the fully degenerate Bell polynomials, the degenerate Fubini polynomials, and the degenerate Stirling numbers of the first and the second kinds. We also show that those polynomials and numbers are connected with the moments of some variants of the degenerate gamma distributions. Moreover, we compare the degenerate derangement polynomials and numbers of the second kind to those of Kim et al.

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Degenerate Derangement Polynomials and Numbers Minyoung Ma Dongkyu Lim doi: 10.3390/fractalfract5030059 Fractal and Fractional 2021-06-22 Fractal and Fractional 2021-06-22 5 3 Article 59 10.3390/fractalfract5030059 https://www.mdpi.com/2504-3110/5/3/59
Fractal Fract, Vol. 5, Pages 58: Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration https://www.mdpi.com/2504-3110/5/2/58 In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0&amp;lt;α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven. Fractal Fract, Vol. 5, Pages 58: Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration

Fractal and Fractional doi: 10.3390/fractalfract5020058

Authors: Tursun K. Yuldashev Bakhtiyar J. Kadirkulov

In the three-dimensional open rectangular domain, the problem of the identification of the redefinition function for a partial differential equation with Gerasimov–Caputo-type fractional operator, degeneration, and integral form condition is considered in the case of the 0&amp;lt;α≤1 order. A positive parameter is present in the mixed derivatives. The solution of this fractional differential equation is studied in the class of regular functions. The Fourier series method is used, and a countable system of ordinary fractional differential equations with degeneration is obtained. The presentation for the redefinition function is obtained using a given additional condition. Using the Cauchy–Schwarz inequality and the Bessel inequality, the absolute and uniform convergence of the obtained Fourier series is proven.

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Inverse Problem for a Partial Differential Equation with Gerasimov–Caputo-Type Operator and Degeneration Tursun K. Yuldashev Bakhtiyar J. Kadirkulov doi: 10.3390/fractalfract5020058 Fractal and Fractional 2021-06-21 Fractal and Fractional 2021-06-21 5 2 Article 58 10.3390/fractalfract5020058 https://www.mdpi.com/2504-3110/5/2/58
Fractal Fract, Vol. 5, Pages 57: Solutions of Bernoulli Equations in the Fractional Setting https://www.mdpi.com/2504-3110/5/2/57 We present a general series representation formula for the local solution of the Bernoulli equation with Caputo fractional derivatives. We then focus on a generalization of the fractional logistic equation and present some related numerical simulations. Fractal Fract, Vol. 5, Pages 57: Solutions of Bernoulli Equations in the Fractional Setting

Fractal and Fractional doi: 10.3390/fractalfract5020057

Authors: Mirko D’Ovidio Anna Chiara Lai Paola Loreti

We present a general series representation formula for the local solution of the Bernoulli equation with Caputo fractional derivatives. We then focus on a generalization of the fractional logistic equation and present some related numerical simulations.

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Solutions of Bernoulli Equations in the Fractional Setting Mirko D’Ovidio Anna Chiara Lai Paola Loreti doi: 10.3390/fractalfract5020057 Fractal and Fractional 2021-06-17 Fractal and Fractional 2021-06-17 5 2 Article 57 10.3390/fractalfract5020057 https://www.mdpi.com/2504-3110/5/2/57
Fractal Fract, Vol. 5, Pages 56: Numerical Simulation for the Treatment of Nonlinear Predator–Prey Equations by Using the Finite Element Optimization Method https://www.mdpi.com/2504-3110/5/2/56 This article aims to introduce an efficient simulation to obtain the solution for a dynamical–biological system, which is called the Lotka–Volterra system, involving predator–prey equations. The finite element method (FEM) is employed to solve this problem. This technique is based mainly upon the appropriate conversion of the proposed model to a system of algebraic equations. The resulting system is then constructed as a constrained optimization problem and optimized in order to get the unknown coefficients and, consequently, the solution itself. We call this combination of the two well-known methods the finite element optimization method (FEOM). We compare the obtained results with the solutions obtained by using the fourth-order Runge–Kutta method (RK4 method). The residual error function is evaluated, which supports the efficiency and the accuracy of the presented procedure. From the given results, we can say that the presented procedure provides an easy and efficient tool to investigate the solution for such models as those investigated in this paper. Fractal Fract, Vol. 5, Pages 56: Numerical Simulation for the Treatment of Nonlinear Predator–Prey Equations by Using the Finite Element Optimization Method

Fractal and Fractional doi: 10.3390/fractalfract5020056

Authors: H. M. Srivastava M. M. Khader

This article aims to introduce an efficient simulation to obtain the solution for a dynamical–biological system, which is called the Lotka–Volterra system, involving predator–prey equations. The finite element method (FEM) is employed to solve this problem. This technique is based mainly upon the appropriate conversion of the proposed model to a system of algebraic equations. The resulting system is then constructed as a constrained optimization problem and optimized in order to get the unknown coefficients and, consequently, the solution itself. We call this combination of the two well-known methods the finite element optimization method (FEOM). We compare the obtained results with the solutions obtained by using the fourth-order Runge–Kutta method (RK4 method). The residual error function is evaluated, which supports the efficiency and the accuracy of the presented procedure. From the given results, we can say that the presented procedure provides an easy and efficient tool to investigate the solution for such models as those investigated in this paper.

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Numerical Simulation for the Treatment of Nonlinear Predator–Prey Equations by Using the Finite Element Optimization Method H. M. Srivastava M. M. Khader doi: 10.3390/fractalfract5020056 Fractal and Fractional 2021-06-16 Fractal and Fractional 2021-06-16 5 2 Article 56 10.3390/fractalfract5020056 https://www.mdpi.com/2504-3110/5/2/56
Fractal Fract, Vol. 5, Pages 55: On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm https://www.mdpi.com/2504-3110/5/2/55 In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm. Fractal Fract, Vol. 5, Pages 55: On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm

Fractal and Fractional doi: 10.3390/fractalfract5020055

Authors: Yang Zhao Shicun Zhao Yi Zhang Da Wang

In this paper, a novel escape-time algorithm is proposed to calculate the connectivity’s degree of Julia sets generated from polynomial maps. The proposed algorithm contains both quantitative analysis and visual display to measure the connectivity of Julia sets. For the quantitative part, a connectivity criterion method is designed by exploring the distribution rule of the connected regions, with an output value Co in the range of [0,1]. The smaller the Co value outputs, the better the connectivity is. For the visual part, we modify the classical escape-time algorithm by highlighting and separating the initial point of each connected area. Finally, the Julia set is drawn into different brightnesses according to different Co values. The darker the color, the better the connectivity of the Julia set. Numerical results are included to assess the efficiency of the algorithm.

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On the Connectivity Measurement of the Fractal Julia Sets Generated from Polynomial Maps: A Novel Escape-Time Algorithm Yang Zhao Shicun Zhao Yi Zhang Da Wang doi: 10.3390/fractalfract5020055 Fractal and Fractional 2021-06-13 Fractal and Fractional 2021-06-13 5 2 Article 55 10.3390/fractalfract5020055 https://www.mdpi.com/2504-3110/5/2/55
Fractal Fract, Vol. 5, Pages 54: Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities https://www.mdpi.com/2504-3110/5/2/54 In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations. Fractal Fract, Vol. 5, Pages 54: Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities

Fractal and Fractional doi: 10.3390/fractalfract5020054

Authors: Rana Safdar Ali Aiman Mukheimer Thabet Abdeljawad Shahid Mubeen Sabila Ali Gauhar Rahman Kottakkaran Sooppy Nisar

In this article, we established a new version of generalized fractional Hadamard and Fejér–Hadamard type integral inequalities. A fractional integral operator (FIO) with a non-singular function (multi-index Bessel function) as its kernel and monotone increasing functions is utilized to obtain the new version of such fractional inequalities. Our derived results are a generalized form of several proven inequalities already existing in the literature. The proven inequalities are useful for studying the stability and control of corresponding fractional dynamic equations.

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Some New Harmonically Convex Function Type Generalized Fractional Integral Inequalities Rana Safdar Ali Aiman Mukheimer Thabet Abdeljawad Shahid Mubeen Sabila Ali Gauhar Rahman Kottakkaran Sooppy Nisar doi: 10.3390/fractalfract5020054 Fractal and Fractional 2021-06-07 Fractal and Fractional 2021-06-07 5 2 Article 54 10.3390/fractalfract5020054 https://www.mdpi.com/2504-3110/5/2/54
Fractal Fract, Vol. 5, Pages 53: Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation https://www.mdpi.com/2504-3110/5/2/53 This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm. Fractal Fract, Vol. 5, Pages 53: Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation

Fractal and Fractional doi: 10.3390/fractalfract5020053

Authors: Waleed Mohamed Abd-Elhameed

This paper is concerned with establishing novel expressions that express the derivative of any order of the orthogonal polynomials, namely, Chebyshev polynomials of the sixth kind in terms of Chebyshev polynomials themselves. We will prove that these expressions involve certain terminating hypergeometric functions of the type 4F3(1) that can be reduced in some specific cases. The derived expressions along with the linearization formula of Chebyshev polynomials of the sixth kind serve in obtaining a numerical solution of the non-linear one-dimensional Burgers’ equation based on the application of the spectral tau method. Convergence analysis of the proposed double shifted Chebyshev expansion of the sixth kind is investigated. Numerical results are displayed aiming to show the efficiency and applicability of the proposed algorithm.

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Novel Expressions for the Derivatives of Sixth Kind Chebyshev Polynomials: Spectral Solution of the Non-Linear One-Dimensional Burgers’ Equation Waleed Mohamed Abd-Elhameed doi: 10.3390/fractalfract5020053 Fractal and Fractional 2021-06-06 Fractal and Fractional 2021-06-06 5 2 Article 53 10.3390/fractalfract5020053 https://www.mdpi.com/2504-3110/5/2/53
Fractal Fract, Vol. 5, Pages 52: Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay https://www.mdpi.com/2504-3110/5/2/52 The existence of solutions for a class of nonlinear neutral Hadamard-type fractional integro-differential equations with infinite delay is researched in this paper. By constructing an appropriate normed space and utilizing the Banach contraction principle, Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions for the existence of solutions. Finally, we provide an example to illustrate the validity of our main results. Fractal Fract, Vol. 5, Pages 52: Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay

Fractal and Fractional doi: 10.3390/fractalfract5020052

Authors: Kaihong Zhao Yue Ma

The existence of solutions for a class of nonlinear neutral Hadamard-type fractional integro-differential equations with infinite delay is researched in this paper. By constructing an appropriate normed space and utilizing the Banach contraction principle, Krasnoselskii’s fixed point theorem, we obtain some sufficient conditions for the existence of solutions. Finally, we provide an example to illustrate the validity of our main results.

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Study on the Existence of Solutions for a Class of Nonlinear Neutral Hadamard-Type Fractional Integro-Differential Equation with Infinite Delay Kaihong Zhao Yue Ma doi: 10.3390/fractalfract5020052 Fractal and Fractional 2021-06-04 Fractal and Fractional 2021-06-04 5 2 Article 52 10.3390/fractalfract5020052 https://www.mdpi.com/2504-3110/5/2/52
Fractal Fract, Vol. 5, Pages 51: Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach https://www.mdpi.com/2504-3110/5/2/51 In recent years, different experimental works with molecular simulation techniques have been developed to study the transport of plasma-generated reactive species in liquid layers. Here, we improve the classical transport model that describes the molecular species movement in liquid layers via considering the fractional reaction–telegraph equation. We have considered the fractional equation to describe a non-Brownian motion of molecular species in a liquid layer, which have different diffusivities. The analytical solution of the fractional reaction–telegraph equation, which is defined in terms of the Caputo fractional derivative, is obtained by using the Laplace–Fourier technique. The profiles of species density with the mean square displacement are discussed in each case for different values of the time-fractional order and relaxation time. Fractal Fract, Vol. 5, Pages 51: Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach

Fractal and Fractional doi: 10.3390/fractalfract5020051

Authors: Ashraf M. Tawfik Mohamed Mokhtar Hefny

In recent years, different experimental works with molecular simulation techniques have been developed to study the transport of plasma-generated reactive species in liquid layers. Here, we improve the classical transport model that describes the molecular species movement in liquid layers via considering the fractional reaction–telegraph equation. We have considered the fractional equation to describe a non-Brownian motion of molecular species in a liquid layer, which have different diffusivities. The analytical solution of the fractional reaction–telegraph equation, which is defined in terms of the Caputo fractional derivative, is obtained by using the Laplace–Fourier technique. The profiles of species density with the mean square displacement are discussed in each case for different values of the time-fractional order and relaxation time.

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Subdiffusive Reaction Model of Molecular Species in Liquid Layers: Fractional Reaction-Telegraph Approach Ashraf M. Tawfik Mohamed Mokhtar Hefny doi: 10.3390/fractalfract5020051 Fractal and Fractional 2021-06-03 Fractal and Fractional 2021-06-03 5 2 Article 51 10.3390/fractalfract5020051 https://www.mdpi.com/2504-3110/5/2/51
Fractal Fract, Vol. 5, Pages 50: Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set https://www.mdpi.com/2504-3110/5/2/50 We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the subordination theory. The fractal functions are suggested for the multi-parametric coefficients type motorboat fractal set. We obtain different formulas of fractal analytic solutions of LDEs. Moreover, we determine the maximum value of the fractal coefficients to obtain the optimal solution. Through the subordination inequality, we determined the upper boundary determination of a class of fractal functions holding multibrot function ϑ(z)=1+3κz+z3. Fractal Fract, Vol. 5, Pages 50: Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set

Fractal and Fractional doi: 10.3390/fractalfract5020050

Authors: Rabha W. Ibrahim Dumitru Baleanu

We present an analytic solvability of a class of Langevin differential equations (LDEs) in the asset of geometric function theory. The analytic solutions of the LDEs are presented by utilizing a special kind of fractal function in a complex domain, linked with the subordination theory. The fractal functions are suggested for the multi-parametric coefficients type motorboat fractal set. We obtain different formulas of fractal analytic solutions of LDEs. Moreover, we determine the maximum value of the fractal coefficients to obtain the optimal solution. Through the subordination inequality, we determined the upper boundary determination of a class of fractal functions holding multibrot function ϑ(z)=1+3κz+z3.

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Analytic Solution of the Langevin Differential Equations Dominated by a Multibrot Fractal Set Rabha W. Ibrahim Dumitru Baleanu doi: 10.3390/fractalfract5020050 Fractal and Fractional 2021-05-25 Fractal and Fractional 2021-05-25 5 2 Article 50 10.3390/fractalfract5020050 https://www.mdpi.com/2504-3110/5/2/50
Fractal Fract, Vol. 5, Pages 49: Fat Tail in the Phytoplankton Movement Patterns and Swimming Behavior: New Insights into the Prey-Predator Interactions https://www.mdpi.com/2504-3110/5/2/49 Phytoplankton movement patterns and swimming behavior are important and basic topics in aquatic biology. Heavy tail distribution exists in diverse taxa and shows theoretical advantages in environments. The fat tails in the movement patterns and swimming behavior of phytoplankton in response to the food supply were studied. The log-normal distribution was used for fitting the probability density values of the movement data of Oxyrrhis marina. Results showed that obvious fat tails exist in the movement patterns of O. marina without and with positive stimulations of food supply. The algal cells tended to show a more chaotic and disorderly movement, with shorter and neat steps after adding the food source. At the same time, the randomness of turning rate, path curvature and swimming speed increased in O. marina cells with food supply. Generally, the responses of phytoplankton movement were stronger when supplied with direct prey cells rather than the cell-free filtrate. The scale-free random movements are considered to benefit the adaption of the entire phytoplankton population to varied environmental conditions. Inferentially, the movement pattern of O. marina should also have the characteristics of long-range dependence, local self-similarity and a system of fractional order. Fractal Fract, Vol. 5, Pages 49: Fat Tail in the Phytoplankton Movement Patterns and Swimming Behavior: New Insights into the Prey-Predator Interactions

Fractal and Fractional doi: 10.3390/fractalfract5020049

Authors: Xi Xiao Caicai Xu Yan Yu Junyu He Ming Li Carlo Cattani

Phytoplankton movement patterns and swimming behavior are important and basic topics in aquatic biology. Heavy tail distribution exists in diverse taxa and shows theoretical advantages in environments. The fat tails in the movement patterns and swimming behavior of phytoplankton in response to the food supply were studied. The log-normal distribution was used for fitting the probability density values of the movement data of Oxyrrhis marina. Results showed that obvious fat tails exist in the movement patterns of O. marina without and with positive stimulations of food supply. The algal cells tended to show a more chaotic and disorderly movement, with shorter and neat steps after adding the food source. At the same time, the randomness of turning rate, path curvature and swimming speed increased in O. marina cells with food supply. Generally, the responses of phytoplankton movement were stronger when supplied with direct prey cells rather than the cell-free filtrate. The scale-free random movements are considered to benefit the adaption of the entire phytoplankton population to varied environmental conditions. Inferentially, the movement pattern of O. marina should also have the characteristics of long-range dependence, local self-similarity and a system of fractional order.

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Fat Tail in the Phytoplankton Movement Patterns and Swimming Behavior: New Insights into the Prey-Predator Interactions Xi Xiao Caicai Xu Yan Yu Junyu He Ming Li Carlo Cattani doi: 10.3390/fractalfract5020049 Fractal and Fractional 2021-05-25 Fractal and Fractional 2021-05-25 5 2 Article 49 10.3390/fractalfract5020049 https://www.mdpi.com/2504-3110/5/2/49
Fractal Fract, Vol. 5, Pages 48: Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions https://www.mdpi.com/2504-3110/5/2/48 In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation. Fractal Fract, Vol. 5, Pages 48: Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions

Fractal and Fractional doi: 10.3390/fractalfract5020048

Authors: Alessandro De Gregorio Roberto Garra

In this paper, we study diffusion equations involving Hadamard-type time-fractional derivatives related to ultra-slow random models. We start our analysis using the abstract fractional Cauchy problem, replacing the classical time derivative with the Hadamard operator. The stochastic meaning of the introduced abstract differential equation is provided, and the application to the particular case of the fractional heat equation is then discussed in detail. The ultra-slow behaviour emerges from the explicit form of the variance of the random process arising from our analysis. Finally, we obtain a particular solution for the nonlinear Hadamard-diffusive equation.

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Hadamard-Type Fractional Heat Equations and Ultra-Slow Diffusions Alessandro De Gregorio Roberto Garra doi: 10.3390/fractalfract5020048 Fractal and Fractional 2021-05-23 Fractal and Fractional 2021-05-23 5 2 Article 48 10.3390/fractalfract5020048 https://www.mdpi.com/2504-3110/5/2/48
Fractal Fract, Vol. 5, Pages 47: Structure, Fractality, Mechanics and Durability of Calcium Silicate Hydrates https://www.mdpi.com/2504-3110/5/2/47 Cement-based materials are widely utilized in infrastructure. The main product of hydrated products of cement-based materials is calcium silicate hydrate (C-S-H) gels that are considered as the binding phase of cement paste. C-S-H gels in Portland cement paste account for 60–70% of hydrated products by volume, which has profound influence on the mechanical properties and durability of cement-based materials. The preparation method of C-S-H gels has been well documented, but the quality of the prepared C-S-H affects experimental results; therefore, this review studies the preparation method of C-S-H under different conditions and materials. The progress related to C-S-H microstructure is explored from the theoretical and computational point of view. The fractality of C-S-H is discussed. An evaluation of the mechanical properties of C-S-H has also been included in this review. Finally, there is a discussion of the durability of C-S-H, with special reference to the carbonization and chloride/sulfate attacks. Fractal Fract, Vol. 5, Pages 47: Structure, Fractality, Mechanics and Durability of Calcium Silicate Hydrates

Fractal and Fractional doi: 10.3390/fractalfract5020047

Authors: Shengwen Tang Yang Wang Zhicheng Geng Xiaofei Xu Wenzhi Yu Hubao A Jingtao Chen

Cement-based materials are widely utilized in infrastructure. The main product of hydrated products of cement-based materials is calcium silicate hydrate (C-S-H) gels that are considered as the binding phase of cement paste. C-S-H gels in Portland cement paste account for 60–70% of hydrated products by volume, which has profound influence on the mechanical properties and durability of cement-based materials. The preparation method of C-S-H gels has been well documented, but the quality of the prepared C-S-H affects experimental results; therefore, this review studies the preparation method of C-S-H under different conditions and materials. The progress related to C-S-H microstructure is explored from the theoretical and computational point of view. The fractality of C-S-H is discussed. An evaluation of the mechanical properties of C-S-H has also been included in this review. Finally, there is a discussion of the durability of C-S-H, with special reference to the carbonization and chloride/sulfate attacks.

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Structure, Fractality, Mechanics and Durability of Calcium Silicate Hydrates Shengwen Tang Yang Wang Zhicheng Geng Xiaofei Xu Wenzhi Yu Hubao A Jingtao Chen doi: 10.3390/fractalfract5020047 Fractal and Fractional 2021-05-17 Fractal and Fractional 2021-05-17 5 2 Review 47 10.3390/fractalfract5020047 https://www.mdpi.com/2504-3110/5/2/47
Fractal Fract, Vol. 5, Pages 46: Design of Fractional-Order Lead Compensator for a Car Suspension System Based on Curve-Fitting Approximation https://www.mdpi.com/2504-3110/5/2/46 An alternative procedure for the implementation of fractional-order compensators is presented in this work. The employment of a curve-fitting-based approximation technique for the approximation of the compensator transfer function offers improved accuracy compared to the Oustaloup and Padé methods. As a design example, a lead compensator intended for usage in car suspension systems is realized. The open-loop and closed-loop behavior of the system is evaluated by post-layout simulation results obtained using the Cadence IC design suite and the Metal Oxide Semiconductor (MOS) transistor models provided by the Austria Mikro Systeme 0.35 μm Complementary Metal Oxide Semiconductor (CMOS) process. The derived results verify the efficient performance of the introduced implementation. Fractal Fract, Vol. 5, Pages 46: Design of Fractional-Order Lead Compensator for a Car Suspension System Based on Curve-Fitting Approximation

Fractal and Fractional doi: 10.3390/fractalfract5020046

Authors: Evisa Memlikai Stavroula Kapoulea Costas Psychalinos Jerzy Baranowski Waldemar Bauer Andrzej Tutaj Paweł Piątek

An alternative procedure for the implementation of fractional-order compensators is presented in this work. The employment of a curve-fitting-based approximation technique for the approximation of the compensator transfer function offers improved accuracy compared to the Oustaloup and Padé methods. As a design example, a lead compensator intended for usage in car suspension systems is realized. The open-loop and closed-loop behavior of the system is evaluated by post-layout simulation results obtained using the Cadence IC design suite and the Metal Oxide Semiconductor (MOS) transistor models provided by the Austria Mikro Systeme 0.35 μm Complementary Metal Oxide Semiconductor (CMOS) process. The derived results verify the efficient performance of the introduced implementation.

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Design of Fractional-Order Lead Compensator for a Car Suspension System Based on Curve-Fitting Approximation Evisa Memlikai Stavroula Kapoulea Costas Psychalinos Jerzy Baranowski Waldemar Bauer Andrzej Tutaj Paweł Piątek doi: 10.3390/fractalfract5020046 Fractal and Fractional 2021-05-15 Fractal and Fractional 2021-05-15 5 2 Article 46 10.3390/fractalfract5020046 https://www.mdpi.com/2504-3110/5/2/46
Fractal Fract, Vol. 5, Pages 45: On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators https://www.mdpi.com/2504-3110/5/2/45 Several extensions of the classical Mittag-Leffler function, including multi-parameter and multivariate versions, have been used to define fractional integral and derivative operators. In this paper, we consider a function of one variable with five parameters, a special case of the Fox–Wright function. It turns out that the most natural way to define a fractional integral based on this function requires considering it as a function of two variables. This gives rise to a model of bivariate fractional calculus, which is useful in understanding fractional differential equations involving mixed partial derivatives. Fractal Fract, Vol. 5, Pages 45: On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators

Fractal and Fractional doi: 10.3390/fractalfract5020045

Authors: Mehmet Ali Özarslan Arran Fernandez

Several extensions of the classical Mittag-Leffler function, including multi-parameter and multivariate versions, have been used to define fractional integral and derivative operators. In this paper, we consider a function of one variable with five parameters, a special case of the Fox–Wright function. It turns out that the most natural way to define a fractional integral based on this function requires considering it as a function of two variables. This gives rise to a model of bivariate fractional calculus, which is useful in understanding fractional differential equations involving mixed partial derivatives.

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On a Five-Parameter Mittag-Leffler Function and the Corresponding Bivariate Fractional Operators Mehmet Ali Özarslan Arran Fernandez doi: 10.3390/fractalfract5020045 Fractal and Fractional 2021-05-14 Fractal and Fractional 2021-05-14 5 2 Article 45 10.3390/fractalfract5020045 https://www.mdpi.com/2504-3110/5/2/45
Fractal Fract, Vol. 5, Pages 44: Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales https://www.mdpi.com/2504-3110/5/2/44 This study investigates the predictions of the real contact area for perfectly elastic rough surfaces using a boundary element method (BEM). Sample surface measurements were used in the BEM to predict the real contact area as a function of load. The surfaces were normalized by the root-mean-square (RMS) slope to evaluate if contact area measurements would collapse onto one master curve. If so, this would confirm that the contact areas of manufactured, real measured surfaces are directly proportional to the root mean square slope and the applied load, which is predicted by fractal diffusion-based rough surface contact theory. The data predicts a complex response that deviates from this behavior. The variation in the RMS slope and the spectrum of the system related to the features in contact are further evaluated to illuminate why this property is seen in some types of surfaces and not others. Fractal Fract, Vol. 5, Pages 44: Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales

Fractal and Fractional doi: 10.3390/fractalfract5020044

Authors: Robert Jackson Yang Xu Swarna Saha Kyle Schulze

This study investigates the predictions of the real contact area for perfectly elastic rough surfaces using a boundary element method (BEM). Sample surface measurements were used in the BEM to predict the real contact area as a function of load. The surfaces were normalized by the root-mean-square (RMS) slope to evaluate if contact area measurements would collapse onto one master curve. If so, this would confirm that the contact areas of manufactured, real measured surfaces are directly proportional to the root mean square slope and the applied load, which is predicted by fractal diffusion-based rough surface contact theory. The data predicts a complex response that deviates from this behavior. The variation in the RMS slope and the spectrum of the system related to the features in contact are further evaluated to illuminate why this property is seen in some types of surfaces and not others.

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Elastic Rough Surface Contact and the Root Mean Square Slope of Measured Surfaces over Multiple Scales Robert Jackson Yang Xu Swarna Saha Kyle Schulze doi: 10.3390/fractalfract5020044 Fractal and Fractional 2021-05-14 Fractal and Fractional 2021-05-14 5 2 Article 44 10.3390/fractalfract5020044 https://www.mdpi.com/2504-3110/5/2/44
Fractal Fract, Vol. 5, Pages 43: Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus https://www.mdpi.com/2504-3110/5/2/43 We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order. Fractal Fract, Vol. 5, Pages 43: Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus

Fractal and Fractional doi: 10.3390/fractalfract5020043

Authors: Gerd Baumann

We shall discuss three methods of inverse Laplace transforms. A Sinc-Thiele approximation, a pure Sinc, and a Sinc-Gaussian based method. The two last Sinc related methods are exact methods of inverse Laplace transforms which allow us a numerical approximation using Sinc methods. The inverse Laplace transform converges exponentially and does not use Bromwich contours for computations. We apply the three methods to Mittag-Leffler functions incorporating one, two, and three parameters. The three parameter Mittag-Leffler function represents Prabhakar’s function. The exact Sinc methods are used to solve fractional differential equations of constant and variable differentiation order.

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Sinc Based Inverse Laplace Transforms, Mittag-Leffler Functions and Their Approximation for Fractional Calculus Gerd Baumann doi: 10.3390/fractalfract5020043 Fractal and Fractional 2021-05-10 Fractal and Fractional 2021-05-10 5 2 Article 43 10.3390/fractalfract5020043 https://www.mdpi.com/2504-3110/5/2/43
Fractal Fract, Vol. 5, Pages 42: Fractal Frames of Functions on the Rectangle https://www.mdpi.com/2504-3110/5/2/42 In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable. Fractal Fract, Vol. 5, Pages 42: Fractal Frames of Functions on the Rectangle

Fractal and Fractional doi: 10.3390/fractalfract5020042

Authors: María A. Navascués Ram Mohapatra Md. Nasim Akhtar

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

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Fractal Frames of Functions on the Rectangle María A. Navascués Ram Mohapatra Md. Nasim Akhtar doi: 10.3390/fractalfract5020042 Fractal and Fractional 2021-05-08 Fractal and Fractional 2021-05-08 5 2 Article 42 10.3390/fractalfract5020042 https://www.mdpi.com/2504-3110/5/2/42
Fractal Fract, Vol. 5, Pages 41: Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative https://www.mdpi.com/2504-3110/5/2/41 In this paper, we consider a problem of continuity fractional-order for pseudo-parabolic equations with the fractional derivative of Caputo. Here, we investigate the stability of the problem with respect to derivative parameters and initial data. We also show that uω′→uω in an appropriate sense as ω′→ω, where ω is the fractional order. Moreover, to test the continuity fractional-order, we present several numerical examples to illustrate this property. Fractal Fract, Vol. 5, Pages 41: Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative

Fractal and Fractional doi: 10.3390/fractalfract5020041

Authors: Ho Duy Binh Luc Nguyen Hoang Dumitru Baleanu Ho Thi Kim Van

In this paper, we consider a problem of continuity fractional-order for pseudo-parabolic equations with the fractional derivative of Caputo. Here, we investigate the stability of the problem with respect to derivative parameters and initial data. We also show that uω′→uω in an appropriate sense as ω′→ω, where ω is the fractional order. Moreover, to test the continuity fractional-order, we present several numerical examples to illustrate this property.

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Continuity Result on the Order of a Nonlinear Fractional Pseudo-Parabolic Equation with Caputo Derivative Ho Duy Binh Luc Nguyen Hoang Dumitru Baleanu Ho Thi Kim Van doi: 10.3390/fractalfract5020041 Fractal and Fractional 2021-05-05 Fractal and Fractional 2021-05-05 5 2 Article 41 10.3390/fractalfract5020041 https://www.mdpi.com/2504-3110/5/2/41
Fractal Fract, Vol. 5, Pages 40: Utilizing Fractals for Modeling and 3D Printing of Porous Structures https://www.mdpi.com/2504-3110/5/2/40 Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores, which is a pressing need as far as the aforementioned application areas are concerned. Thus, more effective porous structure design methods are required. This article presents how to utilize fractal geometry to model porous structures and then print them using 3D printing technology. A mathematical procedure was developed to create stochastic point clouds using the affine maps of a predefined Iterative Function Systems (IFS)-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D-printed porous structures with randomly sized and distributed pores. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research. Fractal Fract, Vol. 5, Pages 40: Utilizing Fractals for Modeling and 3D Printing of Porous Structures

Fractal and Fractional doi: 10.3390/fractalfract5020040

Authors: AMM Sharif Ullah Doriana Marilena D’Addona Yusuke Seto Shota Yonehara Akihiko Kubo

Porous structures exhibiting randomly sized and distributed pores are required in biomedical applications (producing implants), materials science (developing cermet-based materials with desired properties), engineering applications (objects having controlled mass and energy transfer properties), and smart agriculture (devices for soilless cultivation). In most cases, a scaffold-based method is used to design porous structures. This approach fails to produce randomly sized and distributed pores, which is a pressing need as far as the aforementioned application areas are concerned. Thus, more effective porous structure design methods are required. This article presents how to utilize fractal geometry to model porous structures and then print them using 3D printing technology. A mathematical procedure was developed to create stochastic point clouds using the affine maps of a predefined Iterative Function Systems (IFS)-based fractal. In addition, a method is developed to modify a given IFS fractal-generated point cloud. The modification process controls the self-similarity levels of the fractal and ultimately results in a model of porous structure exhibiting randomly sized and distributed pores. The model can be transformed into a 3D Computer-Aided Design (CAD) model using voxel-based modeling or other means for digitization and 3D printing. The efficacy of the proposed method is demonstrated by transforming the Sierpinski Carpet (an IFS-based fractal) into 3D-printed porous structures with randomly sized and distributed pores. Other IFS-based fractals than the Sierpinski Carpet can be used to model and fabricate porous structures effectively. This issue remains open for further research.

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Utilizing Fractals for Modeling and 3D Printing of Porous Structures AMM Sharif Ullah Doriana Marilena D’Addona Yusuke Seto Shota Yonehara Akihiko Kubo doi: 10.3390/fractalfract5020040 Fractal and Fractional 2021-04-30 Fractal and Fractional 2021-04-30 5 2 Article 40 10.3390/fractalfract5020040 https://www.mdpi.com/2504-3110/5/2/40
Fractal Fract, Vol. 5, Pages 39: Fractals Parrondo’s Paradox in Alternated Superior Complex System https://www.mdpi.com/2504-3110/5/2/39 This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria. Fractal Fract, Vol. 5, Pages 39: Fractals Parrondo’s Paradox in Alternated Superior Complex System

Fractal and Fractional doi: 10.3390/fractalfract5020039

Authors: Yi Zhang Da Wang

This work focuses on a kind of fractals Parrondo’s paradoxial phenomenon “deiconnected+diconnected=connected” in an alternated superior complex system zn+1=β(zn2+ci)+(1−β)zn,i=1,2. On the one hand, the connectivity variation in superior Julia sets is explored by analyzing the connectivity loci. On the other hand, we graphically investigate the position relation between superior Mandelbrot set and the Connectivity Loci, which results in the conclusion that two totally disconnected superior Julia sets can originate a new, connected, superior Julia set. Moreover, we present some graphical examples obtained by the use of the escape-time algorithm and the derived criteria.

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Fractals Parrondo’s Paradox in Alternated Superior Complex System Yi Zhang Da Wang doi: 10.3390/fractalfract5020039 Fractal and Fractional 2021-04-28 Fractal and Fractional 2021-04-28 5 2 Article 39 10.3390/fractalfract5020039 https://www.mdpi.com/2504-3110/5/2/39
Fractal Fract, Vol. 5, Pages 38: Generalized Cauchy Process: Difference Iterative Forecasting Model https://www.mdpi.com/2504-3110/5/2/38 The contribution of this article is mainly to develop a new stochastic sequence forecasting model, which is also called the difference iterative forecasting model based on the Generalized Cauchy (GC) process. The GC process is a Long-Range Dependent (LRD) process described by two independent parameters: Hurst parameter H and fractal dimension D. Compared with the fractional Brownian motion (fBm) with a linear relationship between H and D, the GC process can more flexibly describe various LRD processes. Before building the forecasting model, this article demonstrates the GC process using H and D to describe the LRD and fractal properties of stochastic sequences, respectively. The GC process is taken as the diffusion term to establish a differential iterative forecasting model, where the incremental distribution of the GC process is obtained by statistics. The parameters of the forecasting model are estimated by the box dimension, the rescaled range, and the maximum likelihood methods. Finally, a real wind speed data set is used to verify the performance of the GC difference iterative forecasting model. Fractal Fract, Vol. 5, Pages 38: Generalized Cauchy Process: Difference Iterative Forecasting Model

Fractal and Fractional doi: 10.3390/fractalfract5020038

Authors: Jie Xing Wanqing Song Francesco Villecco

The contribution of this article is mainly to develop a new stochastic sequence forecasting model, which is also called the difference iterative forecasting model based on the Generalized Cauchy (GC) process. The GC process is a Long-Range Dependent (LRD) process described by two independent parameters: Hurst parameter H and fractal dimension D. Compared with the fractional Brownian motion (fBm) with a linear relationship between H and D, the GC process can more flexibly describe various LRD processes. Before building the forecasting model, this article demonstrates the GC process using H and D to describe the LRD and fractal properties of stochastic sequences, respectively. The GC process is taken as the diffusion term to establish a differential iterative forecasting model, where the incremental distribution of the GC process is obtained by statistics. The parameters of the forecasting model are estimated by the box dimension, the rescaled range, and the maximum likelihood methods. Finally, a real wind speed data set is used to verify the performance of the GC difference iterative forecasting model.

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Generalized Cauchy Process: Difference Iterative Forecasting Model Jie Xing Wanqing Song Francesco Villecco doi: 10.3390/fractalfract5020038 Fractal and Fractional 2021-04-23 Fractal and Fractional 2021-04-23 5 2 Article 38 10.3390/fractalfract5020038 https://www.mdpi.com/2504-3110/5/2/38
Fractal Fract, Vol. 5, Pages 37: Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations https://www.mdpi.com/2504-3110/5/2/37 A system of nonlinear fractional differential equations with the Riemann–Liouville fractional derivative is considered. Lipschitz stability in time for the studied equations is defined and studied. This stability is connected with the singularity of the Riemann–Liouville fractional derivative at the initial point. Two types of derivatives of Lyapunov functions among the studied fractional equations are applied to obtain sufficient conditions for the defined stability property. Some examples illustrate the results. Fractal Fract, Vol. 5, Pages 37: Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations

Fractal and Fractional doi: 10.3390/fractalfract5020037

Authors: Snezhana Hristova Stepan Tersian Radoslava Terzieva

A system of nonlinear fractional differential equations with the Riemann–Liouville fractional derivative is considered. Lipschitz stability in time for the studied equations is defined and studied. This stability is connected with the singularity of the Riemann–Liouville fractional derivative at the initial point. Two types of derivatives of Lyapunov functions among the studied fractional equations are applied to obtain sufficient conditions for the defined stability property. Some examples illustrate the results.

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Lipschitz Stability in Time for Riemann–Liouville Fractional Differential Equations Snezhana Hristova Stepan Tersian Radoslava Terzieva doi: 10.3390/fractalfract5020037 Fractal and Fractional 2021-04-21 Fractal and Fractional 2021-04-21 5 2 Article 37 10.3390/fractalfract5020037 https://www.mdpi.com/2504-3110/5/2/37
Fractal Fract, Vol. 5, Pages 36: Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models https://www.mdpi.com/2504-3110/5/2/36 We considered relaxation, creep, dissipation, and hysteresis resulting from a six-parameter fractional constitutive model and its particular cases. The storage modulus, loss modulus, and loss factor, as well as their characteristics based on the thermodynamic requirements, were investigated. It was proved that for the fractional Maxwell model, the storage modulus increases monotonically, while the loss modulus has symmetrical peaks for its curve against the logarithmic scale log(ω), and for the fractional Zener model, the storage modulus monotonically increases while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω). The peak values and corresponding stationary points were analytically given. The relaxation modulus and the creep compliance for the six-parameter fractional constitutive model were given in terms of the Mittag–Leffler functions. Finally, the stress–strain hysteresis loops were simulated by making use of the derived creep compliance for the fractional Zener model. These results show that the fractional constitutive models could characterize the relaxation, creep, dissipation, and hysteresis phenomena of viscoelastic bodies, and fractional orders α and β could be used to model real-world physical properties well. Fractal Fract, Vol. 5, Pages 36: Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models

Fractal and Fractional doi: 10.3390/fractalfract5020036

Authors: Jun-Sheng Duan Di-Chen Hu Yang-Quan Chen

We considered relaxation, creep, dissipation, and hysteresis resulting from a six-parameter fractional constitutive model and its particular cases. The storage modulus, loss modulus, and loss factor, as well as their characteristics based on the thermodynamic requirements, were investigated. It was proved that for the fractional Maxwell model, the storage modulus increases monotonically, while the loss modulus has symmetrical peaks for its curve against the logarithmic scale log(ω), and for the fractional Zener model, the storage modulus monotonically increases while the loss modulus and the loss factor have symmetrical peaks for their curves against the logarithmic scale log(ω). The peak values and corresponding stationary points were analytically given. The relaxation modulus and the creep compliance for the six-parameter fractional constitutive model were given in terms of the Mittag–Leffler functions. Finally, the stress–strain hysteresis loops were simulated by making use of the derived creep compliance for the fractional Zener model. These results show that the fractional constitutive models could characterize the relaxation, creep, dissipation, and hysteresis phenomena of viscoelastic bodies, and fractional orders α and β could be used to model real-world physical properties well.

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Simultaneous Characterization of Relaxation, Creep, Dissipation, and Hysteresis by Fractional-Order Constitutive Models Jun-Sheng Duan Di-Chen Hu Yang-Quan Chen doi: 10.3390/fractalfract5020036 Fractal and Fractional 2021-04-20 Fractal and Fractional 2021-04-20 5 2 Article 36 10.3390/fractalfract5020036 https://www.mdpi.com/2504-3110/5/2/36
Fractal Fract, Vol. 5, Pages 35: New Challenges Arising in Engineering Problems with Fractional and Integer Order https://www.mdpi.com/2504-3110/5/2/35 Mathematical models have been frequently studied in recent decades in order to obtain the deeper properties of real-world problems [...] Fractal Fract, Vol. 5, Pages 35: New Challenges Arising in Engineering Problems with Fractional and Integer Order

Fractal and Fractional doi: 10.3390/fractalfract5020035

Authors: Haci Mehmet Baskonus Luis Manuel Sánchez Ruiz Armando Ciancio

Mathematical models have been frequently studied in recent decades in order to obtain the deeper properties of real-world problems [...]

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New Challenges Arising in Engineering Problems with Fractional and Integer Order Haci Mehmet Baskonus Luis Manuel Sánchez Ruiz Armando Ciancio doi: 10.3390/fractalfract5020035 Fractal and Fractional 2021-04-19 Fractal and Fractional 2021-04-19 5 2 Editorial 35 10.3390/fractalfract5020035 https://www.mdpi.com/2504-3110/5/2/35
Fractal Fract, Vol. 5, Pages 34: Some New Results on F-Contractions in 0-Complete Partial Metric Spaces and 0-Complete Metric-Like Spaces https://www.mdpi.com/2504-3110/5/2/34 Within this manuscript we generalize the two recently obtained results of O. Popescu and G. Stan, regarding the F-contractions in complete, ordinary metric space to 0-complete partial metric space and 0-complete metric-like space. As Popescu and Stan we use less conditions than D. Wardovski did in his paper from 2012, and we introduce, with the help of one of our lemmas, a new method of proving the results in fixed point theory. Requiring that the function F only be strictly increasing, we obtain for consequence new families of contractive conditions that cannot be found in the existing literature. Note that our results generalize and complement many well-known results in the fixed point theory. Also, at the end of the paper, we have stated an application of our theoretical results for solving fractional differential equations. Fractal Fract, Vol. 5, Pages 34: Some New Results on F-Contractions in 0-Complete Partial Metric Spaces and 0-Complete Metric-Like Spaces

Fractal and Fractional doi: 10.3390/fractalfract5020034

Authors: Stojan Radenović Nikola Mirkov Ljiljana R. Paunović

Within this manuscript we generalize the two recently obtained results of O. Popescu and G. Stan, regarding the F-contractions in complete, ordinary metric space to 0-complete partial metric space and 0-complete metric-like space. As Popescu and Stan we use less conditions than D. Wardovski did in his paper from 2012, and we introduce, with the help of one of our lemmas, a new method of proving the results in fixed point theory. Requiring that the function F only be strictly increasing, we obtain for consequence new families of contractive conditions that cannot be found in the existing literature. Note that our results generalize and complement many well-known results in the fixed point theory. Also, at the end of the paper, we have stated an application of our theoretical results for solving fractional differential equations.

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Some New Results on F-Contractions in 0-Complete Partial Metric Spaces and 0-Complete Metric-Like Spaces Stojan Radenović Nikola Mirkov Ljiljana R. Paunović doi: 10.3390/fractalfract5020034 Fractal and Fractional 2021-04-19 Fractal and Fractional 2021-04-19 5 2 Article 34 10.3390/fractalfract5020034 https://www.mdpi.com/2504-3110/5/2/34
Fractal Fract, Vol. 5, Pages 33: A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation https://www.mdpi.com/2504-3110/5/2/33 Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion. Fractal Fract, Vol. 5, Pages 33: A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation

Fractal and Fractional doi: 10.3390/fractalfract5020033

Authors: Marylu L. Lagunes Oscar Castillo Fevrier Valdez Jose Soria Patricia Melin

Stochastic fractal search (SFS) is a novel method inspired by the process of stochastic growth in nature and the use of the fractal mathematical concept. Considering the chaotic stochastic diffusion property, an improved dynamic stochastic fractal search (DSFS) optimization algorithm is presented. The DSFS algorithm was tested with benchmark functions, such as the multimodal, hybrid, and composite functions, to evaluate the performance of the algorithm with dynamic parameter adaptation with type-1 and type-2 fuzzy inference models. The main contribution of the article is the utilization of fuzzy logic in the adaptation of the diffusion parameter in a dynamic fashion. This parameter is in charge of creating new fractal particles, and the diversity and iteration are the input information used in the fuzzy system to control the values of diffusion.

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A New Approach for Dynamic Stochastic Fractal Search with Fuzzy Logic for Parameter Adaptation Marylu L. Lagunes Oscar Castillo Fevrier Valdez Jose Soria Patricia Melin doi: 10.3390/fractalfract5020033 Fractal and Fractional 2021-04-17 Fractal and Fractional 2021-04-17 5 2 Article 33 10.3390/fractalfract5020033 https://www.mdpi.com/2504-3110/5/2/33
Fractal Fract, Vol. 5, Pages 32: A Fractional SAIDR Model in the Frame of Atangana–Baleanu Derivative https://www.mdpi.com/2504-3110/5/2/32 It is possible to produce mobile phone worms, which are computer viruses with the ability to command the running of cell phones by taking advantage of their flaws, to be transmitted from one device to the other with increasing numbers. In our day, one of the services to gain currency for circulating these malignant worms is SMS. The distinctions of computers from mobile devices render the existing propagation models of computer worms unable to start operating instantaneously in the mobile network, and this is particularly valid for the SMS framework. The susceptible–affected–infectious–suspended–recovered model with a classical derivative (abbreviated as SAIDR) was coined by Xiao et al., (2017) in order to correctly estimate the spread of worms by means of SMS. This study is the first to implement an Atangana–Baleanu (AB) derivative in association with the fractional SAIDR model, depending upon the SAIDR model. The existence and uniqueness of the drinking model solutions together with the stability analysis are shown through the Banach fixed point theorem. The special solution of the model is investigated using the Laplace transformation and then we present a set of numeric graphics by varying the fractional-order θ with the intention of showing the effectiveness of the fractional derivative. Fractal Fract, Vol. 5, Pages 32: A Fractional SAIDR Model in the Frame of Atangana–Baleanu Derivative

Fractal and Fractional doi: 10.3390/fractalfract5020032

Authors: Esmehan Uçar Sümeyra Uçar Fırat Evirgen Necati Özdemir

It is possible to produce mobile phone worms, which are computer viruses with the ability to command the running of cell phones by taking advantage of their flaws, to be transmitted from one device to the other with increasing numbers. In our day, one of the services to gain currency for circulating these malignant worms is SMS. The distinctions of computers from mobile devices render the existing propagation models of computer worms unable to start operating instantaneously in the mobile network, and this is particularly valid for the SMS framework. The susceptible–affected–infectious–suspended–recovered model with a classical derivative (abbreviated as SAIDR) was coined by Xiao et al., (2017) in order to correctly estimate the spread of worms by means of SMS. This study is the first to implement an Atangana–Baleanu (AB) derivative in association with the fractional SAIDR model, depending upon the SAIDR model. The existence and uniqueness of the drinking model solutions together with the stability analysis are shown through the Banach fixed point theorem. The special solution of the model is investigated using the Laplace transformation and then we present a set of numeric graphics by varying the fractional-order θ with the intention of showing the effectiveness of the fractional derivative.

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A Fractional SAIDR Model in the Frame of Atangana–Baleanu Derivative Esmehan Uçar Sümeyra Uçar Fırat Evirgen Necati Özdemir doi: 10.3390/fractalfract5020032 Fractal and Fractional 2021-04-15 Fractal and Fractional 2021-04-15 5 2 Article 32 10.3390/fractalfract5020032 https://www.mdpi.com/2504-3110/5/2/32
Fractal Fract, Vol. 5, Pages 31: Image Compression Using Fractal Functions https://www.mdpi.com/2504-3110/5/2/31 The rapid growth of geographic information technologies in the field of processing and analysis of spatial data has led to a significant increase in the role of geographic information systems in various fields of human activity. However, solving complex problems requires the use of large amounts of spatial data, efficient storage of data on on-board recording media and their transmission via communication channels. This leads to the need to create new effective methods of compression and data transmission of remote sensing of the Earth. The possibility of using fractal functions for image processing, which were transmitted via the satellite radio channel of a spacecraft, is considered. The information obtained by such a system is presented in the form of aerospace images that need to be processed and analyzed in order to obtain information about the objects that are displayed. An algorithm for constructing image encoding–decoding using a class of continuous functions that depend on a finite set of parameters and have fractal properties is investigated. The mathematical model used in fractal image compression is called a system of iterative functions. The encoding process is time consuming because it performs a large number of transformations and mathematical calculations. However, due to this, a high degree of image compression is achieved. This class of functions has an interesting property—knowing the initial sets of numbers, we can easily calculate the value of the function, but when the values of the function are known, it is very difficult to return the initial set of values, because there are a huge number of such combinations. Therefore, in order to de-encode the image, it is necessary to know fractal codes that will help to restore the raster image. Fractal Fract, Vol. 5, Pages 31: Image Compression Using Fractal Functions

Fractal and Fractional doi: 10.3390/fractalfract5020031

Authors: Olga Svynchuk Oleg Barabash Joanna Nikodem Roman Kochan Oleksandr Laptiev

The rapid growth of geographic information technologies in the field of processing and analysis of spatial data has led to a significant increase in the role of geographic information systems in various fields of human activity. However, solving complex problems requires the use of large amounts of spatial data, efficient storage of data on on-board recording media and their transmission via communication channels. This leads to the need to create new effective methods of compression and data transmission of remote sensing of the Earth. The possibility of using fractal functions for image processing, which were transmitted via the satellite radio channel of a spacecraft, is considered. The information obtained by such a system is presented in the form of aerospace images that need to be processed and analyzed in order to obtain information about the objects that are displayed. An algorithm for constructing image encoding–decoding using a class of continuous functions that depend on a finite set of parameters and have fractal properties is investigated. The mathematical model used in fractal image compression is called a system of iterative functions. The encoding process is time consuming because it performs a large number of transformations and mathematical calculations. However, due to this, a high degree of image compression is achieved. This class of functions has an interesting property—knowing the initial sets of numbers, we can easily calculate the value of the function, but when the values of the function are known, it is very difficult to return the initial set of values, because there are a huge number of such combinations. Therefore, in order to de-encode the image, it is necessary to know fractal codes that will help to restore the raster image.

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Image Compression Using Fractal Functions Olga Svynchuk Oleg Barabash Joanna Nikodem Roman Kochan Oleksandr Laptiev doi: 10.3390/fractalfract5020031 Fractal and Fractional 2021-04-14 Fractal and Fractional 2021-04-14 5 2 Article 31 10.3390/fractalfract5020031 https://www.mdpi.com/2504-3110/5/2/31
Fractal Fract, Vol. 5, Pages 30: Approximate Controllability of Fully Nonlocal Stochastic Delay Control Problems Driven by Hybrid Noises https://www.mdpi.com/2504-3110/5/2/30 In this paper, a class of time-space fractional stochastic delay control problems with fractional noises and Poisson jumps in a bounded domain is considered. The proper function spaces and assumptions are proposed to discuss the existence of mild solutions. In particular, approximate strategy is used to obtain the existence of mild solutions for the problem with linear fractional noises; fixed point theorem is used to achieve the existence of mild solutions for the problem with nonlinear fractional noises. Finally, the approximate controllability of the problems with linear and nonlinear fractional noises is proved by the property of mild solutions. Fractal Fract, Vol. 5, Pages 30: Approximate Controllability of Fully Nonlocal Stochastic Delay Control Problems Driven by Hybrid Noises

Fractal and Fractional doi: 10.3390/fractalfract5020030

Authors: Lixu Yan Yongqiang Fu

In this paper, a class of time-space fractional stochastic delay control problems with fractional noises and Poisson jumps in a bounded domain is considered. The proper function spaces and assumptions are proposed to discuss the existence of mild solutions. In particular, approximate strategy is used to obtain the existence of mild solutions for the problem with linear fractional noises; fixed point theorem is used to achieve the existence of mild solutions for the problem with nonlinear fractional noises. Finally, the approximate controllability of the problems with linear and nonlinear fractional noises is proved by the property of mild solutions.

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Approximate Controllability of Fully Nonlocal Stochastic Delay Control Problems Driven by Hybrid Noises Lixu Yan Yongqiang Fu doi: 10.3390/fractalfract5020030 Fractal and Fractional 2021-04-12 Fractal and Fractional 2021-04-12 5 2 Article 30 10.3390/fractalfract5020030 https://www.mdpi.com/2504-3110/5/2/30
Fractal Fract, Vol. 5, Pages 29: Optimal State Control of Fractional Order Differential Systems: The Infinite State Approach https://www.mdpi.com/2504-3110/5/2/29 Optimal control of fractional order systems is a long established domain of fractional calculus. Nevertheless, it relies on equations expressed in terms of pseudo-state variables which raise fundamental questions. So in order remedy these problems, the authors propose in this paper a new and original approach to fractional optimal control based on a frequency distributed representation of fractional differential equations called the infinite state approach, associated with an original formulation of fractional energy, which is intended to really control the internal system state. In the first step, the fractional calculus of variations is revisited to express appropriate Euler Lagrange equations. Then, the quadratic optimal control of fractional linear systems is formulated. Thanks to a frequency discretization technique, the previous theoretical equations are converted into an equivalent large dimension integer order system which permits the implementation of a feasible optimal solution. A numerical example illustrates the validity of this new approach. Fractal Fract, Vol. 5, Pages 29: Optimal State Control of Fractional Order Differential Systems: The Infinite State Approach

Fractal and Fractional doi: 10.3390/fractalfract5020029

Authors: Jean-Claude Trigeassou Nezha Maamri

Optimal control of fractional order systems is a long established domain of fractional calculus. Nevertheless, it relies on equations expressed in terms of pseudo-state variables which raise fundamental questions. So in order remedy these problems, the authors propose in this paper a new and original approach to fractional optimal control based on a frequency distributed representation of fractional differential equations called the infinite state approach, associated with an original formulation of fractional energy, which is intended to really control the internal system state. In the first step, the fractional calculus of variations is revisited to express appropriate Euler Lagrange equations. Then, the quadratic optimal control of fractional linear systems is formulated. Thanks to a frequency discretization technique, the previous theoretical equations are converted into an equivalent large dimension integer order system which permits the implementation of a feasible optimal solution. A numerical example illustrates the validity of this new approach.

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Optimal State Control of Fractional Order Differential Systems: The Infinite State Approach Jean-Claude Trigeassou Nezha Maamri doi: 10.3390/fractalfract5020029 Fractal and Fractional 2021-04-05 Fractal and Fractional 2021-04-05 5 2 Article 29 10.3390/fractalfract5020029 https://www.mdpi.com/2504-3110/5/2/29
Fractal Fract, Vol. 5, Pages 28: Fractal Interpolation Using Harmonic Functions on the Koch Curve https://www.mdpi.com/2504-3110/5/2/28 The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions. Fractal Fract, Vol. 5, Pages 28: Fractal Interpolation Using Harmonic Functions on the Koch Curve

Fractal and Fractional doi: 10.3390/fractalfract5020028

Authors: Song-Il Ri Vasileios Drakopoulos Song-Min Nam

The Koch curve was first described by the Swedish mathematician Helge von Koch in 1904 as an example of a continuous but nowhere differentiable curve. Such functions are now characterised as fractal since their graphs are in general fractal sets. Furthermore, it can be obtained as the graph of an appropriately chosen iterated function system. On the other hand, a fractal interpolation function can be seen as a special case of an iterated function system thus maintaining all of its characteristics. Fractal interpolation functions are continuous functions that can be used to model continuous signals. An in-depth discussion on the theory of affine fractal interpolation functions generating the Koch Curve by using fractal analysis as well as its recent development including some of the research made by the authors is provided. We ensure that the graph of fractal interpolation functions on the Koch Curve are attractors of an iterated function system constructed by non-constant harmonic functions.

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Fractal Interpolation Using Harmonic Functions on the Koch Curve Song-Il Ri Vasileios Drakopoulos Song-Min Nam doi: 10.3390/fractalfract5020028 Fractal and Fractional 2021-04-05 Fractal and Fractional 2021-04-05 5 2 Article 28 10.3390/fractalfract5020028 https://www.mdpi.com/2504-3110/5/2/28
Fractal Fract, Vol. 5, Pages 27: Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers https://www.mdpi.com/2504-3110/5/2/27 In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented. Fractal Fract, Vol. 5, Pages 27: Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers

Fractal and Fractional doi: 10.3390/fractalfract5020027

Authors: Debasis Sharma Ioannis K. Argyros Sanjaya Kumar Parhi Shanta Kumari Sunanda

In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented.

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Local Convergence and Dynamical Analysis of a Third and Fourth Order Class of Equation Solvers Debasis Sharma Ioannis K. Argyros Sanjaya Kumar Parhi Shanta Kumari Sunanda doi: 10.3390/fractalfract5020027 Fractal and Fractional 2021-04-05 Fractal and Fractional 2021-04-05 5 2 Article 27 10.3390/fractalfract5020027 https://www.mdpi.com/2504-3110/5/2/27
Fractal Fract, Vol. 5, Pages 26: Uncertainty Quantification of Random Microbial Growth in a Competitive Environment via Probability Density Functions https://www.mdpi.com/2504-3110/5/2/26 The Baranyi–Roberts model describes the dynamics of the volumetric densities of two interacting cell populations. We randomize this model by considering that the initial conditions are random variables whose distributions are determined by using sample data and the principle of maximum entropy. Subsequenly, we obtain the Liouville–Gibbs partial differential equation for the probability density function of the two-dimensional solution stochastic process. Because the exact solution of this equation is unaffordable, we use a finite volume scheme to numerically approximate the aforementioned probability density function. From this key information, we design an optimization procedure in order to determine the best growth rates of the Baranyi–Roberts model, so that the expectation of the numerical solution is as close as possible to the sample data. The results evidence good fitting that allows for performing reliable predictions. Fractal Fract, Vol. 5, Pages 26: Uncertainty Quantification of Random Microbial Growth in a Competitive Environment via Probability Density Functions

Fractal and Fractional doi: 10.3390/fractalfract5020026

Authors: Vicente José Bevia Clara Burgos Simón Juan Carlos Cortés Rafael J. Villanueva Micó

The Baranyi–Roberts model describes the dynamics of the volumetric densities of two interacting cell populations. We randomize this model by considering that the initial conditions are random variables whose distributions are determined by using sample data and the principle of maximum entropy. Subsequenly, we obtain the Liouville–Gibbs partial differential equation for the probability density function of the two-dimensional solution stochastic process. Because the exact solution of this equation is unaffordable, we use a finite volume scheme to numerically approximate the aforementioned probability density function. From this key information, we design an optimization procedure in order to determine the best growth rates of the Baranyi–Roberts model, so that the expectation of the numerical solution is as close as possible to the sample data. The results evidence good fitting that allows for performing reliable predictions.

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Uncertainty Quantification of Random Microbial Growth in a Competitive Environment via Probability Density Functions Vicente José Bevia Clara Burgos Simón Juan Carlos Cortés Rafael J. Villanueva Micó doi: 10.3390/fractalfract5020026 Fractal and Fractional 2021-03-24 Fractal and Fractional 2021-03-24 5 2 Article 26 10.3390/fractalfract5020026 https://www.mdpi.com/2504-3110/5/2/26
Fractal Fract, Vol. 5, Pages 25: A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots https://www.mdpi.com/2504-3110/5/1/25 The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials. Fractal Fract, Vol. 5, Pages 25: A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots

Fractal and Fractional doi: 10.3390/fractalfract5010025

Authors: Víctor Galilea José M. Gutiérrez

The purpose of this work is to give a first approach to the dynamical behavior of Schröder’s method, a well-known iterative process for solving nonlinear equations. In this context, we consider equations defined in the complex plane. By using topological conjugations, we characterize the basins of attraction of Schröder’s method applied to polynomials with two roots and different multiplicities. Actually, we show that these basins are half-planes or circles, depending on the multiplicities of the roots. We conclude our study with a graphical gallery that allow us to compare the basins of attraction of Newton’s and Schröder’s method applied to some given polynomials.

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A Characterization of the Dynamics of Schröder’s Method for Polynomials with Two Roots Víctor Galilea José M. Gutiérrez doi: 10.3390/fractalfract5010025 Fractal and Fractional 2021-03-20 Fractal and Fractional 2021-03-20 5 1 Article 25 10.3390/fractalfract5010025 https://www.mdpi.com/2504-3110/5/1/25
Fractal Fract, Vol. 5, Pages 24: A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter https://www.mdpi.com/2504-3110/5/1/24 In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results. Fractal Fract, Vol. 5, Pages 24: A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter

Fractal and Fractional doi: 10.3390/fractalfract5010024

Authors: Ricardo Almeida Natália Martins

In this paper, we present a new fractional variational problem where the Lagrangian depends not only on the independent variable, an unknown function and its left- and right-sided Caputo fractional derivatives with respect to another function, but also on the endpoint conditions and a free parameter. The main results of this paper are necessary and sufficient optimality conditions for variational problems with or without isoperimetric and holonomic restrictions. Our results not only provide a generalization to previous results but also give new contributions in fractional variational calculus. Finally, we present some examples to illustrate our results.

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A Generalization of a Fractional Variational Problem with Dependence on the Boundaries and a Real Parameter Ricardo Almeida Natália Martins doi: 10.3390/fractalfract5010024 Fractal and Fractional 2021-03-18 Fractal and Fractional 2021-03-18 5 1 Article 24 10.3390/fractalfract5010024 https://www.mdpi.com/2504-3110/5/1/24
Fractal Fract, Vol. 5, Pages 23: A Tuning Method via Borges Derivative of a Neural Network-Based Discrete-Time Fractional-Order PID Controller with Hausdorff Difference and Hausdorff Sum https://www.mdpi.com/2504-3110/5/1/23 In this paper, the fractal derivative is introduced into a neural network-based discrete-time fractional-order PID controller in two areas, namely, in the controller’s structure and in the parameter optimization algorithm. The first use of the fractal derivative is to reconstruct the fractional-order PID controller by using the Hausdorff difference and Hausdorff sum derived from the Hausdorff derivative and Hausdorff integral. It can avoid the derivation of the Gamma function for the order updating to realize the parameter and order tuning based on neural networks. The other use is the optimization of order and parameters by using Borges derivative. Borges derivative is a kind of fractal derivative as a local fractional-order derivative. The chain rule of composite function is consistent with the integral-order derivative. It is suitable for updating the parameters and the order of the fractional-order PID controller based on neural networks. This paper improves the neural network-based PID controller in two aspects, which accelerates the response speed and improves the control accuracy. Two illustrative examples are given to verify the effectiveness of the proposed neural network-based discrete-time fractional-order PID control scheme with fractal derivatives. Fractal Fract, Vol. 5, Pages 23: A Tuning Method via Borges Derivative of a Neural Network-Based Discrete-Time Fractional-Order PID Controller with Hausdorff Difference and Hausdorff Sum

Fractal and Fractional doi: 10.3390/fractalfract5010023

Authors: Zhe Gao

In this paper, the fractal derivative is introduced into a neural network-based discrete-time fractional-order PID controller in two areas, namely, in the controller’s structure and in the parameter optimization algorithm. The first use of the fractal derivative is to reconstruct the fractional-order PID controller by using the Hausdorff difference and Hausdorff sum derived from the Hausdorff derivative and Hausdorff integral. It can avoid the derivation of the Gamma function for the order updating to realize the parameter and order tuning based on neural networks. The other use is the optimization of order and parameters by using Borges derivative. Borges derivative is a kind of fractal derivative as a local fractional-order derivative. The chain rule of composite function is consistent with the integral-order derivative. It is suitable for updating the parameters and the order of the fractional-order PID controller based on neural networks. This paper improves the neural network-based PID controller in two aspects, which accelerates the response speed and improves the control accuracy. Two illustrative examples are given to verify the effectiveness of the proposed neural network-based discrete-time fractional-order PID control scheme with fractal derivatives.

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A Tuning Method via Borges Derivative of a Neural Network-Based Discrete-Time Fractional-Order PID Controller with Hausdorff Difference and Hausdorff Sum Zhe Gao doi: 10.3390/fractalfract5010023 Fractal and Fractional 2021-03-14 Fractal and Fractional 2021-03-14 5 1 Article 23 10.3390/fractalfract5010023 https://www.mdpi.com/2504-3110/5/1/23
Fractal Fract, Vol. 5, Pages 22: Analysis of Hilfer Fractional Integro-Differential Equations with Almost Sectorial Operators https://www.mdpi.com/2504-3110/5/1/22 In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory. Fractal Fract, Vol. 5, Pages 22: Analysis of Hilfer Fractional Integro-Differential Equations with Almost Sectorial Operators

Fractal and Fractional doi: 10.3390/fractalfract5010022

Authors: Kulandhaivel Karthikeyan Amar Debbouche Delfim F. M. Torres

In this work, we investigate a class of nonlocal integro-differential equations involving Hilfer fractional derivatives and almost sectorial operators. We prove our results by applying Schauder’s fixed point technique. Moreover, we show the fundamental properties of the representation of the solution by discussing two cases related to the associated semigroup. For that, we consider compactness and noncompactness properties, respectively. Furthermore, an example is given to illustrate the obtained theory.

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Analysis of Hilfer Fractional Integro-Differential Equations with Almost Sectorial Operators Kulandhaivel Karthikeyan Amar Debbouche Delfim F. M. Torres doi: 10.3390/fractalfract5010022 Fractal and Fractional 2021-03-08 Fractal and Fractional 2021-03-08 5 1 Article 22 10.3390/fractalfract5010022 https://www.mdpi.com/2504-3110/5/1/22
Fractal Fract, Vol. 5, Pages 21: Fuel Cell Fractional-Order Model via Electrochemical Impedance Spectroscopy https://www.mdpi.com/2504-3110/5/1/21 The knowledge of the electrochemical processes inside a Fuel Cell (FC) is useful for improving FC diagnostics, and Electrochemical Impedance Spectroscopy (EIS) is one of the most used techniques for electrochemical characterization. This paper aims to propose the identification of a Fractional-Order Transfer Function (FOTF) able to represent the FC behavior in a set of working points. The model was identified by using a data-driven approach. Experimental data were obtained testing a Proton Exchange Membrane Fuel Cell (PEMFC) to measure the cell impedance. A genetic algorithm was firstly used to determine the sets of fractional-order impedance model parameters that best fit the input data in each analyzed working point. Then, a method was proposed to select a single set of parameters, which can represent the system behavior in all the considered working conditions. The comparison with an equivalent circuit model taken from the literature is reported, showing the advantages of the proposed approach. Fractal Fract, Vol. 5, Pages 21: Fuel Cell Fractional-Order Model via Electrochemical Impedance Spectroscopy

Fractal and Fractional doi: 10.3390/fractalfract5010021

Authors: Riccardo Caponetto Fabio Matera Emanuele Murgano Emanuela Privitera Maria Gabriella Xibilia

The knowledge of the electrochemical processes inside a Fuel Cell (FC) is useful for improving FC diagnostics, and Electrochemical Impedance Spectroscopy (EIS) is one of the most used techniques for electrochemical characterization. This paper aims to propose the identification of a Fractional-Order Transfer Function (FOTF) able to represent the FC behavior in a set of working points. The model was identified by using a data-driven approach. Experimental data were obtained testing a Proton Exchange Membrane Fuel Cell (PEMFC) to measure the cell impedance. A genetic algorithm was firstly used to determine the sets of fractional-order impedance model parameters that best fit the input data in each analyzed working point. Then, a method was proposed to select a single set of parameters, which can represent the system behavior in all the considered working conditions. The comparison with an equivalent circuit model taken from the literature is reported, showing the advantages of the proposed approach.

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Fuel Cell Fractional-Order Model via Electrochemical Impedance Spectroscopy Riccardo Caponetto Fabio Matera Emanuele Murgano Emanuela Privitera Maria Gabriella Xibilia doi: 10.3390/fractalfract5010021 Fractal and Fractional 2021-03-06 Fractal and Fractional 2021-03-06 5 1 Article 21 10.3390/fractalfract5010021 https://www.mdpi.com/2504-3110/5/1/21
Fractal Fract, Vol. 5, Pages 20: On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations https://www.mdpi.com/2504-3110/5/1/20 The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class CW(K,a), which is defined here. It is also shown that from the continuity of a resolving family of operators at t=0 the boundedness of A follows. The existence of a resolving family is shown for A∈CW(K,a) and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time. Fractal Fract, Vol. 5, Pages 20: On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations

Fractal and Fractional doi: 10.3390/fractalfract5010020

Authors: Vladimir E. Fedorov Nikolay V. Filin

The aim of this work is to find by the methods of the Laplace transform the conditions for the existence of a strongly continuous resolving family of operators for a linear homogeneous equation in a Banach space with the distributed Gerasimov–Caputo fractional derivative and with a closed densely defined operator A in the right-hand side. It is proved that the existence of a resolving family of operators for such equation implies the belonging of the operator A to the class CW(K,a), which is defined here. It is also shown that from the continuity of a resolving family of operators at t=0 the boundedness of A follows. The existence of a resolving family is shown for A∈CW(K,a) and for the upper limit of the integration in the distributed derivative not greater than 2. As corollary, we obtain the existence of a unique solution for the Cauchy problem to the equation of such class. These results are used for the investigation of the initial boundary value problems unique solvability for a class of partial differential equations of the distributed order with respect to time.

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On Strongly Continuous Resolving Families of Operators for Fractional Distributed Order Equations Vladimir E. Fedorov Nikolay V. Filin doi: 10.3390/fractalfract5010020 Fractal and Fractional 2021-03-02 Fractal and Fractional 2021-03-02 5 1 Article 20 10.3390/fractalfract5010020 https://www.mdpi.com/2504-3110/5/1/20
Fractal Fract, Vol. 5, Pages 19: Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation https://www.mdpi.com/2504-3110/5/1/19 The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with Oh2+τ2 of accuracy on the grids. Here, h and 32h are the step sizes in space variables x1 and x2, respectively and τ is the step size in time. At the second stage, we propose special difference boundary value problems on hexagonal grids for the approximation of first derivatives with respect to spatial variables of which the boundary conditions are defined by using the obtained solution from the first stage. It is proved that the given schemes in the difference problems are unconditionally stable. Further, for r=ωτh2≤37, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order Oh2+τ2 is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures. Fractal Fract, Vol. 5, Pages 19: Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation

Fractal and Fractional doi: 10.3390/fractalfract5010019

Authors: Suzan Cival Buranay Ahmed Hersi Matan Nouman Arshad

The first type of boundary value problem for the heat equation on a rectangle is considered. We propose a two stage implicit method for the approximation of the first order derivatives of the solution with respect to the spatial variables. To approximate the solution at the first stage, the unconditionally stable two layer implicit method on hexagonal grids given by Buranay and Arshad in 2020 is used which converges with Oh2+τ2 of accuracy on the grids. Here, h and 32h are the step sizes in space variables x1 and x2, respectively and τ is the step size in time. At the second stage, we propose special difference boundary value problems on hexagonal grids for the approximation of first derivatives with respect to spatial variables of which the boundary conditions are defined by using the obtained solution from the first stage. It is proved that the given schemes in the difference problems are unconditionally stable. Further, for r=ωτh2≤37, uniform convergence of the solution of the constructed special difference boundary value problems to the corresponding exact derivatives on hexagonal grids with order Oh2+τ2 is shown. Finally, the method is applied on a test problem and the numerical results are presented through tables and figures.

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Two Stage Implicit Method on Hexagonal Grids for Approximating the First Derivatives of the Solution to the Heat Equation Suzan Cival Buranay Ahmed Hersi Matan Nouman Arshad doi: 10.3390/fractalfract5010019 Fractal and Fractional 2021-02-26 Fractal and Fractional 2021-02-26 5 1 Article 19 10.3390/fractalfract5010019 https://www.mdpi.com/2504-3110/5/1/19
Fractal Fract, Vol. 5, Pages 18: The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions https://www.mdpi.com/2504-3110/5/1/18 Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α&amp;lt;q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function. Fractal Fract, Vol. 5, Pages 18: The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions

Fractal and Fractional doi: 10.3390/fractalfract5010018

Authors: Nicos Makris

Motivated from studies on anomalous relaxation and diffusion, we show that the memory function M(t) of complex materials, that their creep compliance follows a power law, J(t)∼tq with q∈R+, is proportional to the fractional derivative of the Dirac delta function, dqδ(t−0)dtq with q∈R+. This leads to the finding that the inverse Laplace transform of sq for any q∈R+ is the fractional derivative of the Dirac delta function, dqδ(t−0)dtq. This result, in association with the convolution theorem, makes possible the calculation of the inverse Laplace transform of sqsα∓λ where α&amp;lt;q∈R+, which is the fractional derivative of order q of the Rabotnov function εα−1(±λ,t)=tα−1Eα,α(±λtα). The fractional derivative of order q∈R+ of the Rabotnov function, εα−1(±λ,t) produces singularities that are extracted with a finite number of fractional derivatives of the Dirac delta function depending on the strength of q in association with the recurrence formula of the two-parameter Mittag–Leffler function.

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The Fractional Derivative of the Dirac Delta Function and Additional Results on the Inverse Laplace Transform of Irrational Functions Nicos Makris doi: 10.3390/fractalfract5010018 Fractal and Fractional 2021-02-25 Fractal and Fractional 2021-02-25 5 1 Article 18 10.3390/fractalfract5010018 https://www.mdpi.com/2504-3110/5/1/18
Fractal Fract, Vol. 5, Pages 17: Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations https://www.mdpi.com/2504-3110/5/1/17 Verified simulation techniques have been investigated intensively by researchers who are dealing with ordinary and partial differential equations. Tasks that have been considered in this context are the solution to initial value problems and boundary value problems, parameter identification, as well as the solution of optimal control problems in cases in which bounded uncertainty in parameters and initial conditions are present. In contrast to system models with integer-order derivatives, fractional-order models have not yet gained the same attention if verified solution techniques are desired. In general, verified simulation techniques rely on interval methods, zonotopes, or Taylor model arithmetic and allow for computing guaranteed outer enclosures of the sets of solutions. As such, not only the influence of uncertain but bounded parameters can be accounted for in a guaranteed way. In addition, also round-off and (temporal) truncation errors that inevitably occur in numerical software implementations can be considered in a rigorous manner. This paper presents novel iterative and series-based solution approaches for the case of initial value problems to fractional-order system models, which will form the basic building block for implementing state estimation schemes in continuous-discrete settings, where the system dynamics is assumed as being continuous but measurements are only available at specific discrete sampling instants. Fractal Fract, Vol. 5, Pages 17: Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations

Fractal and Fractional doi: 10.3390/fractalfract5010017

Authors: Andreas Rauh Luc Jaulin

Verified simulation techniques have been investigated intensively by researchers who are dealing with ordinary and partial differential equations. Tasks that have been considered in this context are the solution to initial value problems and boundary value problems, parameter identification, as well as the solution of optimal control problems in cases in which bounded uncertainty in parameters and initial conditions are present. In contrast to system models with integer-order derivatives, fractional-order models have not yet gained the same attention if verified solution techniques are desired. In general, verified simulation techniques rely on interval methods, zonotopes, or Taylor model arithmetic and allow for computing guaranteed outer enclosures of the sets of solutions. As such, not only the influence of uncertain but bounded parameters can be accounted for in a guaranteed way. In addition, also round-off and (temporal) truncation errors that inevitably occur in numerical software implementations can be considered in a rigorous manner. This paper presents novel iterative and series-based solution approaches for the case of initial value problems to fractional-order system models, which will form the basic building block for implementing state estimation schemes in continuous-discrete settings, where the system dynamics is assumed as being continuous but measurements are only available at specific discrete sampling instants.

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Novel Techniques for a Verified Simulation of Fractional-Order Differential Equations Andreas Rauh Luc Jaulin doi: 10.3390/fractalfract5010017 Fractal and Fractional 2021-02-21 Fractal and Fractional 2021-02-21 5 1 Article 17 10.3390/fractalfract5010017 https://www.mdpi.com/2504-3110/5/1/17
Fractal Fract, Vol. 5, Pages 16: Using Fractal Calculus to Solve Fractal Navier–Stokes Equations, and Simulation of Laminar Static Mixing in COMSOL Multiphysics https://www.mdpi.com/2504-3110/5/1/16 Navier–Stokes equations describe the laminar flow of incompressible fluids. In most cases, one prefers to solve either these equations numerically, or the physical conditions of solving the problem are considered more straightforward than the real situation. In this paper, the Navier–Stokes equations are solved analytically and numerically for specific physical conditions. Using Fα-calculus, the fractal form of Navier–Stokes equations, which describes the laminar flow of incompressible fluids, has been solved analytically for two groups of general solutions. In the analytical section, for just “the single-phase fluid” analytical answers are obtained in a two-dimensional situation. However, in the numerical part, we simulate two fluids’ flow (liquid–liquid) in a three-dimensional case through several fractal structures and the sides of several fractal structures. Static mixers can be used to mix two fluids. These static mixers can be fractal in shape. The Sierpinski triangle, the Sierpinski carpet, and the circular fractal pattern have the static mixer’s role in our simulations. We apply these structures just in zero, first and second iterations. Using the COMSOL software, these equations for “fractal mixing” were solved numerically. For this purpose, fractal structures act as a barrier, and one can handle different types of their corresponding simulations. In COMSOL software, after the execution, we verify the defining model. We may present speed, pressure, and concentration distributions before and after passing fluids through or out of the fractal structure. The parameter for analyzing the quality of fractal mixing is the Coefficient of Variation (CoV). Fractal Fract, Vol. 5, Pages 16: Using Fractal Calculus to Solve Fractal Navier–Stokes Equations, and Simulation of Laminar Static Mixing in COMSOL Multiphysics

Fractal and Fractional doi: 10.3390/fractalfract5010016

Authors: Amir Pishkoo Maslina Darus

Navier–Stokes equations describe the laminar flow of incompressible fluids. In most cases, one prefers to solve either these equations numerically, or the physical conditions of solving the problem are considered more straightforward than the real situation. In this paper, the Navier–Stokes equations are solved analytically and numerically for specific physical conditions. Using Fα-calculus, the fractal form of Navier–Stokes equations, which describes the laminar flow of incompressible fluids, has been solved analytically for two groups of general solutions. In the analytical section, for just “the single-phase fluid” analytical answers are obtained in a two-dimensional situation. However, in the numerical part, we simulate two fluids’ flow (liquid–liquid) in a three-dimensional case through several fractal structures and the sides of several fractal structures. Static mixers can be used to mix two fluids. These static mixers can be fractal in shape. The Sierpinski triangle, the Sierpinski carpet, and the circular fractal pattern have the static mixer’s role in our simulations. We apply these structures just in zero, first and second iterations. Using the COMSOL software, these equations for “fractal mixing” were solved numerically. For this purpose, fractal structures act as a barrier, and one can handle different types of their corresponding simulations. In COMSOL software, after the execution, we verify the defining model. We may present speed, pressure, and concentration distributions before and after passing fluids through or out of the fractal structure. The parameter for analyzing the quality of fractal mixing is the Coefficient of Variation (CoV).

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Using Fractal Calculus to Solve Fractal Navier–Stokes Equations, and Simulation of Laminar Static Mixing in COMSOL Multiphysics Amir Pishkoo Maslina Darus doi: 10.3390/fractalfract5010016 Fractal and Fractional 2021-02-08 Fractal and Fractional 2021-02-08 5 1 Article 16 10.3390/fractalfract5010016 https://www.mdpi.com/2504-3110/5/1/16
Fractal Fract, Vol. 5, Pages 15: Non-Linear First-Order Differential Boundary Problems with Multipoint and Integral Conditions https://www.mdpi.com/2504-3110/5/1/15 This paper considers boundary value problem (BVP) for nonlinear first-order differential problems with multipoint and integral boundary conditions. A suitable Green function was constructed for the first time in order to reduce this problem into a corresponding integral equation. So that by using the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem (SFPT) on the integral equation, we can show that the solution of the multipoint problem exists and it is unique. Fractal Fract, Vol. 5, Pages 15: Non-Linear First-Order Differential Boundary Problems with Multipoint and Integral Conditions

Fractal and Fractional doi: 10.3390/fractalfract5010015

Authors: Misir J. Mardanov Yagub A. Sharifov Yusif S. Gasimov Carlo Cattani

This paper considers boundary value problem (BVP) for nonlinear first-order differential problems with multipoint and integral boundary conditions. A suitable Green function was constructed for the first time in order to reduce this problem into a corresponding integral equation. So that by using the Banach contraction mapping principle (BCMP) and Schaefer’s fixed point theorem (SFPT) on the integral equation, we can show that the solution of the multipoint problem exists and it is unique.

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Non-Linear First-Order Differential Boundary Problems with Multipoint and Integral Conditions Misir J. Mardanov Yagub A. Sharifov Yusif S. Gasimov Carlo Cattani doi: 10.3390/fractalfract5010015 Fractal and Fractional 2021-02-05 Fractal and Fractional 2021-02-05 5 1 Article 15 10.3390/fractalfract5010015 https://www.mdpi.com/2504-3110/5/1/15
Fractal Fract, Vol. 5, Pages 14: Application of Fractal Dimension of Terrestrial Laser Point Cloud in Classification of Independent Trees https://www.mdpi.com/2504-3110/5/1/14 Tree precise classification and identification of forest species is a core issue of forestry resource monitoring and ecological effect assessment. In this paper, an independent tree species classification method based on fractal features of terrestrial laser point cloud is proposed. Firstly, the terrestrial laser point cloud data of an independent tree is preprocessed to obtain terrestrial point clouds of independent tree canopy. Secondly, the multi-scale box-counting dimension calculation algorithm of independent tree canopy dense terrestrial laser point cloud is proposed. Furthermore, a robust box-counting algorithm is proposed to improve the stability and accuracy of fractal dimension expression of independent tree point cloud, which implementing gross error elimination based on Random Sample Consensus. Finally, the fractal dimension of a dense terrestrial laser point cloud of independent trees is used to classify different types of independent tree species. Experiments on nine independent trees of three types show that the fractal dimension can be stabilized under large density variations, proving that the fractal features of terrestrial laser point cloud can stably express tree species characteristics, and can be used for accurate classification and recognition of forest species. Fractal Fract, Vol. 5, Pages 14: Application of Fractal Dimension of Terrestrial Laser Point Cloud in Classification of Independent Trees

Fractal and Fractional doi: 10.3390/fractalfract5010014

Authors: Ju Zhang Qingwu Hu Hongyu Wu Junying Su Pengcheng Zhao

Tree precise classification and identification of forest species is a core issue of forestry resource monitoring and ecological effect assessment. In this paper, an independent tree species classification method based on fractal features of terrestrial laser point cloud is proposed. Firstly, the terrestrial laser point cloud data of an independent tree is preprocessed to obtain terrestrial point clouds of independent tree canopy. Secondly, the multi-scale box-counting dimension calculation algorithm of independent tree canopy dense terrestrial laser point cloud is proposed. Furthermore, a robust box-counting algorithm is proposed to improve the stability and accuracy of fractal dimension expression of independent tree point cloud, which implementing gross error elimination based on Random Sample Consensus. Finally, the fractal dimension of a dense terrestrial laser point cloud of independent trees is used to classify different types of independent tree species. Experiments on nine independent trees of three types show that the fractal dimension can be stabilized under large density variations, proving that the fractal features of terrestrial laser point cloud can stably express tree species characteristics, and can be used for accurate classification and recognition of forest species.

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Application of Fractal Dimension of Terrestrial Laser Point Cloud in Classification of Independent Trees Ju Zhang Qingwu Hu Hongyu Wu Junying Su Pengcheng Zhao doi: 10.3390/fractalfract5010014 Fractal and Fractional 2021-02-01 Fractal and Fractional 2021-02-01 5 1 Article 14 10.3390/fractalfract5010014 https://www.mdpi.com/2504-3110/5/1/14
Fractal Fract, Vol. 5, Pages 13: Cole-Impedance Model Representations of Right-Side Segmental Arm, Leg, and Full-Body Bioimpedances of Healthy Adults: Comparison of Fractional-Order https://www.mdpi.com/2504-3110/5/1/13 The passive electrical properties of a biological tissue, referred to as the tissue bioimpedance, are related to the underlying tissue physiology. These measurements are often well-represented by a fractional-order equivalent circuit model, referred to as the Cole-impedance model. Objective: Identify if there are differences in the fractional-order (&amp;alpha;) of the Cole-impedance parameters that represent the segmental right-body, right-arm, and right-leg of adult participants. Hypothesis: Cole-impedance model parameters often associated with tissue geometry and fluid (R&amp;infin;, R1, C) will be different between body segments, but parameters often associated with tissue type (&amp;alpha;) will not show any statistical differences. Approach: A secondary analysis was applied to a dataset collected for an agreement study between bioimpedance spectroscopy devices and dual-energy X-ray absoptiometry, identifying the Cole-model parameters of the right-side body segments of N=174 participants using a particle swarm optimization approach. Statistical testing was applied to the different groups of Cole-model parameters to evaluate group differences and correlations of parameters with tissue features. Results: All Cole-impedance model parameters showed statistically significant differences between body segments. Significance: The physiological or geometric features of biological tissues that are linked with the fractional-order (&amp;alpha;) of data represented by the Cole-impedance model requires further study to elucidate. Fractal Fract, Vol. 5, Pages 13: Cole-Impedance Model Representations of Right-Side Segmental Arm, Leg, and Full-Body Bioimpedances of Healthy Adults: Comparison of Fractional-Order

Fractal and Fractional doi: 10.3390/fractalfract5010013

Authors: Todd J. Freeborn Shelby Critcher

The passive electrical properties of a biological tissue, referred to as the tissue bioimpedance, are related to the underlying tissue physiology. These measurements are often well-represented by a fractional-order equivalent circuit model, referred to as the Cole-impedance model. Objective: Identify if there are differences in the fractional-order (&amp;alpha;) of the Cole-impedance parameters that represent the segmental right-body, right-arm, and right-leg of adult participants. Hypothesis: Cole-impedance model parameters often associated with tissue geometry and fluid (R&amp;infin;, R1, C) will be different between body segments, but parameters often associated with tissue type (&amp;alpha;) will not show any statistical differences. Approach: A secondary analysis was applied to a dataset collected for an agreement study between bioimpedance spectroscopy devices and dual-energy X-ray absoptiometry, identifying the Cole-model parameters of the right-side body segments of N=174 participants using a particle swarm optimization approach. Statistical testing was applied to the different groups of Cole-model parameters to evaluate group differences and correlations of parameters with tissue features. Results: All Cole-impedance model parameters showed statistically significant differences between body segments. Significance: The physiological or geometric features of biological tissues that are linked with the fractional-order (&amp;alpha;) of data represented by the Cole-impedance model requires further study to elucidate.

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Cole-Impedance Model Representations of Right-Side Segmental Arm, Leg, and Full-Body Bioimpedances of Healthy Adults: Comparison of Fractional-Order Todd J. Freeborn Shelby Critcher doi: 10.3390/fractalfract5010013 Fractal and Fractional 2021-01-28 Fractal and Fractional 2021-01-28 5 1 Article 13 10.3390/fractalfract5010013 https://www.mdpi.com/2504-3110/5/1/13
Fractal Fract, Vol. 5, Pages 12: Acknowledgment to Reviewers of Fractal and Fractional in 2020 https://www.mdpi.com/2504-3110/5/1/12 Peer review is the driving force of journal development, and reviewers are gatekeepers who ensure that Fractal and Fractional maintains its standards for the high quality of its published papers [...] Fractal Fract, Vol. 5, Pages 12: Acknowledgment to Reviewers of Fractal and Fractional in 2020

Fractal and Fractional doi: 10.3390/fractalfract5010012

Authors: Fractal and Fractional Editorial Office Fractal and Fractional Editorial Office

Peer review is the driving force of journal development, and reviewers are gatekeepers who ensure that Fractal and Fractional maintains its standards for the high quality of its published papers [...]

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Acknowledgment to Reviewers of Fractal and Fractional in 2020 Fractal and Fractional Editorial Office Fractal and Fractional Editorial Office doi: 10.3390/fractalfract5010012 Fractal and Fractional 2021-01-28 Fractal and Fractional 2021-01-28 5 1 Editorial 12 10.3390/fractalfract5010012 https://www.mdpi.com/2504-3110/5/1/12
Fractal Fract, Vol. 5, Pages 11: Analysis of the Effects of the Viscous Thermal Losses in the Flute Musical Instruments https://www.mdpi.com/2504-3110/5/1/11 This article presents the third part of a larger project whose final objective is to study and analyse the effects of viscous thermal losses in a flute wind musical instrument. After implementing the test bench in the first phase and modelling and validating the dynamic behaviour of the simulator, based on the previously implemented test bench (without considering the losses in the system) in the second phase, this third phase deals with the study of the viscous thermal losses that will be generated within the resonator of the flute. These losses are mainly due to the friction of the air inside the resonator with its boundaries and the changes of the temperature within this medium. They are mainly affected by the flute geometry and the materials used in the fabrication of this instrument. After modelling these losses in the frequency domain, they will be represented using a system approach where the fractional order part is separated from the system’s transfer function. Thus, this representation allows us to study, in a precise way, the influence of the fractional order behaviour on the overall system. Effectively, the fractional behavior only appears much below the 20 Hz audible frequencies, but it explains the influence of this order on the frequency response over the range [20–20,000] Hz. Some simulations will be proposed to show the effects of the fractional order on the system response. Fractal Fract, Vol. 5, Pages 11: Analysis of the Effects of the Viscous Thermal Losses in the Flute Musical Instruments

Fractal and Fractional doi: 10.3390/fractalfract5010011

Authors: Gaby Abou Haidar Xavier Moreau Roy Abi Zeid Daou

This article presents the third part of a larger project whose final objective is to study and analyse the effects of viscous thermal losses in a flute wind musical instrument. After implementing the test bench in the first phase and modelling and validating the dynamic behaviour of the simulator, based on the previously implemented test bench (without considering the losses in the system) in the second phase, this third phase deals with the study of the viscous thermal losses that will be generated within the resonator of the flute. These losses are mainly due to the friction of the air inside the resonator with its boundaries and the changes of the temperature within this medium. They are mainly affected by the flute geometry and the materials used in the fabrication of this instrument. After modelling these losses in the frequency domain, they will be represented using a system approach where the fractional order part is separated from the system’s transfer function. Thus, this representation allows us to study, in a precise way, the influence of the fractional order behaviour on the overall system. Effectively, the fractional behavior only appears much below the 20 Hz audible frequencies, but it explains the influence of this order on the frequency response over the range [20–20,000] Hz. Some simulations will be proposed to show the effects of the fractional order on the system response.

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Analysis of the Effects of the Viscous Thermal Losses in the Flute Musical Instruments Gaby Abou Haidar Xavier Moreau Roy Abi Zeid Daou doi: 10.3390/fractalfract5010011 Fractal and Fractional 2021-01-19 Fractal and Fractional 2021-01-19 5 1 Article 11 10.3390/fractalfract5010011 https://www.mdpi.com/2504-3110/5/1/11
Fractal Fract, Vol. 5, Pages 10: Signal Propagation in Electromagnetic Media Modelled by the Two-Sided Fractional Derivative https://www.mdpi.com/2504-3110/5/1/10 In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its definition, there is no obvious relation between causality of the derivative and causality of the transfer function induced by this derivative. Hence, causality of the system is investigated; its output is an electromagnetic signal propagating in media described by the time-domain two-sided fractional derivative. It is demonstrated that, for the derivative order in the range [1,+&amp;infin;), the transfer function describing attenuated signal propagation is not causal for any value of the asymmetry parameter of the derivative. On the other hand, it is shown that, for derivative orders in the range (0,1), the transfer function is causal if and only if the asymmetry parameter is equal to certain specific values corresponding to the left-sided Gr&amp;uuml;nwald&amp;ndash;Letnikov derivative. The results are illustrated by numerical simulations and analyses. Some comments on the Kramers&amp;ndash;Kr&amp;ouml;nig relations for logarithm of the transfer function are presented as well. Fractal Fract, Vol. 5, Pages 10: Signal Propagation in Electromagnetic Media Modelled by the Two-Sided Fractional Derivative

Fractal and Fractional doi: 10.3390/fractalfract5010010

Authors: Jacek Gulgowski Dariusz Kwiatkowski Tomasz P. Stefański

In this paper, wave propagation is considered in a medium described by a fractional-order model, which is formulated with the use of the two-sided fractional derivative of Ortigueira and Machado. Although the relation of the derivative to causality is clearly specified in its definition, there is no obvious relation between causality of the derivative and causality of the transfer function induced by this derivative. Hence, causality of the system is investigated; its output is an electromagnetic signal propagating in media described by the time-domain two-sided fractional derivative. It is demonstrated that, for the derivative order in the range [1,+&amp;infin;), the transfer function describing attenuated signal propagation is not causal for any value of the asymmetry parameter of the derivative. On the other hand, it is shown that, for derivative orders in the range (0,1), the transfer function is causal if and only if the asymmetry parameter is equal to certain specific values corresponding to the left-sided Gr&amp;uuml;nwald&amp;ndash;Letnikov derivative. The results are illustrated by numerical simulations and analyses. Some comments on the Kramers&amp;ndash;Kr&amp;ouml;nig relations for logarithm of the transfer function are presented as well.

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Signal Propagation in Electromagnetic Media Modelled by the Two-Sided Fractional Derivative Jacek Gulgowski Dariusz Kwiatkowski Tomasz P. Stefański doi: 10.3390/fractalfract5010010 Fractal and Fractional 2021-01-18 Fractal and Fractional 2021-01-18 5 1 Article 10 10.3390/fractalfract5010010 https://www.mdpi.com/2504-3110/5/1/10
Fractal Fract, Vol. 5, Pages 8: Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm https://www.mdpi.com/2504-3110/5/1/8 This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads. Fractal Fract, Vol. 5, Pages 8: Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm

Fractal and Fractional doi: 10.3390/fractalfract5010008

Authors: Cundi Han Yiming Chen Da-Yan Liu Driss Boutat

This paper applies a numerical method of polynomial function approximation to the numerical analysis of variable fractional order viscoelastic rotating beam. First, the governing equation of the viscoelastic rotating beam is established based on the variable fractional model of the viscoelastic material. Second, shifted Bernstein polynomials and Legendre polynomials are used as basis functions to approximate the governing equation and the original equation is converted to matrix product form. Based on the configuration method, the matrix equation is further transformed into algebraic equations and numerical solutions of the governing equation are obtained directly in the time domain. Finally, the efficiency of the proposed algorithm is proved by analyzing the numerical solutions of the displacement of rotating beam under different loads.

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Numerical Analysis of Viscoelastic Rotating Beam with Variable Fractional Order Model Using Shifted Bernstein–Legendre Polynomial Collocation Algorithm Cundi Han Yiming Chen Da-Yan Liu Driss Boutat doi: 10.3390/fractalfract5010008 Fractal and Fractional 2021-01-13 Fractal and Fractional 2021-01-13 5 1 Article 8 10.3390/fractalfract5010008 https://www.mdpi.com/2504-3110/5/1/8
Fractal Fract, Vol. 5, Pages 9: Electrical Circuits RC, LC, and RLC under Generalized Type Non-Local Singular Fractional Operator https://www.mdpi.com/2504-3110/5/1/9 The current study is of interest when performing a useful extension of a crucial physical problem through a non-local singular fractional operator. We provide solutions that include three arbitrary parameters &amp;alpha;, &amp;rho;, and &amp;gamma; for the Resistance-Capacitance (RC), Inductance-Capacitance (LC), and Resistance-Inductance-Capacitance (RLC) electric circuits utilizing a generalized type fractional operator in the sense of Caputo, called non-local M-derivative. Additionally, to keep the dimensionality of the physical parameter in the proposed model, we use an auxiliary parameter. Owing to the fact that all solutions depend on three parameters unlike the other solutions containing one or two parameters in the literature, the solutions obtained in this study have more general results. On the other hand, in order to observe the advantages of the non-local M-derivative, a comprehensive comparison is carried out in the light of experimental data. We make this comparison for the RC circuit between the non-local M-derivative and Caputo derivative. It is clearly shown on graphs that the fractional M-derivative behaves closer to the experimental data thanks to the added parameters &amp;alpha;, &amp;rho;, and &amp;gamma;. Fractal Fract, Vol. 5, Pages 9: Electrical Circuits RC, LC, and RLC under Generalized Type Non-Local Singular Fractional Operator

Fractal and Fractional doi: 10.3390/fractalfract5010009

Authors: Bahar Acay Mustafa Inc

The current study is of interest when performing a useful extension of a crucial physical problem through a non-local singular fractional operator. We provide solutions that include three arbitrary parameters &amp;alpha;, &amp;rho;, and &amp;gamma; for the Resistance-Capacitance (RC), Inductance-Capacitance (LC), and Resistance-Inductance-Capacitance (RLC) electric circuits utilizing a generalized type fractional operator in the sense of Caputo, called non-local M-derivative. Additionally, to keep the dimensionality of the physical parameter in the proposed model, we use an auxiliary parameter. Owing to the fact that all solutions depend on three parameters unlike the other solutions containing one or two parameters in the literature, the solutions obtained in this study have more general results. On the other hand, in order to observe the advantages of the non-local M-derivative, a comprehensive comparison is carried out in the light of experimental data. We make this comparison for the RC circuit between the non-local M-derivative and Caputo derivative. It is clearly shown on graphs that the fractional M-derivative behaves closer to the experimental data thanks to the added parameters &amp;alpha;, &amp;rho;, and &amp;gamma;.

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Electrical Circuits RC, LC, and RLC under Generalized Type Non-Local Singular Fractional Operator Bahar Acay Mustafa Inc doi: 10.3390/fractalfract5010009 Fractal and Fractional 2021-01-12 Fractal and Fractional 2021-01-12 5 1 Article 9 10.3390/fractalfract5010009 https://www.mdpi.com/2504-3110/5/1/9
Fractal Fract, Vol. 5, Pages 7: A New Approach for the Fractional Integral Operator in Time Scales with Variable Exponent Lebesgue Spaces https://www.mdpi.com/2504-3110/5/1/7 Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-&amp;alpha; integral operator Ma,&amp;delta;c to the norm of the centered fractional maximal diamond-&amp;alpha; integral operator Mac on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales. Fractal Fract, Vol. 5, Pages 7: A New Approach for the Fractional Integral Operator in Time Scales with Variable Exponent Lebesgue Spaces

Fractal and Fractional doi: 10.3390/fractalfract5010007

Authors: Lütfi Akın

Integral equations and inequalities have an important place in time scales and harmonic analysis. The norm of integral operators is one of the important study topics in harmonic analysis. Using the norms in different variable exponent spaces, the boundedness or compactness of the integral operators are examined. However, the norm of integral operators on time scales has been a matter of curiosity to us. In this study, we prove the equivalence of the norm of the restricted centered fractional maximal diamond-&amp;alpha; integral operator Ma,&amp;delta;c to the norm of the centered fractional maximal diamond-&amp;alpha; integral operator Mac on time scales with variable exponent Lebesgue spaces. This study will lead to the study of problems such as the boundedness and compactness of integral operators on time scales.

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A New Approach for the Fractional Integral Operator in Time Scales with Variable Exponent Lebesgue Spaces Lütfi Akın doi: 10.3390/fractalfract5010007 Fractal and Fractional 2021-01-08 Fractal and Fractional 2021-01-08 5 1 Article 7 10.3390/fractalfract5010007 https://www.mdpi.com/2504-3110/5/1/7
Fractal Fract, Vol. 5, Pages 6: Extraction Complex Properties of the Nonlinear Modified Alpha Equation https://www.mdpi.com/2504-3110/5/1/6 This paper applies one of the special cases of auxiliary method, which is named as the Bernoulli sub-equation function method, to the nonlinear modified alpha equation. The characteristic properties of these solutions, such as complex and soliton solutions, are extracted. Moreover, the strain conditions of solutions are also reported in detail. Observing the figures plotted by considering various values of parameters of these solutions confirms the effectiveness of the approximation method used for the governing model. Fractal Fract, Vol. 5, Pages 6: Extraction Complex Properties of the Nonlinear Modified Alpha Equation

Fractal and Fractional doi: 10.3390/fractalfract5010006

Authors: Haci Mehmet Baskonus Muzaffer Ercan

This paper applies one of the special cases of auxiliary method, which is named as the Bernoulli sub-equation function method, to the nonlinear modified alpha equation. The characteristic properties of these solutions, such as complex and soliton solutions, are extracted. Moreover, the strain conditions of solutions are also reported in detail. Observing the figures plotted by considering various values of parameters of these solutions confirms the effectiveness of the approximation method used for the governing model.

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Extraction Complex Properties of the Nonlinear Modified Alpha Equation Haci Mehmet Baskonus Muzaffer Ercan doi: 10.3390/fractalfract5010006 Fractal and Fractional 2021-01-07 Fractal and Fractional 2021-01-07 5 1 Article 6 10.3390/fractalfract5010006 https://www.mdpi.com/2504-3110/5/1/6
Fractal Fract, Vol. 5, Pages 5: A Collocation Method Based on Discrete Spline Quasi-Interpolatory Operators for the Solution of Time Fractional Differential Equations https://www.mdpi.com/2504-3110/5/1/5 In many applications, real phenomena are modeled by differential problems having a time fractional derivative that depends on the history of the unknown function. For the numerical solution of time fractional differential equations, we propose a new method that combines spline quasi-interpolatory operators and collocation methods. We show that the method is convergent and reproduces polynomials of suitable degree. The numerical tests demonstrate the validity and applicability of the proposed method when used to solve linear time fractional differential equations. Fractal Fract, Vol. 5, Pages 5: A Collocation Method Based on Discrete Spline Quasi-Interpolatory Operators for the Solution of Time Fractional Differential Equations

Fractal and Fractional doi: 10.3390/fractalfract5010005

Authors: Enza Pellegrino Laura Pezza Francesca Pitolli

In many applications, real phenomena are modeled by differential problems having a time fractional derivative that depends on the history of the unknown function. For the numerical solution of time fractional differential equations, we propose a new method that combines spline quasi-interpolatory operators and collocation methods. We show that the method is convergent and reproduces polynomials of suitable degree. The numerical tests demonstrate the validity and applicability of the proposed method when used to solve linear time fractional differential equations.

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A Collocation Method Based on Discrete Spline Quasi-Interpolatory Operators for the Solution of Time Fractional Differential Equations Enza Pellegrino Laura Pezza Francesca Pitolli doi: 10.3390/fractalfract5010005 Fractal and Fractional 2021-01-05 Fractal and Fractional 2021-01-05 5 1 Article 5 10.3390/fractalfract5010005 https://www.mdpi.com/2504-3110/5/1/5
Fractal Fract, Vol. 5, Pages 4: The Impact of Anomalous Diffusion on Action Potentials in Myelinated Neurons https://www.mdpi.com/2504-3110/5/1/4 Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases such as epilepsy, multiple sclerosis, and Alzheimer&amp;rsquo;s disease, which have become more prevalent in the latest years. Fractal Fract, Vol. 5, Pages 4: The Impact of Anomalous Diffusion on Action Potentials in Myelinated Neurons

Fractal and Fractional doi: 10.3390/fractalfract5010004

Authors: Corina S. Drapaca

Action potentials in myelinated neurons happen only at specialized locations of the axons known as the nodes of Ranvier. The shapes, timings, and propagation speeds of these action potentials are controlled by biochemical interactions among neurons, glial cells, and the extracellular space. The complexity of brain structure and processes suggests that anomalous diffusion could affect the propagation of action potentials. In this paper, a spatio-temporal fractional cable equation for action potentials propagation in myelinated neurons is proposed. The impact of the ionic anomalous diffusion on the distribution of the membrane potential is investigated using numerical simulations. The results show spatially narrower action potentials at the nodes of Ranvier when using spatial derivatives of the fractional order only and delayed or lack of action potentials when adding a temporal derivative of the fractional order. These findings could reveal the pathological patterns of brain diseases such as epilepsy, multiple sclerosis, and Alzheimer&amp;rsquo;s disease, which have become more prevalent in the latest years.

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The Impact of Anomalous Diffusion on Action Potentials in Myelinated Neurons Corina S. Drapaca doi: 10.3390/fractalfract5010004 Fractal and Fractional 2021-01-05 Fractal and Fractional 2021-01-05 5 1 Article 4 10.3390/fractalfract5010004 https://www.mdpi.com/2504-3110/5/1/4
Fractal Fract, Vol. 5, Pages 3: Optimal V-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems https://www.mdpi.com/2504-3110/5/1/3 Robust stability is a major concern for real-world control applications. Realization of optimal robust stability requires a stabilization scheme, which ensures that the control system is stable and presents robust performance for a predefined range of system perturbations. This study presented an optimal robust stabilization approach for closed-loop fractional order proportional integral derivative (FOPID) control systems with interval parametric uncertainty and uncertain time delay. This stabilization approach, which is carried out in a v-plane, relies on the placement of the minimum angle system pole to a predefined target angle within the stability region of the first Riemann sheet. For this purpose, tuning of FOPID controller coefficients was performed to minimize a root angle error that is defined as the squared difference of minimum angle root of interval characteristic polynomials and the desired target angle within the stability region of the v-plane. To solve this optimization problem, a particle swarm optimization (PSO) algorithm was implemented. Findings of the study reveal that tuning of the target angle can also be used to improve the robust control performance of interval uncertain FOPID control systems. Illustrative examples demonstrated the effectiveness of the proposed v-domain, optimal, robust stabilization of FOPID control systems. Fractal Fract, Vol. 5, Pages 3: Optimal V-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems

Fractal and Fractional doi: 10.3390/fractalfract5010003

Authors: Sevilay Tufenkci Bilal Senol Radek Matušů Baris Baykant Alagoz

Robust stability is a major concern for real-world control applications. Realization of optimal robust stability requires a stabilization scheme, which ensures that the control system is stable and presents robust performance for a predefined range of system perturbations. This study presented an optimal robust stabilization approach for closed-loop fractional order proportional integral derivative (FOPID) control systems with interval parametric uncertainty and uncertain time delay. This stabilization approach, which is carried out in a v-plane, relies on the placement of the minimum angle system pole to a predefined target angle within the stability region of the first Riemann sheet. For this purpose, tuning of FOPID controller coefficients was performed to minimize a root angle error that is defined as the squared difference of minimum angle root of interval characteristic polynomials and the desired target angle within the stability region of the v-plane. To solve this optimization problem, a particle swarm optimization (PSO) algorithm was implemented. Findings of the study reveal that tuning of the target angle can also be used to improve the robust control performance of interval uncertain FOPID control systems. Illustrative examples demonstrated the effectiveness of the proposed v-domain, optimal, robust stabilization of FOPID control systems.

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Optimal V-Plane Robust Stabilization Method for Interval Uncertain Fractional Order PID Control Systems Sevilay Tufenkci Bilal Senol Radek Matušů Baris Baykant Alagoz doi: 10.3390/fractalfract5010003 Fractal and Fractional 2021-01-03 Fractal and Fractional 2021-01-03 5 1 Article 3 10.3390/fractalfract5010003 https://www.mdpi.com/2504-3110/5/1/3
Fractal Fract, Vol. 5, Pages 2: 5G Poor and Rich Novel Control Scheme Based Load Frequency Regulation of a Two-Area System with 100% Renewables in Africa https://www.mdpi.com/2504-3110/5/1/2 Remote farms in Africa are cultivated lands planned for 100% sustainable energy and organic agriculture in the future. This paper presents the load frequency control of a two-area power system feeding those farms. The power system is supplied by renewable technologies and storage facilities only which are photovoltaics, biogas, biodiesel, solar thermal, battery storage and flywheel storage systems. Each of those facilities has 150-kW capacity. This paper presents a model for each renewable energy technology and energy storage facility. The frequency is controlled by using a novel non-linear fractional order proportional integral derivative control scheme (NFOPID). The novel scheme is compared to a non-linear PID controller (NPID), fractional order PID controller (FOPID), and conventional PID. The effect of the different degradation factors related to the communication infrastructure, such as the time delay and packet loss, are modeled and simulated to assess the controlled system performance. A new cost function is presented in this research. The four controllers are tuned by novel poor and rich optimization (PRO) algorithm at different operating conditions. PRO controller design is compared to other state of the art techniques in this paper. The results show that the PRO design for a novel NFOPID controller has a promising future in load frequency control considering communication delays and packet loss. The simulation and optimization are applied on MATLAB/SIMULINK 2017a environment. Fractal Fract, Vol. 5, Pages 2: 5G Poor and Rich Novel Control Scheme Based Load Frequency Regulation of a Two-Area System with 100% Renewables in Africa

Fractal and Fractional doi: 10.3390/fractalfract5010002

Remote farms in Africa are cultivated lands planned for 100% sustainable energy and organic agriculture in the future. This paper presents the load frequency control of a two-area power system feeding those farms. The power system is supplied by renewable technologies and storage facilities only which are photovoltaics, biogas, biodiesel, solar thermal, battery storage and flywheel storage systems. Each of those facilities has 150-kW capacity. This paper presents a model for each renewable energy technology and energy storage facility. The frequency is controlled by using a novel non-linear fractional order proportional integral derivative control scheme (NFOPID). The novel scheme is compared to a non-linear PID controller (NPID), fractional order PID controller (FOPID), and conventional PID. The effect of the different degradation factors related to the communication infrastructure, such as the time delay and packet loss, are modeled and simulated to assess the controlled system performance. A new cost function is presented in this research. The four controllers are tuned by novel poor and rich optimization (PRO) algorithm at different operating conditions. PRO controller design is compared to other state of the art techniques in this paper. The results show that the PRO design for a novel NFOPID controller has a promising future in load frequency control considering communication delays and packet loss. The simulation and optimization are applied on MATLAB/SIMULINK 2017a environment.

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5G Poor and Rich Novel Control Scheme Based Load Frequency Regulation of a Two-Area System with 100% Renewables in Africa Hady H. Fayek doi: 10.3390/fractalfract5010002 Fractal and Fractional 2020-12-23 Fractal and Fractional 2020-12-23 5 1 Article 2 10.3390/fractalfract5010002 https://www.mdpi.com/2504-3110/5/1/2
Fractal Fract, Vol. 5, Pages 1: Boundary Value Problem for Fractional Order Generalized Hilfer-Type Fractional Derivative with Non-Instantaneous Impulses https://www.mdpi.com/2504-3110/5/1/1 This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and Krasnosel’skii’s fixed point theorem. To illustrate the results, an example is provided. Fractal Fract, Vol. 5, Pages 1: Boundary Value Problem for Fractional Order Generalized Hilfer-Type Fractional Derivative with Non-Instantaneous Impulses

Fractal and Fractional doi: 10.3390/fractalfract5010001

Authors: Abdelkrim Salim Mouffak Benchohra John R. Graef Jamal Eddine Lazreg

This manuscript is devoted to proving some results concerning the existence of solutions to a class of boundary value problems for nonlinear implicit fractional differential equations with non-instantaneous impulses and generalized Hilfer fractional derivatives. The results are based on Banach’s contraction principle and Krasnosel’skii’s fixed point theorem. To illustrate the results, an example is provided.

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Boundary Value Problem for Fractional Order Generalized Hilfer-Type Fractional Derivative with Non-Instantaneous Impulses Abdelkrim Salim Mouffak Benchohra John R. Graef Jamal Eddine Lazreg doi: 10.3390/fractalfract5010001 Fractal and Fractional 2020-12-22 Fractal and Fractional 2020-12-22 5 1 Article 1 10.3390/fractalfract5010001 https://www.mdpi.com/2504-3110/5/1/1
Fractal Fract, Vol. 4, Pages 60: Correction: Gürel Yılmaz, Ö., et al. On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative. Fractal Fract. 2020, 4, 48 https://www.mdpi.com/2504-3110/4/4/60 The authors wish to make the following corrections to this paper [...] Fractal Fract, Vol. 4, Pages 60: Correction: Gürel Yılmaz, Ö., et al. On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative. Fractal Fract. 2020, 4, 48

Fractal and Fractional doi: 10.3390/fractalfract4040060

Authors: Övgü Gürel Yılmaz Rabia Aktaş Fatma Taşdelen

The authors wish to make the following corrections to this paper [...]

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Correction: Gürel Yılmaz, Ö., et al. On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative. Fractal Fract. 2020, 4, 48 Övgü Gürel Yılmaz Rabia Aktaş Fatma Taşdelen doi: 10.3390/fractalfract4040060 Fractal and Fractional 2020-12-21 Fractal and Fractional 2020-12-21 4 4 Correction 60 10.3390/fractalfract4040060 https://www.mdpi.com/2504-3110/4/4/60
Fractal Fract, Vol. 4, Pages 59: Fractional-Fractal Modeling of Filtration-Consolidation Processes in Saline Saturated Soils https://www.mdpi.com/2504-3110/4/4/59 To study the peculiarities of anomalous consolidation processes in saturated porous (soil) media in the conditions of salt transfer, we present a new mathematical model developed on the base of the fractional-fractal approach that allows considering temporal non-locality of transfer processes in media of fractal structure. For the case of the finite thickness domain with permeable boundaries, a finite-difference technique for numerical solution of the corresponding one-dimensional non-linear boundary value problem is developed. The paper also presents a fractional-fractal model of a filtration-consolidation process in clay soils of fractal structure saturated with salt solutions. An analytical solution is found for the corresponding one-dimensional boundary value problem in the domain of finite thickness with permeable upper and impermeable lower boundaries. Fractal Fract, Vol. 4, Pages 59: Fractional-Fractal Modeling of Filtration-Consolidation Processes in Saline Saturated Soils

Fractal and Fractional doi: 10.3390/fractalfract4040059

Authors: Vsevolod Bohaienko Volodymyr Bulavatsky

To study the peculiarities of anomalous consolidation processes in saturated porous (soil) media in the conditions of salt transfer, we present a new mathematical model developed on the base of the fractional-fractal approach that allows considering temporal non-locality of transfer processes in media of fractal structure. For the case of the finite thickness domain with permeable boundaries, a finite-difference technique for numerical solution of the corresponding one-dimensional non-linear boundary value problem is developed. The paper also presents a fractional-fractal model of a filtration-consolidation process in clay soils of fractal structure saturated with salt solutions. An analytical solution is found for the corresponding one-dimensional boundary value problem in the domain of finite thickness with permeable upper and impermeable lower boundaries.

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Fractional-Fractal Modeling of Filtration-Consolidation Processes in Saline Saturated Soils Vsevolod Bohaienko Volodymyr Bulavatsky doi: 10.3390/fractalfract4040059 Fractal and Fractional 2020-12-16 Fractal and Fractional 2020-12-16 4 4 Article 59 10.3390/fractalfract4040059 https://www.mdpi.com/2504-3110/4/4/59
Fractal Fract, Vol. 4, Pages 58: LMI Criteria for Admissibility and Robust Stabilization of Singular Fractional-Order Systems Possessing Poly-Topic Uncertainties https://www.mdpi.com/2504-3110/4/4/58 The issue of robust admissibility and control for singular fractional-order systems (FOSs) with polytopic uncertainties is investigated in this paper. Firstly, a new method based on linear matrix inequalities (LMIs) is presented to solve the admissibility problems of uncertain linear systems. Then, a solid criterion of robust admissibility and a corresponding state feedback controller are derived, which overcome the conservatism of the existing results. Finally, for the sake of demonstrating the validity of proposed results, some relevant examples are provided. Fractal Fract, Vol. 4, Pages 58: LMI Criteria for Admissibility and Robust Stabilization of Singular Fractional-Order Systems Possessing Poly-Topic Uncertainties

Fractal and Fractional doi: 10.3390/fractalfract4040058

Authors: Xuefeng Zhang Jia Dong

The issue of robust admissibility and control for singular fractional-order systems (FOSs) with polytopic uncertainties is investigated in this paper. Firstly, a new method based on linear matrix inequalities (LMIs) is presented to solve the admissibility problems of uncertain linear systems. Then, a solid criterion of robust admissibility and a corresponding state feedback controller are derived, which overcome the conservatism of the existing results. Finally, for the sake of demonstrating the validity of proposed results, some relevant examples are provided.

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LMI Criteria for Admissibility and Robust Stabilization of Singular Fractional-Order Systems Possessing Poly-Topic Uncertainties Xuefeng Zhang Jia Dong doi: 10.3390/fractalfract4040058 Fractal and Fractional 2020-12-15 Fractal and Fractional 2020-12-15 4 4 Article 58 10.3390/fractalfract4040058 https://www.mdpi.com/2504-3110/4/4/58
Fractal Fract, Vol. 4, Pages 57: An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations https://www.mdpi.com/2504-3110/4/4/57 A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction&amp;ndash;diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the PageRank algorithm. Some numerical examples are given. Fractal Fract, Vol. 4, Pages 57: An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations

Fractal and Fractional doi: 10.3390/fractalfract4040057

Authors: Moreno Concezzi Renato Spigler

A numerical method for solving fractional partial differential equations (fPDEs) of the diffusion and reaction&amp;ndash;diffusion type, subject to Dirichlet boundary data, in three dimensions is developed. Such fPDEs may describe fluid flows through porous media better than classical diffusion equations. This is a new, fractional version of the Alternating Direction Implicit (ADI) method, where the source term is balanced, in that its effect is split in the three space directions, and it may be relevant, especially in the case of anisotropy. The method is unconditionally stable, second-order in space, and third-order in time. A strategy is devised in order to improve its speed of convergence by means of an extrapolation method that is coupled to the PageRank algorithm. Some numerical examples are given.

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An ADI Method for the Numerical Solution of 3D Fractional Reaction-Diffusion Equations Moreno Concezzi Renato Spigler doi: 10.3390/fractalfract4040057 Fractal and Fractional 2020-12-14 Fractal and Fractional 2020-12-14 4 4 Article 57 10.3390/fractalfract4040057 https://www.mdpi.com/2504-3110/4/4/57
Fractal Fract, Vol. 4, Pages 56: Generalized Differentiability of Continuous Functions https://www.mdpi.com/2504-3110/4/4/56 Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated. Fractal Fract, Vol. 4, Pages 56: Generalized Differentiability of Continuous Functions

Fractal and Fractional doi: 10.3390/fractalfract4040056

Authors: Dimiter Prodanov

Many physical phenomena give rise to mathematical models in terms of fractal, non-differentiable functions. The paper introduces a broad generalization of the derivative in terms of the maximal modulus of continuity of the primitive function. These derivatives are called indicial derivatives. As an application, the indicial derivatives are used to characterize the nowhere monotonous functions. Furthermore, the non-differentiability set of such derivatives is proven to be of measure zero. As a second application, the indicial derivative is used in the proof of the Lebesgue differentiation theorem. Finally, the connection with the fractional velocities is demonstrated.

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Generalized Differentiability of Continuous Functions Dimiter Prodanov doi: 10.3390/fractalfract4040056 Fractal and Fractional 2020-12-10 Fractal and Fractional 2020-12-10 4 4 Article 56 10.3390/fractalfract4040056 https://www.mdpi.com/2504-3110/4/4/56
Fractal Fract, Vol. 4, Pages 55: Optimal Modelling of (1 + α) Order Butterworth Filter under the CFE Framework https://www.mdpi.com/2504-3110/4/4/55 This paper presents the optimal rational approximation of (1+&amp;alpha;) order Butterworth filter, where &amp;alpha; ∊ (0,1) under the continued fraction expansion framework, by employing a new cost function. Two simple techniques based on the constrained optimization and the optimal pole-zero placements are proposed to model the magnitude-frequency response of the fractional-order lowpass Butterworth filter (FOLBF). The third-order FOLBF approximants achieve good agreement to the ideal characteristic for six decades of design bandwidth. Circuit realization using the current feedback operational amplifier is presented, and the modelling efficacy is validated in the OrCAD PSPICE platform. Fractal Fract, Vol. 4, Pages 55: Optimal Modelling of (1 + α) Order Butterworth Filter under the CFE Framework

Fractal and Fractional doi: 10.3390/fractalfract4040055

Authors: Shibendu Mahata Rajib Kar Durbadal Mandal

This paper presents the optimal rational approximation of (1+&amp;alpha;) order Butterworth filter, where &amp;alpha; ∊ (0,1) under the continued fraction expansion framework, by employing a new cost function. Two simple techniques based on the constrained optimization and the optimal pole-zero placements are proposed to model the magnitude-frequency response of the fractional-order lowpass Butterworth filter (FOLBF). The third-order FOLBF approximants achieve good agreement to the ideal characteristic for six decades of design bandwidth. Circuit realization using the current feedback operational amplifier is presented, and the modelling efficacy is validated in the OrCAD PSPICE platform.

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Optimal Modelling of (1 + α) Order Butterworth Filter under the CFE Framework Shibendu Mahata Rajib Kar Durbadal Mandal doi: 10.3390/fractalfract4040055 Fractal and Fractional 2020-12-03 Fractal and Fractional 2020-12-03 4 4 Article 55 10.3390/fractalfract4040055 https://www.mdpi.com/2504-3110/4/4/55
Fractal Fract, Vol. 4, Pages 54: Realization of Cole–Davidson Function-Based Impedance Models: Application on Plant Tissues https://www.mdpi.com/2504-3110/4/4/54 The Cole&amp;ndash;Davidson function is an efficient tool for describing the tissue behavior, but the conventional methods of approximation are not applicable due the form of this function. In order to overcome this problem, a novel scheme for approximating the Cole&amp;ndash;Davidson function, based on the utilization of a curve fitting procedure offered by the MATLAB software, is introduced in this work. The derived rational transfer function is implemented using the conventional Cauer and Foster RC networks. As an application example, the impedance model of the membrane of mesophyll cells is realized, with simulation results verifying the validity of the introduced procedure. Fractal Fract, Vol. 4, Pages 54: Realization of Cole–Davidson Function-Based Impedance Models: Application on Plant Tissues

Fractal and Fractional doi: 10.3390/fractalfract4040054

Authors: Stavroula Kapoulea Costas Psychalinos Ahmed S. Elwakil

The Cole&amp;ndash;Davidson function is an efficient tool for describing the tissue behavior, but the conventional methods of approximation are not applicable due the form of this function. In order to overcome this problem, a novel scheme for approximating the Cole&amp;ndash;Davidson function, based on the utilization of a curve fitting procedure offered by the MATLAB software, is introduced in this work. The derived rational transfer function is implemented using the conventional Cauer and Foster RC networks. As an application example, the impedance model of the membrane of mesophyll cells is realized, with simulation results verifying the validity of the introduced procedure.

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Realization of Cole–Davidson Function-Based Impedance Models: Application on Plant Tissues Stavroula Kapoulea Costas Psychalinos Ahmed S. Elwakil doi: 10.3390/fractalfract4040054 Fractal and Fractional 2020-11-30 Fractal and Fractional 2020-11-30 4 4 Article 54 10.3390/fractalfract4040054 https://www.mdpi.com/2504-3110/4/4/54
Fractal Fract, Vol. 4, Pages 53: A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems https://www.mdpi.com/2504-3110/4/4/53 In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to &amp;alpha; and &amp;beta; are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms. Fractal Fract, Vol. 4, Pages 53: A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems

Fractal and Fractional doi: 10.3390/fractalfract4040053

Authors: Mehmet Giyas Sakar Onur Saldır

In this article, a new reproducing kernel approach is developed for obtaining a numerical solution of multi-order fractional nonlinear three-point boundary value problems. This approach is based on a reproducing kernel, which is constructed by shifted Legendre polynomials (L-RKM). In the considered problem, fractional derivatives with respect to &amp;alpha; and &amp;beta; are defined in the Caputo sense. This method has been applied to some examples that have exact solutions. In order to show the robustness of the proposed method, some examples are solved and numerical results are given in tabulated forms.

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A New Reproducing Kernel Approach for Nonlinear Fractional Three-Point Boundary Value Problems Mehmet Giyas Sakar Onur Saldır doi: 10.3390/fractalfract4040053 Fractal and Fractional 2020-11-24 Fractal and Fractional 2020-11-24 4 4 Article 53 10.3390/fractalfract4040053 https://www.mdpi.com/2504-3110/4/4/53
Fractal Fract, Vol. 4, Pages 52: Fractional Diffusion to a Cantor Set in 2D https://www.mdpi.com/2504-3110/4/4/52 A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set F&amp;mu; is considered. This search-like problem is treated in the framework of a diffusion&amp;ndash;reaction equation, when an absorbing term is included inside a Fokker&amp;ndash;Planck equation as a reaction term. This macroscopic approach for the 2D transport in the R2 space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink. Fractal Fract, Vol. 4, Pages 52: Fractional Diffusion to a Cantor Set in 2D

Fractal and Fractional doi: 10.3390/fractalfract4040052

Authors: Alexander Iomin Trifce Sandev

A random walk on a two dimensional square in R2 space with a hidden absorbing fractal set F&amp;mu; is considered. This search-like problem is treated in the framework of a diffusion&amp;ndash;reaction equation, when an absorbing term is included inside a Fokker&amp;ndash;Planck equation as a reaction term. This macroscopic approach for the 2D transport in the R2 space corresponds to the comb geometry, when the random walk consists of 1D movements in the x and y directions, respectively, as a direct-Cartesian product of the 1D movements. The main value in task is the first arrival time distribution (FATD) to sink points of the fractal set, where travelling particles are absorbed. Analytical expression for the FATD is obtained in the subdiffusive regime for both the fractal set of sinks and for a single sink.

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Fractional Diffusion to a Cantor Set in 2D Alexander Iomin Trifce Sandev doi: 10.3390/fractalfract4040052 Fractal and Fractional 2020-11-07 Fractal and Fractional 2020-11-07 4 4 Article 52 10.3390/fractalfract4040052 https://www.mdpi.com/2504-3110/4/4/52
Fractal Fract, Vol. 4, Pages 51: Biased Continuous-Time Random Walks with Mittag-Leffler Jumps https://www.mdpi.com/2504-3110/4/4/51 We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process &amp;lsquo;space-time Mittag-Leffler process&amp;rsquo;. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a &amp;ldquo;well-scaled&amp;rdquo; diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the &amp;lsquo;state density kernel&amp;rsquo; solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs. Fractal Fract, Vol. 4, Pages 51: Biased Continuous-Time Random Walks with Mittag-Leffler Jumps

Fractal and Fractional doi: 10.3390/fractalfract4040051

Authors: Thomas M. Michelitsch Federico Polito Alejandro P. Riascos

We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process &amp;lsquo;space-time Mittag-Leffler process&amp;rsquo;. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a &amp;ldquo;well-scaled&amp;rdquo; diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the &amp;lsquo;state density kernel&amp;rsquo; solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.

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Biased Continuous-Time Random Walks with Mittag-Leffler Jumps Thomas M. Michelitsch Federico Polito Alejandro P. Riascos doi: 10.3390/fractalfract4040051 Fractal and Fractional 2020-10-31 Fractal and Fractional 2020-10-31 4 4 Article 51 10.3390/fractalfract4040051 https://www.mdpi.com/2504-3110/4/4/51
Fractal Fract, Vol. 4, Pages 50: Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties https://www.mdpi.com/2504-3110/4/4/50 This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear terms of SFOSs. Firstly, by expanding the dimension of the SFOS, a novel sliding surface was constructed. A necessary and sufficient condition was given to ensure the admissibility of the SFOS while the system state moves on the sliding surface. The obtained results are linear matrix inequalities (LMIs), which are more general than the existing research. Then, the adaptive control law based on the RBF neural network was organized to guarantee that the SFOS reaches the sliding surface in a finite time. Finally, a simulation example is proposed to verify the validity of the designed procedures. Fractal Fract, Vol. 4, Pages 50: Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties

Fractal and Fractional doi: 10.3390/fractalfract4040050

Authors: Xuefeng Zhang Wenkai Huang

This paper focuses on the sliding mode control (SMC) problem for a class of uncertain singular fractional order systems (SFOSs). The uncertainties occur in both state and derivative matrices. A radial basis function (RBF) neural network strategy was utilized to estimate the nonlinear terms of SFOSs. Firstly, by expanding the dimension of the SFOS, a novel sliding surface was constructed. A necessary and sufficient condition was given to ensure the admissibility of the SFOS while the system state moves on the sliding surface. The obtained results are linear matrix inequalities (LMIs), which are more general than the existing research. Then, the adaptive control law based on the RBF neural network was organized to guarantee that the SFOS reaches the sliding surface in a finite time. Finally, a simulation example is proposed to verify the validity of the designed procedures.

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Adaptive Neural Network Sliding Mode Control for Nonlinear Singular Fractional Order Systems with Mismatched Uncertainties Xuefeng Zhang Wenkai Huang doi: 10.3390/fractalfract4040050 Fractal and Fractional 2020-10-22 Fractal and Fractional 2020-10-22 4 4 Article 50 10.3390/fractalfract4040050 https://www.mdpi.com/2504-3110/4/4/50
Fractal Fract, Vol. 4, Pages 49: Numerical Simulation of the Fractal-Fractional Ebola Virus https://www.mdpi.com/2504-3110/4/4/49 In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter &amp;rho; is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of &amp;rho;=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of &amp;rho; and k. All calculations in this work are accomplished by using the Mathematica package. Fractal Fract, Vol. 4, Pages 49: Numerical Simulation of the Fractal-Fractional Ebola Virus

Fractal and Fractional doi: 10.3390/fractalfract4040049

Authors: H. M. Srivastava Khaled M. Saad

In this work we present three new models of the fractal-fractional Ebola virus. We investigate the numerical solutions of the fractal-fractional Ebola virus in the sense of three different kernels based on the power law, the exponential decay and the generalized Mittag-Leffler function by using the concepts of the fractal differentiation and fractional differentiation. These operators have two parameters: The first parameter &amp;rho; is considered as the fractal dimension and the second parameter k is the fractional order. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions. In the case of &amp;rho;=k=1, all of the numerical solutions based on the power kernel, the exponential kernel and the generalized Mittag-Leffler kernel are found to be close to each other and, therefore, one of the kernels is compared with such numerical methods as the finite difference methods. This has led to an excellent agreement. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values of &amp;rho; and k. All calculations in this work are accomplished by using the Mathematica package.

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Numerical Simulation of the Fractal-Fractional Ebola Virus H. M. Srivastava Khaled M. Saad doi: 10.3390/fractalfract4040049 Fractal and Fractional 2020-09-29 Fractal and Fractional 2020-09-29 4 4 Article 49 10.3390/fractalfract4040049 https://www.mdpi.com/2504-3110/4/4/49
Fractal Fract, Vol. 4, Pages 48: On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative https://www.mdpi.com/2504-3110/4/4/48 Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on the k-analogues of F1Appell function introduced by Mubeen et al. 2015 and the k-generalizations of F2 and F3 Appell functions indicated in Kıymaz et al. 2017. we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions. Fractal Fract, Vol. 4, Pages 48: On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative

Fractal and Fractional doi: 10.3390/fractalfract4040048

Authors: Övgü Gürel Yılmaz Rabia Aktaş Fatma Taşdelen

Our present investigation is mainly based on the k-hypergeometric functions which are constructed by making use of the Pochhammer k-symbol in Diaz et al. 2007, which are one of the vital generalizations of hypergeometric functions. In this study, we focus on the k-analogues of F1Appell function introduced by Mubeen et al. 2015 and the k-generalizations of F2 and F3 Appell functions indicated in Kıymaz et al. 2017. we present some important transformation formulas and some reduction formulas which show close relation not only with k-Appell functions but also with k-hypergeometric functions. Employing the theory of Riemann–Liouville k-fractional derivative from Rahman et al. 2020, and using the relations which we consider in this paper, we acquire linear and bilinear generating relations for k-analogue of hypergeometric functions and Appell functions.

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On Some Formulas for the k-Analogue of Appell Functions and Generating Relations via k-Fractional Derivative Övgü Gürel Yılmaz Rabia Aktaş Fatma Taşdelen doi: 10.3390/fractalfract4040048 Fractal and Fractional 2020-09-24 Fractal and Fractional 2020-09-24 4 4 Article 48 10.3390/fractalfract4040048 https://www.mdpi.com/2504-3110/4/4/48
Fractal Fract, Vol. 4, Pages 47: Integral Representation of Fractional Derivative of Delta Function https://www.mdpi.com/2504-3110/4/3/47 Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise. Fractal Fract, Vol. 4, Pages 47: Integral Representation of Fractional Derivative of Delta Function

Fractal and Fractional doi: 10.3390/fractalfract4030047

Authors: Ming Li

Delta function is a widely used generalized function in various fields, ranging from physics to mathematics. How to express its fractional derivative with integral representation is a tough problem. In this paper, we present an integral representation of the fractional derivative of the delta function. Moreover, we provide its application in representing the fractional Gaussian noise.

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Integral Representation of Fractional Derivative of Delta Function Ming Li doi: 10.3390/fractalfract4030047 Fractal and Fractional 2020-09-20 Fractal and Fractional 2020-09-20 4 3 Article 47 10.3390/fractalfract4030047 https://www.mdpi.com/2504-3110/4/3/47
Fractal Fract, Vol. 4, Pages 46: Fractal and Fractional Derivative Modelling of Material Phase Change https://www.mdpi.com/2504-3110/4/3/46 An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations in liquid/solid transitions in physical processes. Three types of transformation are tested experimentally, whipping of cream (rheopexy), solidification of gelatine and melting of ethyl vinyl acetate (EVA). A liquid-type model is used throughout the cream whipping process while liquid and solid models are required for gelatine and EVA to capture the yield characteristic of these materials. Fractal Fract, Vol. 4, Pages 46: Fractal and Fractional Derivative Modelling of Material Phase Change

Fractal and Fractional doi: 10.3390/fractalfract4030046

Authors: Harry Esmonde

An iterative approach is taken to develop a fractal topology that can describe the material structure of phase changing materials. Transfer functions and frequency response functions based on fractional calculus are used to describe this topology and then applied to model phase transformations in liquid/solid transitions in physical processes. Three types of transformation are tested experimentally, whipping of cream (rheopexy), solidification of gelatine and melting of ethyl vinyl acetate (EVA). A liquid-type model is used throughout the cream whipping process while liquid and solid models are required for gelatine and EVA to capture the yield characteristic of these materials.

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Fractal and Fractional Derivative Modelling of Material Phase Change Harry Esmonde doi: 10.3390/fractalfract4030046 Fractal and Fractional 2020-09-14 Fractal and Fractional 2020-09-14 4 3 Article 46 10.3390/fractalfract4030046 https://www.mdpi.com/2504-3110/4/3/46
Fractal Fract, Vol. 4, Pages 45: Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus https://www.mdpi.com/2504-3110/4/3/45 Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper. Fractal Fract, Vol. 4, Pages 45: Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus

Fractal and Fractional doi: 10.3390/fractalfract4030045

Authors: Arran Fernandez Iftikhar Husain

Mittag-Leffler functions and their variations are a popular topic of study at the present time, mostly due to their applications in fractional calculus and fractional differential equations. Here we propose a modification of the usual Mittag-Leffler functions of one, two, or three parameters, which is ideally suited for extending certain fractional-calculus operators into the complex plane. Complex analysis has been underused in combination with fractional calculus, especially with newly developed operators like those with Mittag-Leffler kernels. Here we show the natural analytic continuations of these operators using the modified Mittag-Leffler functions defined in this paper.

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Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus Arran Fernandez Iftikhar Husain doi: 10.3390/fractalfract4030045 Fractal and Fractional 2020-09-12 Fractal and Fractional 2020-09-12 4 3 Article 45 10.3390/fractalfract4030045 https://www.mdpi.com/2504-3110/4/3/45
Fractal Fract, Vol. 4, Pages 44: Fractional SIS Epidemic Models https://www.mdpi.com/2504-3110/4/3/44 In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (&amp;alpha;-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order &amp;alpha; converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the &amp;alpha;-SIS models. Fractal Fract, Vol. 4, Pages 44: Fractional SIS Epidemic Models

Fractal and Fractional doi: 10.3390/fractalfract4030044

Authors: Caterina Balzotti Mirko D’Ovidio Paola Loreti

In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (&amp;alpha;-SIS model) in the case of constant population size. We provide a representation of the explicit solution to the fractional model and we illustrate the results by numerical schemes. A comparison with the limit case when the fractional order &amp;alpha; converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the &amp;alpha;-SIS models.

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Fractional SIS Epidemic Models Caterina Balzotti Mirko D’Ovidio Paola Loreti doi: 10.3390/fractalfract4030044 Fractal and Fractional 2020-08-31 Fractal and Fractional 2020-08-31 4 3 Article 44 10.3390/fractalfract4030044 https://www.mdpi.com/2504-3110/4/3/44
Fractal Fract, Vol. 4, Pages 43: Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact https://www.mdpi.com/2504-3110/4/3/43 The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop a new stable financial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with fixed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system&amp;rsquo;s actual macroeconomic behavior. Specifically component of its application to the large scale and smaller scale forms, just as the utilization of specific strategies and instruments such fractal stochastic procedures and expectation. Fractal Fract, Vol. 4, Pages 43: Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact

Fractal and Fractional doi: 10.3390/fractalfract4030043

The main objective of this paper is to construct and test fractional order derivatives for the management and simulation of a fractional order disorderly finance system. In the developed system, we add the critical minimum interest rate d parameter in order to develop a new stable financial model. The new emerging paradigm increases the demand for innovation, which is the gateway to the knowledge economy. The derivatives are characterized in the Caputo fractional order derivative and Atangana-Baleanu derivative. We prove the existence and uniqueness of the solutions with fixed point theorem and an iterative scheme. The interest rate begins to rise according to initial conditions as investment demand and price exponent begin to fall, which shows the financial system&amp;rsquo;s actual macroeconomic behavior. Specifically component of its application to the large scale and smaller scale forms, just as the utilization of specific strategies and instruments such fractal stochastic procedures and expectation.

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Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact Muhammad Farman Ali Akgül Dumitru Baleanu Sumaiyah Imtiaz Aqeel Ahmad doi: 10.3390/fractalfract4030043 Fractal and Fractional 2020-08-21 Fractal and Fractional 2020-08-21 4 3 Article 43 10.3390/fractalfract4030043 https://www.mdpi.com/2504-3110/4/3/43
Fractal Fract, Vol. 4, Pages 42: Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators https://www.mdpi.com/2504-3110/4/3/42 The approach based on fractional advection&amp;ndash;diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick&amp;rsquo;s law containing the Riemann&amp;ndash;Liouville fractional derivative is related to the well-known fractional Fokker&amp;ndash;Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker&amp;ndash;Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed. Fractal Fract, Vol. 4, Pages 42: Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators

Fractal and Fractional doi: 10.3390/fractalfract4030042

Authors: Renat T. Sibatov HongGuang Sun

The approach based on fractional advection&amp;ndash;diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick&amp;rsquo;s law containing the Riemann&amp;ndash;Liouville fractional derivative is related to the well-known fractional Fokker&amp;ndash;Planck equation, and it is consistent with the universal characteristics of dispersive transport observed in the time-of-flight experiment (ToF). In the present paper, we consider the generalized Fick laws containing other forms of fractional time operators with singular and non-singular kernels and find out features of ToF transient currents that can indicate the presence of such fractional dynamics. Solutions of the corresponding fractional Fokker&amp;ndash;Planck equations are expressed through solutions of integer-order equation in terms of an integral with the subordinating function. This representation is used to calculate the ToF transient current curves. The physical reasons leading to the considered fractional generalizations are elucidated and discussed.

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Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators Renat T. Sibatov HongGuang Sun doi: 10.3390/fractalfract4030042 Fractal and Fractional 2020-08-17 Fractal and Fractional 2020-08-17 4 3 Article 42 10.3390/fractalfract4030042 https://www.mdpi.com/2504-3110/4/3/42
Fractal Fract, Vol. 4, Pages 41: Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation https://www.mdpi.com/2504-3110/4/3/41 This manuscript focuses on the application of the (m+1/G&amp;prime;)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schr&amp;ouml;dinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted. Fractal Fract, Vol. 4, Pages 41: Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation

Fractal and Fractional doi: 10.3390/fractalfract4030041

Authors: Hulya Durur Esin Ilhan Hasan Bulut

This manuscript focuses on the application of the (m+1/G&amp;prime;)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schr&amp;ouml;dinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are derived. Various figures such as 3D and 2D surfaces with the selecting the suitable of parameter values are plotted.

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Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation Hulya Durur Esin Ilhan Hasan Bulut doi: 10.3390/fractalfract4030041 Fractal and Fractional 2020-08-16 Fractal and Fractional 2020-08-16 4 3 Article 41 10.3390/fractalfract4030041 https://www.mdpi.com/2504-3110/4/3/41
Fractal Fract, Vol. 4, Pages 40: Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive? https://www.mdpi.com/2504-3110/4/3/40 In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations. Fractal Fract, Vol. 4, Pages 40: Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?

Fractal and Fractional doi: 10.3390/fractalfract4030040

Authors: Jocelyn Sabatier

In the field of fractional calculus and applications, a current trend is to propose non-singular kernels for the definition of new fractional integration and differentiation operators. It was recently claimed that fractional-order derivatives defined by continuous (in the sense of non-singular) kernels are too restrictive. This note shows that this conclusion is wrong as it arises from considering the initial conditions incorrectly in (partial or not) fractional differential equations.

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Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive? Jocelyn Sabatier doi: 10.3390/fractalfract4030040 Fractal and Fractional 2020-08-11 Fractal and Fractional 2020-08-11 4 3 Viewpoint 40 10.3390/fractalfract4030040 https://www.mdpi.com/2504-3110/4/3/40
Fractal Fract, Vol. 4, Pages 39: Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation https://www.mdpi.com/2504-3110/4/3/39 This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm&amp;rsquo;s accuracy. Fractal Fract, Vol. 4, Pages 39: Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation

Fractal and Fractional doi: 10.3390/fractalfract4030039

Authors: Rafał Brociek Agata Chmielowska Damian Słota

This paper presents the application of the swarm intelligence algorithm for solving the inverse problem concerning the parameter identification. The paper examines the two-dimensional Riesz space fractional diffusion equation. Based on the values of the function (for the fixed points of the domain) which is the solution of the described differential equation, the order of the Riesz derivative and the diffusion coefficient are identified. The paper includes numerical examples illustrating the algorithm&amp;rsquo;s accuracy.

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Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation Rafał Brociek Agata Chmielowska Damian Słota doi: 10.3390/fractalfract4030039 Fractal and Fractional 2020-08-06 Fractal and Fractional 2020-08-06 4 3 Article 39 10.3390/fractalfract4030039 https://www.mdpi.com/2504-3110/4/3/39
Fractal Fract, Vol. 4, Pages 38: A Stochastic Fractional Calculus with Applications to Variational Principles https://www.mdpi.com/2504-3110/4/3/38 We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler&amp;ndash;Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation. Fractal Fract, Vol. 4, Pages 38: A Stochastic Fractional Calculus with Applications to Variational Principles

Fractal and Fractional doi: 10.3390/fractalfract4030038

Authors: Houssine Zine Delfim F. M. Torres

We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler&amp;ndash;Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.

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A Stochastic Fractional Calculus with Applications to Variational Principles Houssine Zine Delfim F. M. Torres doi: 10.3390/fractalfract4030038 Fractal and Fractional 2020-08-01 Fractal and Fractional 2020-08-01 4 3 Article 38 10.3390/fractalfract4030038 https://www.mdpi.com/2504-3110/4/3/38
Fractal Fract, Vol. 4, Pages 37: Design of Cascaded and Shifted Fractional-Order Lead Compensators for Plants with Monotonically Increasing Lags https://www.mdpi.com/2504-3110/4/3/37 This paper concerns cascaded, shifted, fractional-order, lead compensators made by the serial connection of two stages introducing their respective phase leads in shifted adjacent frequency ranges. Adding up leads in these intervals gives a flat phase in a wide frequency range. Moreover, the simple elements of the cascade can be easily realized by rational transfer functions. On this basis, a method is proposed in order to design a robust controller for a class of benchmark plants that are difficult to compensate due to monotonically increasing lags. The simulation experiments show the efficiency, performance and robustness of the approach. Fractal Fract, Vol. 4, Pages 37: Design of Cascaded and Shifted Fractional-Order Lead Compensators for Plants with Monotonically Increasing Lags

Fractal and Fractional doi: 10.3390/fractalfract4030037

Authors: Guido Maione

This paper concerns cascaded, shifted, fractional-order, lead compensators made by the serial connection of two stages introducing their respective phase leads in shifted adjacent frequency ranges. Adding up leads in these intervals gives a flat phase in a wide frequency range. Moreover, the simple elements of the cascade can be easily realized by rational transfer functions. On this basis, a method is proposed in order to design a robust controller for a class of benchmark plants that are difficult to compensate due to monotonically increasing lags. The simulation experiments show the efficiency, performance and robustness of the approach.

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Design of Cascaded and Shifted Fractional-Order Lead Compensators for Plants with Monotonically Increasing Lags Guido Maione doi: 10.3390/fractalfract4030037 Fractal and Fractional 2020-07-27 Fractal and Fractional 2020-07-27 4 3 Article 37 10.3390/fractalfract4030037 https://www.mdpi.com/2504-3110/4/3/37