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Fractal Fract
Volume 4
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Fractal Fract., Volume 4, Issue 3 (September 2020) – 18 articles

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6 pages, 580 KiB  
Article
Integral Representation of Fractional Derivative of Delta Function
by Ming Li
Fractal Fract. 2020, 4(3), 47; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030047 - 20 Sep 2020
Cited by 8 | Viewed by 2387
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
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15 pages, 6290 KiB  
Article
Fractal and Fractional Derivative Modelling of Material Phase Change
by Harry Esmonde
Fractal Fract. 2020, 4(3), 46; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030046 - 14 Sep 2020
Cited by 8 | Viewed by 2087
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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15 pages, 296 KiB  
Article
Modified Mittag-Leffler Functions with Applications in Complex Formulae for Fractional Calculus
by Arran Fernandez and Iftikhar Husain
Fractal Fract. 2020, 4(3), 45; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030045 - 12 Sep 2020
Cited by 13 | Viewed by 2520
Abstract
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, [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
18 pages, 554 KiB  
Article
Fractional SIS Epidemic Models
by Caterina Balzotti, Mirko D’Ovidio and Paola Loreti
Fractal Fract. 2020, 4(3), 44; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030044 - 31 Aug 2020
Cited by 18 | Viewed by 3044
Abstract
In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-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 [...] Read more.
In this paper, we consider the fractional SIS (susceptible-infectious-susceptible) epidemic model (α-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 α converges to 1 (the SIS model) is also given. We analyze the effects of the fractional derivatives by comparing the SIS and the α-SIS models. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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19 pages, 609 KiB  
Article
Analysis of Fractional Order Chaotic Financial Model with Minimum Interest Rate Impact
by Muhammad Farman, Ali Akgül, Dumitru Baleanu, Sumaiyah Imtiaz and Aqeel Ahmad
Fractal Fract. 2020, 4(3), 43; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030043 - 21 Aug 2020
Cited by 31 | Viewed by 2760
Abstract
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 [...] Read more.
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’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. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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11 pages, 349 KiB  
Article
Dispersive Transport Described by the Generalized Fick Law with Different Fractional Operators
by Renat T. Sibatov and HongGuang Sun
Fractal Fract. 2020, 4(3), 42; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030042 - 17 Aug 2020
Cited by 4 | Viewed by 2375
Abstract
The approach based on fractional advection–diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick’s law containing the Riemann–Liouville fractional derivative is related to the well-known fractional Fokker–Planck equation, [...] Read more.
The approach based on fractional advection–diffusion equations provides an effective and meaningful tool to describe the dispersive transport of charge carriers in disordered semiconductors. A fractional generalization of Fick’s law containing the Riemann–Liouville fractional derivative is related to the well-known fractional Fokker–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–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. Full article
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8 pages, 2887 KiB  
Article
Novel Complex Wave Solutions of the (2+1)-Dimensional Hyperbolic Nonlinear Schrödinger Equation
by Hulya Durur, Esin Ilhan and Hasan Bulut
Fractal Fract. 2020, 4(3), 41; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030041 - 16 Aug 2020
Cited by 57 | Viewed by 3397
Abstract
This manuscript focuses on the application of the (m+1/G′)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation. With the help of projected method, the periodic and singular complex wave solutions to the considered model are [...] Read more.
This manuscript focuses on the application of the (m+1/G′)-expansion method to the (2+1)-dimensional hyperbolic nonlinear Schrödinger 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. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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5 pages, 218 KiB  
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Fractional-Order Derivatives Defined by Continuous Kernels: Are They Really Too Restrictive?
by Jocelyn Sabatier
Fractal Fract. 2020, 4(3), 40; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030040 - 11 Aug 2020
Cited by 22 | Viewed by 2967
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue 2020 Selected Papers from Fractal Fract’s Editorial Board Members)
10 pages, 391 KiB  
Article
Parameter Identification in the Two-Dimensional Riesz Space Fractional Diffusion Equation
by Rafał Brociek, Agata Chmielowska and Damian Słota
Fractal Fract. 2020, 4(3), 39; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030039 - 06 Aug 2020
Cited by 5 | Viewed by 1870
Abstract
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) [...] Read more.
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’s accuracy. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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10 pages, 294 KiB  
Article
A Stochastic Fractional Calculus with Applications to Variational Principles
by Houssine Zine and Delfim F. M. Torres
Fractal Fract. 2020, 4(3), 38; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030038 - 01 Aug 2020
Cited by 9 | Viewed by 2574
Abstract
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–Lagrange equation is obtained, extending those available in the literature for the classical, [...] Read more.
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–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. Full article
(This article belongs to the Special Issue Fractional Calculus and Special Functions with Applications)
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20 pages, 947 KiB  
Article
Design of Cascaded and Shifted Fractional-Order Lead Compensators for Plants with Monotonically Increasing Lags
by Guido Maione
Fractal Fract. 2020, 4(3), 37; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030037 - 27 Jul 2020
Cited by 6 | Viewed by 2135
Abstract
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Fractional Calculus in Control and Modelling)
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15 pages, 474 KiB  
Article
Mathematical Description of the Groundwater Flow and that of the Impurity Spread, which Use Temporal Caputo or Riemann–Liouville Fractional Partial Derivatives, Is Non-Objective
by Agneta M. Balint and Stefan Balint
Fractal Fract. 2020, 4(3), 36; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030036 - 21 Jul 2020
Cited by 13 | Viewed by 2540
Abstract
In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of [...] Read more.
In this paper, it is shown that the mathematical description of the bulk fluid flow and that of content impurity spread, which uses temporal Caputo or temporal Riemann–Liouville fractional order partial derivatives, having integral representation on a finite interval, in the case of a horizontal unconfined aquifer is non-objective. The basic idea is that different observers using this type of description obtain different results which cannot be reconciled, in other words, transformed into each other using only formulas that link the numbers representing a moment in time for two different choices from the origin of time measurement. This is not an academic curiosity; it is rather a problem to find which one of the obtained results is correct. Full article
(This article belongs to the Special Issue Fractional Behavior in Nature 2019)
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22 pages, 904 KiB  
Article
Stability Analysis and Numerical Computation of the Fractional Predator–Prey Model with the Harvesting Rate
by Mehmet Yavuz and Ndolane Sene
Fractal Fract. 2020, 4(3), 35; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030035 - 16 Jul 2020
Cited by 109 | Viewed by 4945
Abstract
In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral [...] Read more.
In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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11 pages, 1131 KiB  
Article
Hardware Implementation and Performance Study of Analog PIλDμ Controllers on DC Motor
by Dina A. John, Saket Sehgal and Karabi Biswas
Fractal Fract. 2020, 4(3), 34; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030034 - 15 Jul 2020
Cited by 5 | Viewed by 2802
Abstract
In this paper, the performance of an analog PI λ D μ controller is done for speed regulation of a DC motor. The circuits for the fractional integrator and differentiator of PI λ D μ controller are designed by optimal pole-zero interlacing algorithm. [...] Read more.
In this paper, the performance of an analog PI λ D μ controller is done for speed regulation of a DC motor. The circuits for the fractional integrator and differentiator of PI λ D μ controller are designed by optimal pole-zero interlacing algorithm. The performance of the controller is compared with another PI λ D μ controller—in which the fractional integrator circuit employs a solid-state fractional capacitor. It can be verified from the results that using PI λ D μ controllers, the speed response of the DC motor has improved with reduction in settling time ( T s ), steady state error (SS error) and % overshoot (% M p ). Full article
(This article belongs to the Special Issue Fractional Calculus in Control and Modelling)
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11 pages, 279 KiB  
Article
On the Volterra-Type Fractional Integro-Differential Equations Pertaining to Special Functions
by Yudhveer Singh, Vinod Gill, Jagdev Singh, Devendra Kumar and Kottakkaran Sooppy Nisar
Fractal Fract. 2020, 4(3), 33; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030033 - 09 Jul 2020
Cited by 2 | Viewed by 2155
Abstract
In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce [...] Read more.
In this article, we apply an integral transform-based technique to solve the fractional order Volterra-type integro-differential equation (FVIDE) involving the generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function in terms of several complex variables in the kernel. We also investigate and introduce the Elazki transform of Hilfer-derivative, generalized Lorenzo-Hartely function and generalized Lauricella confluent hypergeometric function. In this article, we have established three results that are present in the form of lemmas, which give us new results on the above mentioned three functions, and by using these results we have derived our main results that are given in the form of theorems. Our main results are very general in nature, which gives us some new and known results as a particular case of results established here. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
18 pages, 439 KiB  
Article
Transition from Diffusion to Wave Propagation in Fractional Jeffreys-Type Heat Conduction Equation
by Emilia Bazhlekova and Ivan Bazhlekov
Fractal Fract. 2020, 4(3), 32; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030032 - 08 Jul 2020
Cited by 10 | Viewed by 2114
Abstract
The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized [...] Read more.
The heat conduction equation with a fractional Jeffreys-type constitutive law is studied. Depending on the value of a characteristic parameter, two fundamentally different types of behavior are established: diffusion regime and propagation regime. In the first case, the considered equation is a generalized diffusion equation, while in the second it is a generalized wave equation. The corresponding memory kernels are expressed in both cases in terms of Mittag–Leffler functions. Explicit representations for the one-dimensional fundamental solution and the mean squared displacement are provided and analyzed analytically and numerically. The one-dimensional fundamental solution is shown to be a spatial probability density function evolving in time, which is unimodal in the diffusion regime and bimodal in the propagation regime. The multi-dimensional fundamental solutions are probability densities only in the diffusion case, while in the propagation case they can have negative values. In addition, two different types of subordination principles are formulated for the two regimes. The Bernstein functions technique is extensively employed in the theoretical proofs. Full article
(This article belongs to the Special Issue The Craft of Fractional Modelling in Science and Engineering III)
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19 pages, 364 KiB  
Article
Existence Results for Fractional Order Single-Valued and Multi-Valued Problems with Integro-Multistrip-Multipoint Boundary Conditions
by Sotiris K. Ntouyas, Bashir Ahmad and Ahmed Alsaedi
Fractal Fract. 2020, 4(3), 31; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030031 - 05 Jul 2020
Cited by 4 | Viewed by 1767
Abstract
We study the existence of solutions for a new class of boundary value problems of arbitrary order fractional differential equations and inclusions, supplemented with integro-multistrip-multipoint boundary conditions. Suitable fixed point theorems are applied to prove some new existence results. The inclusion problem is [...] Read more.
We study the existence of solutions for a new class of boundary value problems of arbitrary order fractional differential equations and inclusions, supplemented with integro-multistrip-multipoint boundary conditions. Suitable fixed point theorems are applied to prove some new existence results. The inclusion problem is discussed for convex valued as well as non-convex valued multi-valued map. Examples are also constructed to illustrate the main results. The results presented in this paper are not only new in the given configuration but also provide some interesting special cases. Full article
9 pages, 307 KiB  
Article
Laplace Transform Method for Economic Models with Constant Proportional Caputo Derivative
by Esra Karatas Akgül, Ali Akgül and Dumitru Baleanu
Fractal Fract. 2020, 4(3), 30; https://0-doi-org.brum.beds.ac.uk/10.3390/fractalfract4030030 - 03 Jul 2020
Cited by 33 | Viewed by 4011
Abstract
In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag–Leffler [...] Read more.
In this study, we solved the economic models based on market equilibrium with constant proportional Caputo derivative using the Laplace transform. We proved the accuracy and efficiency of the method. We constructed the relations between the solutions of the problems and bivariate Mittag–Leffler functions. Full article
(This article belongs to the Special Issue New Challenges Arising in Engineering Problems with Fractional and Integer Order)
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