Fractal and Fractional

In this paper, we establish the existence and uniqueness of a strong solution to a fractional mean field games system with non-separable Hamiltonians, where the fractional exponent $\sigma \in (\frac{1}{2},1)$. Our result is new for fractional mean field games with non-separable Hamiltonians, which generalizes the work of D.M. Ambrose for the integral case. The important step is to choose the new appropriate fractional order function spaces and use the Banach fixed-point theorem under stronger assumptions for the Hamiltonians. Full article

Discrimination of various microseismic (MS) events induced by blasting and mining in coal mines is significant for the evaluation and forecasting of rock bursts. In this paper, multifractal and moment tensor inversion methods were used to investigate the waveform characteristics and focal mechanisms of different MS events in a more quantitative way. The multifractal spectrum calculation results indicate that the three types of MS waveform have different distribution ranges in the multifractal parameters of ∆α and Δf(α). The results show that the blasting schemes also have a great influence on MS waveform characteristics. Consequently, the multifractal parameters of ∆α and Δf(α) can be used to discriminate different MS events. Further, the focal mechanisms of MS events were calculated by seismic moment tensor inversion. The results show that an explosion is not the dominant mechanism of deep-hole blasting MS events, and the CLVD and DC components account for an important proportion, indicating that some additional processes occur during blasting. Moreover, the coal-rock fracture MS events are characterized by compression implosion or compression/shear implosion mixed focal mechanisms, while the overburden movement MS events are tensile explosion or tensile/shear explosion mixed focal mechanisms. The focal mechanisms and nodal plane parameters have close relationships with the inducing factors and occurrence processes of MS events. Full article

By using Lucas polynomials, we define a new subclass of analytic bi-univalent functions, class $\mathsf{\Sigma }$, in the open unit disc with respect to symmetric conjugate points connected with the combination Binomial series and Babalola operator. The bounds on the initial coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for the functions in this new subclass of $\mathsf{\Sigma }$ are investigated. Moreover, we obtain an estimation for the Fekete–Szego problem for the function subclass defined in this paper. Relevant connections of these results are presented here as corollaries. Full article

In this current manuscript, we study a fractional-order modificatory hybrid optical model (FOMHO model). Experiments manifest that under appropriate parameter conditions, the fractional-order modificatory hybrid optical model will generate chaotic behavior. In order to eliminate the chaotic phenomenon of the (FOMHO model), we devise two different control techniques. First of all, a suitable delayed feedback controller is designed to control chaos in the (FOMHO model). A sufficient condition ensuring the stability and the occurrence of Hopf bifurcation of the fractional-order controlled modificatory hybrid optical model is set up. Next, a suitable delayed mixed controller which includes state feedback and parameter perturbation is designed to suppress chaos in the (FOMHO model). A sufficient criterion guaranteeing the stability and the onset of Hopf bifurcation of the fractional-order controlled modificatory hybrid optical model is derived. In the end, software simulations are implemented to verify the accuracy of the devised controllers. The acquired results of this manuscript are completely new and have extremely vital significance in suppressing chaos in physics. Furthermore, the exploration idea can also be utilized to control chaos in many other differential chaotic dynamical models. Full article

We study, in this paper, the Cauchy problem for matrix factorizations of the Helmholtz equation in the space ${\mathbb{R}}^m$. Based on the constructed Carleman matrix, we find an explicit form of the approximate solution of this problem and prove the stability of the solutions. Full article

The functional implications of substances, such as retardation and relaxation, can be studied for magnetized diffusion coefficient based on the relative increase throughout magnetization is a well-known realization. In this context, we have explored the Oldroyd-B hybrid nanofluid flowing through a pored oscillating plate along with an inclined applied magnetics effect. The slipping effect and sinusoidal heating conditions are also supposed to be under consideration. An innovative and current classification of fractional derivatives, i.e., Prabhakar fractional derivative and Laplace transform, are implemented for the result of transformed leading equations. The graphical representation is also described to understand the physical implementation of all effecting parameters. In order to justify and physically examine the considered problem, some limiting cases, the rate of heat and mass transfer, and friction factors are also analyzed. As a result, we have concluded that the thermal enhancement can be improved more progressively with the interaction of silver-water-based nanofluid suspension compared to copper-nanoparticles mixed nanofluid. Furthermore, It has examined the impact of both parameters, i.e., time relaxation and retardation is opposite of the momentum field. Full article

In this work we study variational problems, where ordinary derivatives are replaced by a generalized proportional fractional derivative. This fractional operator depends on a fixed parameter, acting as a weight over the state function and its first-order derivative. We consider the problem with and without boundary conditions, and with additional restrictions like isoperimetric and holonomic. Herglotz’s variational problem and when in presence of time delays are also considered. Full article

A framework of distributed interval observers is introduced for fractional-order multiagent systems in the presence of nonlinearity. First, a frame was designed to construct the upper and lower bounds of the system state. By using monotone system theory, the positivity of the error dynamics could be ensured, which implies that the bounds could trap the original state. Second, a sufficient condition was applied to guarantee the boundedness of distributed interval observers. Then, an extension of Lyapunov function in the fractional calculus field was the basis of the sufficient condition. An algorithm associated with the procedure of the observer design is also provided. Lastly, a numerical simulation is used to demonstrate the effectiveness of the distributed interval observer. Full article

We used the concept of quantum calculus (Jackson’s calculus) in a recent note to develop an extended class of multivalent functions on the open unit disk. Convexity and star-likeness properties are obtained by establishing conditions for this class. The most common inequalities of the proposed functions are geometrically investigated. Our approach was influenced by the theory of differential subordination. As a result, we called attention to a few well-known corollaries of our main conclusions. Full article

Fractal characteristics and the fractal dimension are widely used in the description and characterization of rock fracture networks. They are important tools for coal mining, oil and gas transportation, and other engineering problems. However, due to the complexity of rock fracture networks and the difficulty in directly applying the limit definition of the fractal dimension, the definition and application of the fractal dimension have become hot topics in related projects. In this paper, the traditional fractal calculation methods were reviewed. Using the traditional fractal theory and the head/tail breaks method, a new fractal dimension quantization model was established as a simple method of fractal calculation. This simple method of fractal calculation was used to calculate the fractal dimensions of three rock fracture networks. Through comparison with the box-counting dimension calculation results, it was verified that the model could calculate the fractal dimension of the fracture length of rock fracture networks, as well as quantify it accurately and effectively. In addition, we found a number of similarities between rock fracture networks and urban road traffic networks in GIS. The application of the space syntax metric to rock fracture networks prevents controversy with respect to the definition of the axis and it showed a good effect. Using the space syntax metric as a parameter can better reflect the space relationship of rock fractures than length. Through the calculation of the fractal dimension of the connection value and control value, it was found that the trend of the length fractal dimension was the same as that of the control value, whereas the fractal dimension of the connection value was the opposite. This further verifies the applicability of the space syntax metric in rock fracture networks. Full article

By developing the direct method of moving planes, we study the radial symmetry of nonnegative solutions for a fractional Laplacian system with different negative powers: ${(-\Delta )}^{\frac{\alpha }{2}}u\left(x\right)+u^{-\gamma }\left(x\right)+v^{-q}\left(x\right)=0,x\in R^N,$ ${(-\Delta )}^{\frac{\beta }{2}}v\left(x\right)+v^{-\sigma }\left(x\right)+u^{-p}\left(x\right)=0,x\in R^N,$ $u\left(x\right)\gtrsim {\left|x\right|}^a,v\left(x\right)\gtrsim {\left|x\right|}^b\mathrm{as}\left|x\right|\to \infty ,$ where $\alpha ,\beta \in (0,2)$, and $a,b>0$ are constants. We study the decay at infinity and narrow region principle for the fractional Laplacian system with different negative powers. The same results hold for nonlinear Hénon-type fractional Laplacian systems with different negative powers. Full article

The revised pore–solid fractal (PSF) model is presented by using the largest inscribed circle-based geometries of squares or cubes to replace the original pore or solid subregions as the new pore or solid phase in porous media. The revised PSF model changes the discrete lacunar pore and solid phases in the original PSF model to integrated. Permeability is an intrinsic property of geomaterials and has broad applications in exploring fluid flow and species transport. Based on the revised PSF model and critical path analysis, a fractal model for predicting the permeability of saturated geomaterials is proposed. The permeability prediction model is verified by comparison with the existing predicted model and the laboratory testing. The results show that the predicted permeabilities match the measured values very well. This work provides a theoretical framework for the revised PSF model and its application in predicting the permeability of geomaterials. Full article

This paper emphasized on studying the asymptotic synchronization and finite synchronization of fractional-order memristor-based inertial neural networks with time-varying latency. The fractional-order memristor-based inertial neural network model is offered as a more general and flexible alternative to the integer-order inertial neural network. By utilizing the properties of fractional calculus, two lemmas on asymptotic stability and finite-time stability are provided. Based on the two lemmas and the constructed Lyapunov functionals, some updated and valid criteria have been developed to achieve asymptotic and finite-time synchronization of the addressed systems. Finally, the effectiveness of the proposed method is demonstrated by a number of examples and simulations. Full article

This paper aims to numerically study the time-fractional Allen-Cahn equation, where the time-fractional derivative is in the sense of Caputo with order $\alpha \in (0,1)$. Considering the weak singularity of the solution $u(\mathbf{x},t)$ at the starting time, i.e., its first and/or second derivatives with respect to time blowing-up as $t\to 0^{+}$ albeit the function itself being right continuous at $t=0$, two well-known difference formulas, including the nonuniform L1 formula and the nonuniform L2-$1_{\sigma }$ formula, which are used to approximate the Caputo time-fractional derivative, respectively, and the local discontinuous Galerkin (LDG) method is applied to discretize the spatial derivative. With the help of discrete fractional Gronwall-type inequalities, the stability and optimal error estimates of the fully discrete numerical schemes are demonstrated. Numerical experiments are presented to validate the theoretical results. Full article

In recent decades, fractional order calculus has become an important mathematical tool for effectively solving complex problems through better modeling with the introduction of fractional differential/integral operators; fractional order swarming heuristics are also introduced and applied for better performance in different optimization tasks. This study investigates the nonlinear system identification problem of the input nonlinear control autoregressive (IN-CAR) model through the novel implementation of fractional order particle swarm optimization (FO-PSO) heuristics; further, the key term separation technique (KTST) is introduced in the FO-PSO to solve the over-parameterization issue involved in the parameter estimation of the IN-CAR model. The proposed KTST-based FO-PSO, i.e., KTST-FOPSO accurately estimates the parameters of an unknown IN-CAR system with robust performance in cases of different noise scenarios. The performance of the KTST-FOPSO is investigated exhaustively for different fractional orders as well as in comparison with the standard counterpart. The results of statistical indices through Monte Carlo simulations endorse the reliability and stability of the KTST-FOPSO for IN-CAR identification. Full article

Particle breakage was reported to have great influence on the mechanical property of granular materials. However, limited studies were conducted to quantify the detailed effects of relative density and initial grading on the particle breakage behaviour of granular materials under different confining pressures. In this study, a series of monotonic drained triaxial tests were performed on isotropically consolidated granular materials with four different initial gradings and relative densities. It is observed that particle breakage increases as the confining pressure or relative density increases, whereas it decreases with the increasing coefficient of uniformity. Due to particle breakage, the grading curves of granular materials after triaxial tests can be simulated by a power-law function with fractal dimension. As the confining pressure increases, the fractal dimension approaches the limit of granular materials, i.e., 2.6. A unique normalized relation between the particle breakage extent and confining pressure by considering relative density and grading index was found. Full article

The infection dynamics of COVID-19 is difficult to contain due to the mutation nature of the SARS-CoV-2 virus. This has been a public health concern globally with the impact of the pandemic on the world’s economy and mode of living. In the present work, we formulate and examine a fractional model of COVID-19 considering the two variants of concern on the disease transmission pathways, namely SARS-CoV-2 and D614G on our model formulation. The existence and uniqueness of our model solutions were analyzed using the fixed point theory. Mathematical analyses were presented, and the model’s basic reproduction numbers ${\mathcal{R}}_{01}$ and ${\mathcal{R}}_{02}$ were determined. The model has three equilibria: the disease-free equilibrium, that endemic for strain 1, and that endemic for strain 2. The locally asymptotic stability of the equilibria was established based on the ${\mathcal{R}}_{01}$ and ${\mathcal{R}}_{02}$ values. Caputo fractional operator was used to simulate the model to study the dynamics of the model solution. Results from numerical simulations envisaged that an increase in the transmission parameters of strain 1 leads to an increase in the number of infected individuals. On the other hand, an increase in the strain 2 transmission rate gives rise to more infection. Furthermore, it was established that there is an increased number of infections with a negative impact of strain 1 on strain 2 dynamics and vice versa. Full article

Slope entropy (SlEn) is a time series complexity indicator proposed in recent years, which has shown excellent performance in the fields of medical and hydroacoustics. In order to improve the ability of SlEn to distinguish different types of signals and solve the problem of two threshold parameters selection, a new time series complexity indicator on the basis of SlEn is proposed by introducing fractional calculus and combining particle swarm optimization (PSO), named PSO fractional SlEn (PSO-FrSlEn). Then we apply PSO-FrSlEn to the field of fault diagnosis and propose a single feature extraction method and a double feature extraction method for rolling bearing fault based on PSO-FrSlEn. The experimental results illustrated that only PSO-FrSlEn can classify 10 kinds of bearing signals with 100% classification accuracy by using double features, which is at least 4% higher than the classification accuracies of the other four fractional entropies. Full article

In this paper, we reconstruct the time-dependent volatility function of the underlying asset and the mean-reverting parameter $\gamma$ of the interest rate for European options under the fractional Chan–Karolyi–Longstaff–Sanders (CKLS) stochastic interest rate model. Tikhonov regularization is used to solve the ill-posedness of the inverse problem. The existence and stability of the solution of the regularization problem are given. We employ the alternating direction method of multipliers (ADMM) to iteratively optimize the volatility function and the parameter $\gamma$. Finally, numerical simulations and the empirical analysis are presented to illustrate the efficiency of the proposed method. Full article