Matlab Code for Lyapunov Exponents of Fractional-Order Systems, Part II: The Noncommensurate Case

In this paper, the Benettin–Wolf algorithm for determining all Lyapunov exponents of noncommensurate fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. The paper continues the work started in [ Danca & Kuznetsov, 2018 ], where the Matlab code of commensurate fractional-order systems is given. To integrate the extended systems, the Adams–Bashforth–Moulton scheme for fractional differential equations is utilized. Like the Matlab program for commensurate-order systems, the program presented in this paper prints and plots all Lyapunov exponents as function of time. The program can be simply adapted to plot the evolution of the Lyapunov exponents as a function of orders, or a function of a bifurcation parameter. Special attention is paid to the periodicity of fractional-order systems and its influences. The case of noncommensurate Lorenz system is demonstrated.

An existence result for the fractional Kelvin–Voigt’s model on time-dependent cracked domains

We prove an existence result for the fractional Kelvin–Voigt’s model involving Caputo’s derivative on time-dependent cracked domains. We first show the existence of a solution to a regularized version of this problem. Then, we use a compactness argument to derive that the fractional Kelvin–Voigt’s model admits a solution which satisfies an energy-dissipation inequality. Finally, we prove that when the crack is not moving, the solution is unique.

Series solution to fractional contact problem using Caputo’s derivative

In this article, contact problem with fractional derivatives is studied. We use fractional derivative in the sense of Caputo. We deploy penalty function method to degenerate the obstacle problem into a system of fractional boundary value problems (FBVPs). The series solution of this system of FBVPs is acquired by using the variational iteration method (VIM). The performance as well as precision of the applied method is gauged by means of significant numerical tests. We further study the convergence and residual errors of the solutions by giving variation to the fractional parameter, and graphically present the solutions and residual errors accordingly. The outcomes thus obtained witness the high effectiveness of VIM for solving FBVPs.

Continuous dependence of uncertain fractional differential equations with Caputo’s derivative

Uncertain fractional differential equation driven by Liu process plays an important role in describing uncertain dynamic systems. This paper investigates the continuous dependence of solution on the parameters and initial values, respectively, for uncertain fractional differential equations involving the Caputo fractional derivative in measure sense. Several continuous dependence theorems are obtained based on uncertainty theory by employing the generalized Gronwall inequality, in which the coefficients of uncertain fractional differential equation are required to satisfy the Lipschitz conditions. Several illustrative examples are provided to verify the validity of the obtained results.

Homotopy Transforms Analysis Method for Solving Fractional Navier- Stokes Equations with Applications

The presented work includes the Homotopy Transforms of Analysis Method (HTAM). By this method, the approximate solution of nonlinear Navier- Stokes equations of fractional order derivative was obtained. The Caputo's derivative was used in the proposed method. The desired solution was calculated by using the convergent power series to the components. The obtained results are demonstrated by comparison with the results of Adomain decomposition method, Homotopy Analysis method and exact solution, as explained in examples (4.1) and (4.2). The comparison shows that the used method is powerful and efficient.

A Note on Caputo’s Derivative Operator Interpretation in Economy

We propound the economic idea in terms of fractional derivatives, which involves the modified Caputo’s fractional derivative operator. The suggested economic interpretation is based on a generalization of average count and marginal value of economic indicators. We use the concepts ofT-indicatorswhich analyses the economic performance with the presence of memory. The reaction of economic agents due to recurrence identical alteration is minimized by using the modified Caputo’s derivative operator of orderλinstead of integer order derivativen. The two sides of Caputo’s derivative are expressed by a brief time-line. The degree of attenuation is further depressed by involving the modified Caputo’s operator.

Matlab Code for Lyapunov Exponents of Fractional-Order Systems

In this paper, the Benettin–Wolf algorithm to determine all Lyapunov exponents for a class of fractional-order systems modeled by Caputo’s derivative and the corresponding Matlab code are presented. First, it is proved that the considered class of fractional-order systems admits the necessary variational system necessary to find the Lyapunov exponents. The underlying numerical method to solve the extended system of fractional order, composed of the initial value problem and the variational system, is the predictor-corrector Adams–Bashforth–Moulton for fractional differential equations. The Matlab program prints and plots the Lyapunov exponents as function of time. Also, the programs to obtain Lyapunov exponents as function of the bifurcation parameter and as function of the fractional order are described. The Matlab program for Lyapunov exponents is developed from an existing Matlab program for Lyapunov exponents of integer order. To decrease the computing time, a fast Matlab program which implements the Adams–Bashforth–Moulton method, is utilized. Four representative examples are considered.

IMPLEMENTATION OF THE HALF-SWEEP AOR ITERATIVE ALGORITHM FOR SPACE-FRACTIONAL DIFFUSION EQUATIONS

In this paper, we consider the numerical solution of one dimensional space-fractional diffusion equation. The half-sweep AOR (HSAOR) iterative method is applied to solve linear system generated from discretization of one dimensional space-fractional diffusion equation using Caputo’s derivative operator and half-sweep implicit finite difference scheme. Furthermore, the formulation and implementation of HSAOR iterative method to solve the problem are also presented. Two examples and comparisons with FSAOR iterative method are given to show the effectiveness of the proposed method. From numerical results obtained, it has shown that the HSAOR iterative method is superior as compared with the FSAOR methods.