Fractional Differential Equations

Introduction to linear and nonlinear programming

Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet

We propose and analyze a time-stepping discontinuous Petrov--Galerkin method combined with the continuous conforming finite element method in space for the numerical solution of time-fractional subdiffusion problems. We prove the existence, uniqueness, and stability of approximate solutions and derive error estimates. To achieve high order convergence rates from the time discretizations, the time mesh is graded appropriately near $t=0$ to compensate for the singular (temporal) behavior of the exact solution near $t=0$ caused by the weakly singular kernel, but the spatial mesh is quasi uniform. In the $L_\infty((0,T);L_2(\Omega))$-norm, ($(0,T)$ is the time domain and $\Omega$ is the spatial domain); for sufficiently graded time meshes, a global convergence of order $k^{m+\alpha/2}+h^{r+1}$ is shown, where $0<\alpha<1$ is the fractional exponent, $k$ is the maximum time step, $h$ is the maximum diameter of the elements of the spatial mesh, and $m$ and $r$ are the degrees of approximate solutions in time an...

A Discontinuous Petrov--Galerkin Method for Time-Fractional Diffusion Equations

Asymptotic stability of delayed fractional-order neural networks with impulsive effects

A numerical method for solving the fractional-order logistic differential equation with two different delays (FOLE) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based upon Chebyshev approximations. The properties of Chebyshev polynomials are utilized to reduce FOLE to a system of algebraic equations. Special attention is given to study the convergence and the error estimate of the presented method. Numerical illustrations are presented to demonstrate utility of the proposed method. Chaotic behavior is observed and the smallest fractional order for the chaotic behavior is obtained. Also, FOLE is studied using variational iteration method (VIM) and the fractional complex transform is introduced to convert fractional Logistic equation to its differential partner, so that its variational iteration algorithm can be simply constructed. Numerical experiment is presented to illustrate the validity and the great potential of both proposed techniques.

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Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays

In this paper, we develop the Crank-Nicolson nite dierence method (C-N-FDM) to solve the linear time-fractional diusion equation, for- mulated with Caputo's fractional derivative. Special attention is given to study the stability of the proposed method which is introduced by means of a recently proposed procedure akin to the standard Von-Neumann stable analysis. Some numerical examples are presented and the behavior of the solution is examined to verify stability of the proposed method. It is found that the C-N-FDM is applicable, simple and ecient for such problems.

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Crank-nicolson finite difference method for solving time-fractional diffusion equation

The main goal of this paper is to develop a mathematical model for the piezoelectric elastic-semiconductor medium. The medium is homogeneous and isotropic that is exposed to photothermal excitation...

Thermal-piezoelectric problem of a semiconductor medium during photo-thermal excitation

In this paper, Generalized Mittag-Leffler function method (GMLFM) and Sumudu transform method (STM) are applied to study and solve the fractional order smoking model, where the derivatives are defined in the Caputo fractional sense. The disease free equilibrium (DFE) and stability of equilibrium point are studied. Lyapunov exponents and  Poincare map of the model are drawn to ensure the stability of the model. The obtained results show that the proposed methods are very effective and convenient. In addition, the model is presented by its signal flow graph and simulated using Matlab/Simulink.

Approximate solution for solving nonlinear fractional order smoking model

This paper is devoted with numerical solution of the system fractional diffe rential equations (FDEs) which are generated by optimization problem using the Chebyshev collocation method. The fractional derivatives are presented in terms of Caputo sense. The application of the proposed method to the generated system of FDEs leads to algebraic system which can be solved by the Newton iteration method. The method introduces a promising tool for solving many systems of non-linear FDEs. Two numerical examples are provided to confirm the accuracy and the effectiveness of the pro posed methods. Comparisons with the fractional finite difference method (FDM) and the fourth order Runge-Kutta (RK4) are given.

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Amr M. S. Mahdy

Papers

Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays

Crank-nicolson finite difference method for solving time-fractional diffusion equation

Thermal-piezoelectric problem of a semiconductor medium during photo-thermal excitation

Approximate solution for solving nonlinear fractional order smoking model

Numerical Study for the Fractional Differential Equations Generated by Optimization Problem Using Chebyshev Collocation Method and FDM