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

Abstract

This paper is devoted with numerical solution of the system fractional differential 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 proposed methods. Comparisons with the fractional finite difference method (FDM) and the fourth order Runge-Kutta (RK4) are given.

Numerical solution of nonlinear fractional integro-differential equation by Collocation method

“…[12] The error in approximating y(x) by the sum of its first n terms is bounded by the sum of the absolute values of all the neglected coefficients. If…”

Application of Homotopy Perturbation Sumudu Transform Method for Solving Nonlinear Fractional Heat-like Equations

“…(8) and (9) D t w(r; t) = Lw(r; t) + Nw(r; t) + q(r; t); n 1 < n; (12) subject to the initial conditions:…”

“…Most FDEs do not have exact solutions; so, approximate and numerical techniques [2][3][4] must be used. Recently, several numerical methods for solving FDEs have been given, such as homotopy perturbation method [5], Adomian decomposition method [6], and collocation method [7][8][9][10][11][12][13][14][15].…”

Block-pulse functions method for solving three-dimensional fractional Poisson type equations with Neumann boundary conditions

“…However, since the kernel of these differential equations is fractional, it is extremely difficult to obtain exact solutions. Therefore, extensive research has been performed on the development of numerical methods for fractional differential equations such as the Chebyshev collocation method [9,10], the Laplace transform method [11,12], DTM [13,14], ADM [15,16], the operational matrices method [17][18][19][20], and the wavelets method [21][22][23].…”