Numerical Investigation of Space Fractional Order Diffusion Equation by the Chebyshev Collocation Method of the Fourth Kind and Compact Finite Difference Scheme

Abstract

This paper develops a numerical scheme for finding the approximate solution of space fractional order of the diffusion equation (SFODE). Firstly, the compact finite difference (CFD) with convergence order O(delta tau 2) is used for discretizing time derivative. Afterwards, the spatial fractional derivative is approximated by the Chebyshev collocation method of the fourth kind. Furthermore, time-discrete stability and convergence analysis are presented. Finally, two examples are numerically investigated by the proposed method. The examples illustrate the performance and accuracy of our method compared to existing methods presented in the literature. 1. Introduction. One of the issues which have garnered researchers' attention these days is the fractional differential equations (FDEs) and have been numerically investigated by a huge number of authors [2, 3, 8, 9, 16, 21, 23, 25, 28, 29]. Fractional calculus is involved in many applications of science and engineering such as economics, physics, optimal control, and other applications, see [10, 11, 13, 19, 22, 26, 33, 34, 35]. A case in point is the diffusion and reaction-diffusion models in