Finite Integration Method via Chebyshev Polynomial Expansion for Solving 2-D Linear Time-Dependent and Linear Space-Fractional Differential Equations

Ratinan Boonklurb, Ampol Duangpan, Arnont Saengsiritongchai

Abstract

In this paper, we modify the finite integration method (FIM) using Chebyshev polynomial, to construct a numerical algorithm for finding approximate solutions of two-dimensional linear time-dependent differential equations. Comparing with the traditional FIMs using trapezoidal and Simpson’s rules, the numerical results demonstrate that our proposed algorithm give a better accuracy even for a large time step. In addition, we also devise a numerical algorithm based on the idea of FIM using Chebyshev polynomial to find approximate solutions of a linear space-fractional differential equations under Riemann-Liouville definition of fractional order derivative. Several numerical examples are given and accuracy of our numerical method is demonstrated comparing to their exact solutions. It is shown that the our proposed method can give a good accuracy as high as 10^{-5} even with a few numbers of computational node.