Crank-Nicolson finite difference method for two-dimensional fractional sub-diffusion equation

Umair Ali,Farah Aini Abdullah,Ahmad Izani Ismail

doi:10.5899/2017/jiasc-00117

Abstract

A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion equation in the case where the Grunwald-Letnikov definition is used for the time-fractional derivative. The stability and convergence of the proposed Crank-Nicolson scheme are also analyzed. Finally, numerical examples are presented to test that the numerical scheme is accurate and feasible.

Highlights

Fractional calculus is essentially arbitrary order differentiation and integration
A Crank-Nicolson finite difference method is presented to solve the time fractional two-dimensional sub-diffusion equation in the case where the Grunwald-Letnikov definition is used for the time-fractional derivative
Certain phenomena and processes can best be described by the fractional diffusion equation having fractional order derivatives in time or space or space-time [5]

Summary

Introduction

Fractional calculus is essentially arbitrary order differentiation and integration. Comprehensive studies on fractional calculus and its applications can be found in [1,2,3,4]. Most papers on the numerical solution of the time fractional sub-diffusion equation have utilized the Caputo definition for the time fractional derivative [6, 7, 8, 9]. This paper discusses the use of a Crank-Nicolson scheme for solving the two-dimensional time fractional sub-diffusion equation is constructed by applying the Grunwald-Letnikov definition instead of the Caputo definition for the timefractional derivative. The Grunwald-Letnikov time fractional derivative formula is defined by [15]. The relation between Caputo and Reimann-Liouville fractional derivative is [16]: Dt1−α u(x, y, t). These two fractional derivatives are equivalent if and only if u(x, y, 0) = 0. International Scientific Publications and Consulting Services http://www.ispacs.com/journals/jiasc/2017/jiasc-00117/

Crank-Nicolson method

Stability analysis of Crank-Nicolson method

Convergence analysis of Crank-Nicolson method

Numerical Experiments

Conclusions