A simple numerical method for two-dimensional nonlinear fractional anomalous sub-diffusion equations

摘要

Recently, many numerical techniques were presented to solve the fractional anomalous sub-diffusion equations, and the results were excellent. In this paper, we study a simple numerical technique to solve two important types of fractional anomalous sub-diffusion equations that appear strongly in chemical reactions and spiny neuronal dendrites, which are the two-dimensional fractional Cable equation and the two-dimensional fractional reaction sub-diffusion equation. The proposed technique is a simple one which is an extension of the weighted average finite difference technique. The stability analysis of the proposed method is studied by means of John von Neumann stability analysis technique. An accurate stability criterion which is valid for different discretization schemes of the fractional derivative and arbitrary weight factor is introduced. Four numerical examples are presented (two for the Cable equation and two for the reaction sub-diffusion equation) to demonstrate the effectiveness and the accuracy of the presented method.