Algorithm solution for space-fractional diffusion equations

A Sunarto,J Sulaiman

doi:10.1088/1757-899x/725/1/012086

Abstract

In this study, we propose approximate algorithm solution of the space-fractional diffusion equation (SFDE’s) based on a quarter-sweep (QS) implicit finite difference approximation equation. To derive this approximation equation, the Caputo’s space-fractional derivative has been used to discretize the proposed problems. By using the Caputo’s finite difference approximation equation, a linear system will be generated and solved iteratively. In addition to that, formulation and implementation algorithm the Quarter-Sweep AOR (QSAOR) iterative method are also presented. Based on numerical results of the proposed iterative method, it can be concluded that the proposed iterative method is superior to the FSAOR and HSAOR iterative method.

Highlights

In this paper we focus on numerical solution for one-dimensional SFDE’s
We propose approximate algorithm solution of the spacefractional diffusion equation (SFDE’s) based on a quarter-sweep (QS) implicit finite difference approximation equation
We describe some necessary definitions and mathematical preliminaries of the fractional derivative theory which are required for our subsequent development of the approximation equation for the problem in Eq(1)

Summary

Introduction

We describe some necessary definitions and mathematical preliminaries of the fractional derivative theory which are required for our subsequent development of the approximation equation for the problem in Eq(1). Definition 1.[1,2] The Riemann-Liouville fractional integral operator , Jβ of order- β is defined as. Definition 2.[2, 3] The Caputo’s fractional partial derivative operator, Dβ of order - β is defined as. We discretized SFDE’s equation using implicit finite difference scheme with Caputo’s derivative operator in order to examine the implementation of QSAOR iteration method in solving the resultant linear system of equations. Method known as the FSAOR iterative method and HSAOR is implemented as control method in order to investigate the performance of QSAOR iterative method

Quarter-Sweep Caputo’s Implicit Finite Difference Approximation Equations

Formulation of QSAOR Iterative Method

Numerical Experiments

Conclusion