Finite Element Solutions for the Space Fractional Diffusion Equation with a Nonlinear Source Term

Y J Choi,S K Chung

doi:10.1155/2012/596184

Abstract

We consider finite element Galerkin solutions for the space fractional diffusion equation with a nonlinear source term. Existence, stability, and order of convergence of approximate solutions for the backward Euler fully discrete scheme have been discussed as well as for the semidiscrete scheme. The analytical convergent orders are obtained as , where is a constant depending on the order of fractional derivative. Numerical computations are presented, which confirm the theoretical results when the equation has a linear source term. When the equation has a nonlinear source term, numerical results show that the diffusivity depends on the order of fractional derivative as we expect.

Highlights

Fractional calculus is an old mathematical topic but it has not been attracted enough for almost three hundred years
Galerkin finite element methods are considered for the space fractional diffusion equation with a nonlinear source term
We have derived the variational formula of the semidiscrete scheme by using the Galerkin finite element method in space

Summary

Introduction

Fractional calculus is an old mathematical topic but it has not been attracted enough for almost three hundred years. In this paper we discuss Galerkin approximate solutions for the space fractional diffusion equation with a nonlinear source term. Finite difference methods have been studied in 14–16 for linear space fractional diffusion problems. They used the right-shifted Gruwald-Letnikov approximate for the fractional derivative since the standard Gruwald-Letnikov approximate gives the unconditional instability even for the implicit method. For the space fractional diffusion problems with a nonlinear source term, Lynch et al 17 used the so-called L2 and L2C methods in 6 and compared computational accuracy of them. Choi et al 18 have shown existence and stability of numerical solutions of an implicit finite difference equation obtained by using the right-shifted Gruwald-Letnikov approximation. We will see that numerical solutions of fractional diffusion equations diffuse more slowly than that of the classical diffusion problem and diffusivity depends on the order of fractional derivatives

The Variational Form

The Semidiscrete Variational Form

The Fully Discrete Variational Form

Numerical Experiments

Concluding Remarks