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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 596184, 25 pages
http://dx.doi.org/10.1155/2012/596184
Research Article

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

Department of Mathematics Education, Seoul National University, Seoul 151-748, Republic of Korea

Received 26 April 2012; Revised 6 July 2012; Accepted 17 July 2012

Academic Editor: Bashir Ahmad

Copyright © 2012 Y. J. Choi and S. K. Chung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑂 ( 𝑘 + ̃ 𝛾 ) , where ̃ 𝛾 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.