International Journal of Mathematics and Mathematical Sciences
Volume 20 (1997), Issue 1, Pages 147-163
doi:10.1155/S0161171297000215

Finite difference approximations for a class of non-local parabolic equations

Yanping Lin,1 Shuzhan Xu,1 and Hong-Ming Yin2

1Department of Mathematical Sciences, University of Alberta Edmonton, Alberta T6G 2G1, Canada
2Department of Mathematics, University of Notre Dame, Notre Dame 46556-0398, Indiana, USA

Received 4 October 1994

Copyright © 1997 Yanping Lin et al. 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

In this paper we study finite difference procedures for a class of parabolic equations with non-local boundary condition. The semi-implicit and fully implicit backward Euler schemes are studied. It is proved that both schemes preserve the maximum principle and monotonicity of the solution of the original equation, and fully-implicit scheme also possesses strict monotonicity. It is also proved that finite difference solutions approach to zero as t→∞ exponentially. The numerical results of some examples are presented, which support our theoretical justifications.