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

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.