Mathematical Problems in Engineering
Volume 2003 (2003), Issue 2, Pages 81-92
doi:10.1155/S1024123X03111015
On the numerical solution of the diffusion equation with a
nonlocal boundary condition
Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, Tehran 15914, Iran
Received 15 November 2001
Copyright © 2003 Mehdi Dehghan. 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
Parabolic partial differential equations with
nonlocal boundary specifications feature in the mathematical
modeling of many phenomena. In this paper, numerical schemes are
developed for obtaining approximate solutions to the initial
boundary value problem for one-dimensional diffusion equation
with a nonlocal constraint in place of one of the standard
boundary conditions. The method of lines (MOL) semidiscretization
approach is used to transform the model partial differential
equation into a system of first-order linear ordinary differential
equations (ODEs). The partial derivative with respect to the space
variable is approximated by a second-order finite-difference
approximation. The solution of the resulting system of first-order
ODEs satisfies a recurrence relation which involves a matrix
exponential function. Numerical techniques are developed by
approximating the exponential matrix function in this recurrence
relation. We use a partial fraction expansion to compute the
matrix exponential function via Pade approximations, which is
particularly useful in parallel processing. The algorithm is
tested on a model problem from the literature.