Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2003 / Article

Open Access

Volume 2003 |Article ID 129473 | https://doi.org/10.1155/S1024123X03111015

Mehdi Dehghan, "On the numerical solution of the diffusion equation with a nonlocal boundary condition", Mathematical Problems in Engineering, vol. 2003, Article ID 129473, 12 pages, 2003. https://doi.org/10.1155/S1024123X03111015

On the numerical solution of the diffusion equation with a nonlocal boundary condition

Received15 Nov 2001

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.

Copyright © 2003 Hindawi Publishing Corporation. 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.


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