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

Mehdi Dehghan

doi:10.1155/s1024123x03111015

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.

Highlights

Over the last few years, many physical phenomena were formulated into nonlocal mathematical models [1, 2, 3, 4, 5, 6, 7, 8]
An algorithm was applied to the one-dimensional diffusion equation with a nonlocal condition replacing one standard boundary condition
The second-order spatial derivative was discretized to result in an approximating system of ordinary differential equations (ODEs)

Summary

Introduction

Over the last few years, many physical phenomena were formulated into nonlocal mathematical models [1, 2, 3, 4, 5, 6, 7, 8]. Such problems were not well studied in general. We are concerned with one-dimensional parabolic equations with a nonlocal condition: the so-called energy specification. This is a linear constraint having the form b 0 u(x, t)dx =. Coupled with a one-dimensional parabolic equation, this condition is quite different from the classical boundary condition

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Methods

Conclusion