Vol. LXXIX, 1 (2010)
p. 89 - 104

Fifth-order numerical methods for heat equation subject
to a boundary integral specification

M. A. Rehman, M. S. A. Taj and M. M. Butt

Received: December 24, 2008;   Revised: June 8, 2009;   Accepted: June 11, 2009

Abstract.   In this paper a fifth-order numerical scheme is developed and implemented for the solution of homogeneous heat equation ut = a uxx with nonlocal boundary condition as well as for inhomogeneous heat equation ut = a uxx + s(x,t) with nonlocal boundary condition. The results obtained show that the numerical method based on the proposed technique is fifth-order accurate as well as L-acceptable. In the development of this method second-order spatial derivative are approximated by fifth-order finite-difference approximations which give a system of first order, linear, ordinary differential equations whose solution satisfies a recurrence relation which leads to the development of algorithm. The algorithm is tested on various heat equations and no oscillations are observed in the experiments. This method is based on partial fraction technique which is useful in parrel processing and it does not require complex arithmetic.

Keywords:  heat equation; non-local boundary condition; fifth-order numerical methods; method of lines; parallel algorithm.  

AMS Subject classification: Primary:  65M06, 65N40  

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Acta Mathematica Universitatis Comenianae
ISSN 0862-9544   (Printed edition)

Faculty of Mathematics, Physics and Informatics
Comenius University
842 48 Bratislava, Slovak Republic  

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