### Fifth order numerical method for heat equation with nonlocal boundary conditions

#### Abstract

This paper deals with numerical method for the approximate solution of one dimensional heat equation $ u_t=u_{xx}+q(x,t)$ with integral boundary conditions. The integral conditions are approximated by Simpson's $\frac{1}{3}$ rule while the space derivatives are approximated by fifth-order difference approximations. The method of lines, semi discretization approach is used to transform the model partial differential equation into a system of first-order linear ordinary differential equations whose solution satisfies a recurrence relation involving matrix exponential function. The method developed is L-acceptable, fifth-order accurate in space and time and do not required the use of complex arithmetic. A parallel algorithm is also developed and implemented on several problems from literature and found highly accurate when compared with the exact ones and alternative techniques.

**How to Cite this Article:**M.A. Rehman, M.S.A. Taj, S.A. Mardan, Fifth order numerical method for heat equation with nonlocal boundary conditions, Journal of Mathematical and Computational Science, Vol 4, No 6 (2014), 1044-1054 Copyright © 2014 M.A. Rehman, M.S.A. Taj, S.A. Mardan. 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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