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.