Numerical solution of a parabolic equation with non-local boundary specifications

Abstract

The parabolic partial differential equations with non-local boundary specifications model various physical problems. Numerical schemes are developed for obtaining approximate solutions to the initial boundary-value problem for one-dimensional second-order linear parabolic partial differential equation with non-local boundary specifications replacing boundary conditions. The method of lines semi-discretization approach will be used to transform the model partial differential equation into a system of first-order linear ordinary differential equations (ODEs). The spatial derivative in the PDE is approximated by a 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. The new algorithms are tested on two problems from the literature. The central processor unit times needed are also considered.