Numerical techniques for a parabolic equation subject to an overspecified boundary condition

摘要

An inverse problem concerning diffusion equation with a control parameter is considered. A new numerical scheme is developed for obtaining approximate solutions to an initial-boundary-value problem for the semi-linear parabolic partial differential equation subject to an overspecified boundary condition. The space derivative in the PDE is approximated by finite difference replacements. The solution of the resulting system of first-order ordinary differential equations (ODEs) satisfies a recurrence relation which involves a matrix exponential function. The accuracy in time can be controlled by choosing several Pade approximants to replace this matrix exponential term. A numerical technique is developed to compute the required solution using a splitting method, leading to algorithms for sequential and parallel implementation. The algorithms are tested on a model problem from the literature. The results of a numerical experiment are presented, and the accuracy and Central Processor (CPU) time needed for this inverse problem are discussed.