Numerical methods for reconstruction of source term of heat equation from the final overdetermination

TL;DR

This paper develops numerical methods to reconstruct the spatial source term in a heat equation from final data, addressing both free boundary and Neumann boundary cases, with validation through numerical experiments.

Contribution

It introduces numerical techniques for inverse heat source problems with a specific source form, extending to free boundary and Neumann boundary conditions.

Findings

Numerical methods successfully reconstruct the source term from final data.

The extension method effectively handles free boundary conditions.

Numerical experiments validate the proposed approaches.

Abstract

This paper deals with the numerical methods for the reconstruction of source term in linear parabolic equation from final overdetermination. We assume that the source term has the form f(x)h(t) and h(t) is given, which guarantees the uniqueness of the inverse problem for determining the source term $f(x)$ from final overdetermination. We present the numerical methods for both free boundary and Neumann boundary situations. Moreover, we show that the solution of the boundary conditions problem has the form of the free boundary solution problem by using the extension method. Numerical experiments are done for the inverse problem with the boundary conditions.