A quasi‐boundary semi‐analytical method for backward heat conduction problems

Abstract

In this paper, we propose a semi‐analytical method to deal with the backward heat conduction problem due to a quasi‐boundary idea. First of all, the Fourier series expansion technique is used to calculate the temperature field u(x, t) at any time t < T. Second, we consider a direct regularization by adding the term αu(x, 0) into the final time condition to obtain a second kind Fredholm integral equation for u(x, 0). The termwise separable property of the kernel function allows us to transform the backward problem into a two‐point boundary value problem and therefore, a closed‐form solution is derived. The uniform convergence and error estimation of the regularized solution uα (x, t) are provided and a tactic to choose the regularization parameter is suggested. When several numerical examples are amenable, we discover that the present approach can retrieve all the past data very well and is robust even for seriously noised final data.