Analytic series solutions of 2D forward and backward heat conduction problems in rectangles and a new regularization

Abstract

In the paper, we solve a non-homogeneous heat conduction equation with non-homogeneous boundary conditions in a 2D rectangle. First, we derive the domain/boundary integral equations for both the forward and backward heat conduction problems. Then, by using the technique of homogenization, inserting the adjoint Trefftz test functions into the derived integral equations and expanding the solutions in terms of eigenfunctions, we can obtain the expansion coefficients in closed form. Hence, the analytic series solutions of forward heat conduction problems (FHCPs) and backward heat conduction problems (BHCPs) are available. For the FHCPs, only a few terms in the series render very high-order accurate solutions at any time, with errors of the order . For the BHCPs, we require to modify the closed-form series solutions via a new spring-damping regularization technique. Numerical tests for the BHCPs in a large space-time domain reveal that the present analytic series solution is very accurate to recover the initial temperature with an error of the order , although the measured final time temperature is very small when is large up to 100 and is even polluted by a large relative noise up to the level .