A regularized Fourier sine series solution of a 3D backward heat conduction problem with extremal long time span

Abstract

In the article, we solve an extremal long time span backward heat conduction problem (BHCP) of a 3D nonhomogeneous heat conduction equation with nonhomogeneous boundary conditions in a cuboid. We first derive a time-dependent 3D homogenization function, such that in terms of the new variable by a variable transformation we can find the expansion coefficients in a closed form by using the Fourier sine series method. After a simple regularization technique, a stable analytic series solution of the 3D BHCP is available. We also develop a regularized Fourier sine series solution of temperature in the whole space-time domain. Numerical tests for the BHCPs in a large space-time domain reveal that the present method is very accurate to recover the initial temperature and the whole solution.