An Analysis of Backward Heat Conduction Problems Using the Time Evolution Method of Fundamental Solutions

摘要

The time evolution method of fundamental solutions (MFS) is proposed to solve backward heat conduction problems (BHCPs). The time evolution MFS belongs to one of the mesh-free numerical methods and is essentially composed of a sequence of diffusion fundamental solutions which exactly satisfy the heat conduction equations. Through correct treatment of temporal evolution, the resulting system of the time evolution MFS is smaller, and effectively decreases the possibility of ill-conditioning induced by such strongly ill-posed problems. Both one-dimensional and two-dimensional BHCPs are examined in this study, and the numerical results demonstrate the accuracy and stability of the MFS, especially for the BHCPs with high levels of noise. Conclusively, time evolution MFS is a stable and powerful numerical scheme, and is especially suitable for the numerical solution of BHCPs.