基本解方法求解各向异性材料中热传导方程的时间反向问题

最近，HON和WEI给出了求解各向同性热传导反问题的基本解方法．该方法提供了一种在整个时间空间区域上的行之有效的数值格式．本文尝试将该无网格方法推广应用于求解各向异性材料中热传导方程的时间反向问题．首先，通过变量转换得到该问题的控制方程的基本解．接着，应用截断奇异值分解和L-曲线准则求解所得的高度病态的线性方程组．最后给出几个数值例子展示本方法的有效性，并分析了解的精度跟参数T、最终时刻的关系．

Method of fundamental solutions to backward heat conduction problem in an anisotropic medium

Recently, HON and WEI proposed a method of fundamental solutions to solving isotropic inverse heat conduction problems, It provides an efficient global approximation scheme in both spatial and time domains. In this paper, the inherently meshless and integration-free method is extended to solve 2D backward heat conduction problem in an anisotropic medium, Firstly, the fundamental solution to the governing equation is obtained through variables transformation, Then the truncated singular value decomposition and the L-curve criterion are applied to solve the resulting matrix equation which is highly ill-conditioned, Results for several numerical examples are presented to demonstrate its efficiency; the relationships between the accuracy of the numerical solutions and the value of the parameter T and the final time are also investigated.