广义基本解方法求解一维非齐次热传导方程的反边界值问题

An extended method of fundamental solutions for inverse boundary value problems associated with one-dimensional nonhomogenous heat conduction

给出一种求解一维非齐次热传导方程反边界值问题的无网格方法,即广义基本解方法.该方法将问题的解分成特解和相应齐次问题的解两个部分:齐次解用基本解方法求解,而特解则是利用相应的特征方程的基本解近似得到.鉴于所考虑问题的不适定性,应用截断奇异值分解和L曲线准则求解离散后得到的高度病态的线性方程组.最后给出数值例子说明该方法的稳定性和有效性,并分析了数值解精度与各参数之间的关系.

A meshless method, the extended method of fundamental solutions, is proposed to solve inverse boundary value problems associated with one-dimensional nonhomogenous heat conduction. In the method, the solution is split into two parts, the particular and homogenous solutions.- the homogenous solution is evaluated by the method of fundamental solutions while an approximation of the particular solution is derived by using the fundamental solutions of the associated eigenvalue equations. Since the inverse problem is ill-posed, the truncated singular value decomposition with the regularization parameter given by the L-curve method is employed to sovle the resulting highly ill-conditioned matrix equation. Numerical results are presented to verify the reliability and efficacy of the proposed method. The relationships between the accuracy of the numerical solutions and the parameters are also investigated.