Numerical Computation Based on the Method of Fundamental Solutions for a Cauchy Problem of Heat Equation

In this article, we consider a Cauchy problem of heat equation, that is, determining the unknown temperature and heat flux at an inaccessible boundary from scattered temperature measurements on an accessible boundary or in some interior locations. This method is similar to the boundary control approach proposed by Leevan Ling and Tomoya Takeuchi in [1] where the authors considered a Cauchy problem for the Laplace equation. We use the standard integral equation method coupled with the method of fundamental solutions to solve the Cauchy problem for heat equation. This kind of inverse heat conduction problem arises in some industrial and engineering applications, such as crystal growing [2] and material structure [3]. The Cauchy problem of heat equation is a highly ill-posed problem, because the solution does not depend continuously on the boundary date, ie, any small change on the input data can result in a dramatic change to the solution. So it is difficult to obtain an accurate and stable approximate solution. Usually one regularization strategy is necessary. In order to solve such problem, one can employ the boundary element method (BEM) [4], finite difference method(FDM) [5], finite element method(FEM) [6], and so on. Among these methods, the FDM and the FEM depend critically on the quality of mesh. However, generating a good quality mesh for complicated geometries could be time-consuming. Using the BEM can reduce the computational time and storage requirement but the problem of numerical in stability still persists. Recently, several meshless and integration-free methods have been proposed. One of the most commonly used technique is the method of fundamental solutions. Hon and Wei have already successfully applied this method to solve One-dimensional and multidimensional inverse heat conduction problems in [7,8]. In this paper, the difference from one method in [7,8] is that we use the method of fundamental solutions to solve a sequence of direct problems instead of solving the inverse problem directly.