The Method Of Fundamental Solution With Double Layer Potential

Many problems in the engineering and science can come to boundary value problems of partial differential equations, Besides some special problems,it’s not possible obtain its analytical solution,normally only get numerical solution.In the family of scientific computing, The main numerical methods based on grid are Finite Element Method(FEM), Finite Difference Method(FDM) and Boundary Element Method(BEM). The Method of Fundamental Solution(MFS) is a simple, highly accurate mesh-less method, without dividing its boundary and the region. For its excellent characteristics of high computing accuracy, fast convergence, simple programming, the MFS become a method which get the favour of many scientific workers. However, traditional MFS based on single-layer potential and superposition principle, in solving some limited domain problems, it has some limited get the virtual boundary, when solving some infinite domain problems can not solved. This paper present method of fundamental solution with double layer potential for the problems of 2D steady temperature field, it’s different from the traditional MFS based on single-layer. But, in practice, the MFS that based on double layer potential and superposition principle can avoid the problems of the traditional MFS in solving the 2D infinite domain potential problems,and it still have some problems in solving the 2D infinite potential problems. Therefore, this paper present an improved method of fundamental solution with double layer potential and superposition principle. Examples in this paper show that the improved MFS not only avoids the promlems of the traditional MFS in solving the limited domain problems, also applies in solving any two-dimensional infinite domain problems and the inverse problem s.In this work, we present the MFS based on double layer potential to solve the 2D potential problems and inverse problems in chapter three and chapter four. Then improved the method to investigate the 2D infinite potential problems and inverse problems in chapter five and six. In chapter seven we present an improved method of fundamental solutions and apply it to 2D elasticity Cauchy inverse problem. In the study of the inverse problems, the ill-conditioned linear system of equations discreted by the MFS with double layer potential is solved by TSVD method and the Tikhonov regular method. And the parameters used in TSVD and Tikhonov are determine d by the L-curve and GCV method, good effect has been achieved. It not only ensured the precision, also greatly expanded the virtual boundary range of options. Numerical examples in this paper show that the improved MFS is applied in solving any problems of the boundary value.