WIT Press

Solving Cauchy Problems Of Elliptic Equations By The Method Of Fundamental Solutions


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Y. C. Hon & T.Wei1


The method of fundamental solutions is used to solve various Cauchy problems of Laplace, Helmholtz and modified Helmholtz equations. The resultant linear system of equations for the undetermined coefficients are known to be severely illconditioned. The use of the Tikhonov regularization technique with two different strategies for the choice of regularization parameters successfully provides a stable numerical approximation to the solution of the Cauchy problems. Numerical verification of the proposed methods are also given. 1 Introduction Themethod of fundamental solutions (MFS) has been extensively applied for solving partial differential equations. The idea of MFS is to represent the solution by a linear combination of fundamental solutions with source points located outside the computational domain. The expansion coefficients are then determined by solving a resultant linear system of equations or a least squares problem. MFS does not involve the evaluation of the integrals and thus provides an efficient numerical method for solving high dimensional problems with irregular domains. More details can be found in the review papers of Fairweather and Karageorghis [2] and Golberg and Chen [4]. All these studies focus on the well-posed direct problems in which the Dirichlet or Neumann data on the whole boundary have been given. But for inverse problems, boundary conditions are usually complicated and incomplete. The truly meshless MFS is suitable to solve inverse problems. In this paper we apply MFS to solve various Cauchy problems for elliptic equations. The resultant linear systems of equations for the undetermined coefficients are severely ill-conditioned. Most standard numerical schemes fail to give an