Solving Cauchy problems of elliptic equations by the method of fundamental solutions

摘要

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 ill-conditioned. 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.