The method of fundamental solutions (MFS) is first proposed in 1964 by Kupradze and theoretical basis of this method is constructed at the end of 1980’s. As a meshless method, no domain meshing is required for MFS. Fundamental solutions are used to solve problems without coping with the singularity on the boundary because of the fictitious boundary defined containing the domain of the problem. In this paper effectiveness of the MFS will be introduced by two test problem for the homogeneous and inhomogeneous modified helmholtz equations. In-homogeneous terms are approximated by using the method of particular solutions through the dual reciprocity method. The conduction heat transfer problem is defined and transformed to the corresponding elliptic partial differential equation by using finite difference and the method of lines method which gives an inhomogeneous helmholtz equation. Then the problem is solved iteratively by using the MFS. Two test problem are solved by both the finite element method (FEM) and MFS and compared in the figures. It can be seen that as a meshless method, MFS gives very good results for the test problems. The thermal shock problem presented here also gives accurate solutions by using MFS and agrees well with the FEM solution.

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