A Modified Method of Fundamental Solutions with Source on the Boundary for Solving Laplace Equations with Circular and Arbitrary Domains

A boundary-type method for solving the Laplace problems using the modified method of fundamental solutions (MMFS) is proposed. The present method (MMFS) implements the sin- gular fundamental solutions to evaluate the solu- tions, and it can locate the source points on the real boundary as contrasted to the conventional MFS, where a fictitious boundary is needed to avoid thesingularityofdiagonal term ofinfluence matrices. The diagonal term of influence matrices for arbitrary domain can be novelly determined by relating the MFS with the indirect BEM and are also solved forcircular domain analyticallyby using separable kernels and circulants. The major difficultyofthecoincidenceofthesourceand col- locationpointsintheconventionalMFS isthereby overcome. The off-diagonal coefficients of influ- ence matrices can be easily determined by using thetwo-pointfunction. Theill-posednatureofthe conventional MFS then disappears. Finally, we provide numerical evidences that the presentmethod improves theaccuracy of the solu- tion after comparing with the conventional MFS, in particular for complicated boundaries in which the conventional MFS may encounter difficulties. Good agreements are observed as comparing with analytic or other numerical solutions.