A numerical scheme for solving a class of logarithmic integral equations arisen from two-dimensional Helmholtz equations using local thin plate splines

摘要

This paper presents a numerical method for solving logarithmic Fredholm integral equations which occur as a reformulation of two-dimensional Helmholtz equations over the unit circle with the Robin boundary conditions. The method approximates the solution utilizing the discrete collocation method based on the locally supported thin plate splines as a type of free shape parameter radial basis functions. The local thin plate splines establish an efficient and stable technique to estimate an unknown function by a small set of nodes instead of all points over the solution domain. To compute logarithm-like singular integrals appeared in the method, we use a particular nonuniform Gauss–Legendre quadrature rule. Since the scheme does not require any mesh generations on the domain, it can be identified as a meshless method. The error estimate of the proposed method is presented. Numerical results are included to show the validity and efficiency of the new technique. These results also confirm that the proposed method uses much less computer memory in comparison with the method established on the globally supported thin plate splines. Moreover, it seems that the algorithm of the presented approach is attractive and easy to implement on computers.