A meshless local discrete collocation (MLDC) scheme for solving 2‐dimensional singular integral equations with logarithmic kernels

Abstract

AbstractIn the current investigation, we study a numerical technique to solve 2‐dimensional singular integral equations of the second kind with logarithmic kernels. These integral equations are a general form of the problem of obtaining the cross‐sectional separation of current in an infinitely long thin filament of conduct bar which arises in electrical engineering. The moving least squares scheme estimates a function without mesh generation on the domain that includes a locally weighted least squares polynomial fitting. The discrete collocation method in addition to the shape functions of moving least squares established on scattered points is used to approximate the solution of integral equations. To compute logarithm‐like singular integrals appeared in the process of the scheme, we use a particular dual nonuniform composite Gauss‐Legendre integration rule on normal domains. The proposed scheme does not require any meshes, so it is meshless and does not depend to the domain form. Error analysis is also given for the method. Illustrative examples are shown clearly the reliability and efficiency of the new scheme and confirm the theoretical error estimates.