Interpolation method for evaluating weakly singular kernels

Abstract

The solution of initial, boundary, and mixed value problems through the integral equation method yields certain boundary singular integral equations. In many scientific and industrial applications in artificial intelligence, biological systems, scattering, radiation, and image processing, there is a need to evaluate such types of singular integrals, especially the weakly singular kernels. This study presents a new numerical method for evaluating weakly singular kernels based on some advanced matrix-vector barycentric Lagrange interpolation formulas. We developed these formulas to be applied to numerically evaluating singular integrals. At the same time, we created three computational rules to determine the optimal locations for the distribution of the interpolation nodes to be within the integration domain and never be outside for any value of the interpolant degree. These rules are devised so that the equidistant nodes depend on the step-sizes, which are defined as functions of the interpolant degree by some small real number greater than or equal to zero. Thus, we overcame the singularity of the kernels on the whole integration domain and obtained uniform interpolation. Moreover, the presented method gives the kernel's values and the kernel's integral values at the singular points, whereas the numerical or exact values do not exist. The solutions to the illustrated four examples are shown in the given tables and figures. The interpolant solutions which we obtain by low-degree interpolants are faster to converge to the numerical or exact ones (if they exist). This confirms the originality of the presented method and its effectiveness in obtaining high-precision results.