Abstract

In this study, we applied an advanced barycentric Lagrange interpolation formula to find the interpolate solutions of weakly singular Fredholm integral equations of the second kind. The kernel is interpolated twice concerning both variables and then is transformed into the product of five matrices; two of them are monomial basis matrices. To isolate the singularity of the kernel, we developed two techniques based on a good choice of different two sets of nodes to be distributed over the integration domain. Each set is specific to one of the kernel arguments so that the kernel values never become zero or imaginary. The significant advantage of thetwo presented techniques is the ability to gain access to an algebraic linear system equivalent to the interpolant solution without applying the collocation method. Moreover, the convergence in the mean of the interpolant solution and the maximum error norm estimation are studied. The interpolate solutions of the illustrated four examples are found strongly converging uniformly to the exact solutions.

Highlights

  • The solutions of the initial, boundary, or mixed value problems have become common to be obtained through the integral equation method

  • The Dirichlet boundary value problems for the Laplace equation for an open arc in the plane is predominantly reduced to the solution of a weakly singular Fredholm integral equation of the first kind whose unknown function is singular at the Shoukralla et al Adv

  • This study focuses on the application of two advanced barycentric interpolation formulas to solve the weakly singular Fredholm integral equation of the second kind with two innovative techniques for the treatment of the kernel’s singularity depending on the perfect choice of the node distribution rules

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Summary

Introduction

The solutions of the initial, boundary, or mixed value problems have become common to be obtained through the integral equation method. This study focuses on the application of two advanced barycentric interpolation formulas to solve the weakly singular Fredholm integral equation of the second kind with two innovative techniques for the treatment of the kernel’s singularity depending on the perfect choice of the node distribution rules.

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