A Fully Discrete Galerkin Method for Abel-type Integral Equations

TL;DR

This paper introduces a fully discrete Galerkin method for solving Abel-type integral equations, providing stability, convergence analysis, and efficient quadrature techniques, supported by numerical experiments.

Contribution

It develops a novel fully-discrete Galerkin approach with tensor-Gauss quadrature for Abel-type equations, including stability, convergence, and perturbation analysis.

Findings

Numerical results confirm theoretical stability and convergence estimates.

The method is efficient with a small number of quadrature points.

Solution sensitivity to parameters is demonstrated through experiments.

Abstract

In this paper, we present a Galerkin method for Abel-type integral equation with a general class of kernel. Stability and quasi-optimal convergence estimates are derived in ractional-order Sobolev norms. The fully-discrete Galerkin method is defined by employing simple tensor-Gauss quadrature. We develop a corresponding perturbation analysis which allows to keep the number of quadrature points small. Numerical experiments have been performed which illustrate the sharpness of the theoretical estimates and the sensitivity of the solution with respect to some parameters in the equation.