Control of condition number in spectral Galerkin implementation for solving generalized Abel integral equation

Numerical discretization of functional equation using spectral methods usually leads to a linear system of algebraic equations with a high conditional and full coefficient matrix. This drawback destroys the accuracy specially for large values of approximation degree. In this paper, we propose a new spectral Galerkin implementation for solving a class of generalized Abel integral equations, with the purpose of controlling condition number and recovering the familiar spectral accuracy.