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.