Gauss quadrature rules for integrals involving weight functions with variable exponents and an application to weakly singular Volterra integral equations

Abstract

This paper presents a numerical integration approach that can be used to approximate on a finite interval, the integrals of functions that contain Jacobi weights with variable exponents. A modification of the integrand close to the singularities is needed, and a new modification is proposed. An application of such a rule to the numerical solution of variable-exponent weakly singular Volterra integral equations of the second kind is also explored. In the space of continuous functions, the stability and the error estimates are demonstrated, and numerical tests that validate these estimates are conducted.