On the approximation of highly oscillatory Volterra integral equations of the first kind via Laplace transform

Abstract

This paper focuses on approximation of Volterra integral equations of the first kind with highly oscillatory Bessel kernel. We first give a lemma about the coefficient relations among several different interpolation polynomials of continuous functions g(x), and present the existence and uniqueness theorem of the solution of the Volterra integral equation. Based on the Laplace transform and inverse Laplace transform, we derive the explicit formulas for the solution of the first kind integral equation. Furthermore, based on the asymptotic of the solution for large values of the parameters, we deduce some simpler formulas for approximating the solution. Finally, we present two numerical methods that is the efficient quadrature rule and the Clenshaw–Curtis-type method, which can efficiently calculate highly oscillatory integrals in the solution of the Volterra integral equation.