Efficient Nyström-type method for the solution of highly oscillatory Volterra integral equations of the second kind

Abstract

Highly oscillatory Volterra integral equations are frequently encountered in engineering applications. The Nyström-type method is an important numerical approach for solving such problems. However, there remains scope to further optimize and accelerate the Nyström method. This paper presents a novel Nyström-type method to efficiently approximate solutions to second-kind Volterra integral equations with highly oscillatory kernels. First, the unknown function is interpolated at Chebyshev points. Then the integral equation is solved using the Nyström-type method, which leads to a problem of solving a system of linear equations. A key contribution is the technique to express the fundamental Lagrange polynomial in matrix form. The elements of the matrix, which involves highly oscillatory integrals, are calculated by using the classical Fejér quadrature formula with a dilation technique. The proposed method is more efficient than the one proposed in the recent literature. Numerical examples verify the efficiency and accuracy of the proposed method.