Laplace transforms for evaluation of Volterra integral equation of the first kind with highly oscillatory kernel

This paper focuses on the numerical solution for Volterra integral equations of the first kind with highly oscillatory Bessel kernel and highly oscillatory triangle function on the right-hand side. We first establish a new existence theorem of solutions for such equations, and then, the explicit formulas of the solution are derived based on Laplace and inverse Laplace transforms. Furthermore, high-order accurate numerical solutions for approximating the explicit solution are further deduced by applying the Clenshaw–Curtis–Filon method and other effective numerical methods. Preliminary numerical results not only show the exact formulas of the solution, but also present the accuracy of the approximations.