Laplace transforms for approximation of highly oscillatory Volterra integral equations of the first kind

This paper focuses on Laplace and inverse Laplace transforms for approximation of Volterra integral equations of the first kind with highly oscillatory Bessel kernels, where the explicit formulae for the solution of the first kind integral equations are derived, from which the integral equations can also be efficiently calculated by the Clenshaw-Curtis-Filon-type methods. Furthermore, by applying the asymptotics of the solution, some simpler formulas for approximating the solution for large values of the parameters are deduced. Preliminary numerical results are presented based on the approximate formulae and the explicit formulae, which are compared with the convolution quadrature and numerical inverse Laplace transform methods. All these methods share that the costs the same independent of large values of frequencies.