On the approximation of Volterra integral equations with highly oscillatory Bessel kernels via Laplace transform and quadrature

Abstract

The present work focuses on formulating a numerical scheme for approximation of Volterra integral equations with highly oscillatory Bessel kernels. The application of Laplace transform reduces integral equations into algebraic equations. By application of inverse Laplace transform solution is presented as an integral along a smooth curve extending into the left half of the complex plane, which is then evaluated by quadrature. Some model problems are solved and the results are compared with other available methods. The supremacy of the present method is that the transformed problem becomes non oscillatory. Consequently such types of integral equations with highly oscillatory kernels can be approximated very effectively with large values of oscillation parameter. A small amount of work such as Clenshaw-Curtis-Filon type methods are available for efficient approximation of highly oscillatory integral equations. Keywords: Laplace transform, Numerical inverse Laplace transform, Oscillatory kernels of convolution type