Numerical method for solving Volterra integral equations with oscillatory kernels using a transform

In the present work, a numerical scheme is constructed for the approximation of a class of Volterra integral equations of the convolution type with highly oscillatory kernels. The proposed numerical technique transforms the Volterra integral equations of the convolution type into simple algebraic equations. By an inverse transform the problem is converted into an integral representation in the complex plane, and then computed by a suitable quadrature formula. The numerical scheme is applied for a class of linear and nonlinear Volterra integral equations of the convolution type with highly oscillatory kernels, and some of the obtained results are compared with the methods available in the literature. The main advantage of the present scheme is the transformation of a highly oscillatory problem to a non-oscillatory and simple problem. So a large class of a similar type of integral equations having kernels of a highly oscillatory type can be very effectively approximated.