Efficient methods for some highly oscillatory integrals and integral equations

Abstract

A number of mathematical and physical problems occur in the areas of electromagnetics and acoustic scattering simulations, which are of important theoretical values and have wide applications. High oscillation plays a crucial role and represents formidable mathematical and computational challenge. Highly oscillatory differential equations and integral equations are two fundamental models for these problems, whose computations are difficult and of many challenging problems. From the view of reformulation by means of integral equations, this paper gives a survey on the new developments on highly oscillatory problems, particularly, the details on generalized Fourier transforms, Bessel transforms and Volterra integral equations with highly oscillatory kernels. These methods share that the higher the frequency the more accurate of the numerical solution, which provides a new way of solving these kinds of equations.