Numerical approximations for highly oscillatory Bessel transforms and applications

This paper presents an efficient numerical method for approximating highly oscillatory Bessel transforms. Based on analytic continuation, we transform the integrals into the problems of integrating the forms on [0,+∞)[0,+∞) with the integrand that does not oscillate and decays exponentially fast, which can be efficiently computed by using Gauss–Laguerre quadrature rule. We then derive the error of the method depending on the frequency and the node number. Moreover, we apply the scheme for studying the approximations of the solutions of two kinds of highly oscillatory integral equations. Preliminary numerical results show the efficiency and accuracy of numerical approximations.