An efficient and accurate numerical method for the Bessel transform with an irregular oscillator and its error analysis

Abstract

In this paper we present and analyze efficient numerical schemes for computing the Bessel transforms with an irregular oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points for the general oscillator g(t), we derive a series of new quadrature formulae for such transforms and carry out rigorous analysis for the proposed numerical methods. The error analysis demonstrates that this methods exhibit high asymptotic order, and the accuracy improves drastically with either increasing the frequency ω or adding more nodes. Compared with the existing modified Filon-type method, the established approaches show higher precision and order of the error estimate at the same computational cost. The extensive numerical examples are provided to verify the theoretical results and illustrate the efficiency and accuracy of the proposed method.