An Efficient and Accurate Numerical Method for the Bessel Transform with an Irregular Oscillator and Its Error Analysis

In this paper we present and analyze an efficient numerical scheme for computing the Bessel transform with an irregular oscillator. Especially in the presence of critical points, e.g., endpoints, zeros and stationary points for the general oscillator g(x), we derive a series of new quadrature formulae for such transform and carry out rigorous analysis for the proposed numerical method. The error analysis demonstrates that this method exhibits high asymptotic order, and the accuracy improves drastically with either increasing the frequency $\omega$ or adding more nodes. Compared with the existing modified Filon-type method, the established approach shows higher precision and error order at the same computational cost. The abundant numerical examples are provided to verify the theoretical results and illustrate the efficiency and accuracy of the proposed method. Mathematics Subject Classification (2000) 65D30 · 65D32 · 65R10 · 41A60