Efficient quadrature rules for the singularly oscillatory Bessel transforms and their error analysis

Abstract

In this paper, we focus on the computation and analysis of the highly oscillatory Bessel transforms with endpoint singularities of algebraic and logarithmic type. Based on the modification of the numerical steepest descent method, we present a new and efficient quadrature rule. Firstly, we divide the considered integrals into two parts by $J_{m}(z)=\frac {1}{2}\left [H_{m}^{(1)}(z)+H_{m}^{(2)}(z)\right ]$ , where each part can be transformed into the Fourier-type integrals. Then, we use the Cauchy’s residue theorem to convert these Fourier-type integrals into the infinite integrals on $[0,+\infty )$ . Next, the resulting infinite integrals can be efficiently calculated by constructing some appropriate Gaussian quadrature rules. In addition, we conduct error analysis in inverse powers of the frequency parameter. Finally, several numerical examples are provided to show the efficiency and accuracy of the proposed method.