Numerical methods of oscillatory Bessel transforms with algebraic and Cauchy singularities

This article proposes and analyzes fast and precise numerical methods for calculating the Bessel integral, which exhibits rapid oscillations and includes algebraic and Cauchy singularities. When $a>0$, we utilize the numerical steepest descent method with the Gauss-Laguerre quadrature formula to solve it. If $a=0$, we partition the integral into two parts, solving each part using the modified Filon-type method and the numerical steepest descent method, respectively. Moreover, the strict error analysis with respect to the frequency parameter ω is provided via a plenty of theoretical analysis. Finally, the efficiency and precision of these proposed methods are validated by numerical examples.