Numerical steepest descent method for computing oscillatory-type Bessel integral transforms

Abstract

In this paper, numerical steepest descent method is implemented to approximate highly oscillatory Bessel-type integral transforms. We begin our analysis by utilizing an important relationship between Bessel function of the first kind and modified Bessel function of the second kind. Subsequently, we transform new integrals into the forms on the interval $[0,+\infty )$, where the integrands do not oscillate and decay exponentially fast. These integrals can then be efficiently computed using Gauss–Laguerre quadrature rule. Furthermore, we derive the theoretical error estimates that depend on the frequency $\omega$ and the number of nodes $n$. Numerical examples based on the theoretical results are provided to demonstrate the effectiveness of these methods.