On numerical evaluation of integrals involving oscillatory Bessel and Hankel functions

A mixed-type formulation composed of Gauss-Laguerre quadrature and meshless collocation is presented for approximation of oscillatory integrals containing Hankel function of the first kind. An adaptive splitting procedure is implemented to resolve singularity of the transformed integral at x = 0. The singularity ridden integral is computed by the multi-resolution quadrature. Bessel function of the second kind is transformed into a Bessel function of the first kind before being approximated by the meshless collocation method (Zaman and Siraj-ul-Islam, J. Comput. Appl. Math. 315, 161–174, 2017). Theoretical error bounds of the proposed methods are established. Numerical results obtained from the benchmark problems are presented in the tabular and graphical forms for verification of theoretical error bounds and accuracy of the methods.