Efficient numerical methods for Bessel type of oscillatory integrals

In this paper, new quadrature rules are proposed for numerical evaluation of highly oscillatory integrals containing first kind of Bessel functions Jυ(κx). Meshless procedure with uniform and scattered nodes is used to cope with frequent irregular oscillations caused by Bessel and Bessel-trigonometric functions. In addition, multi-resolution quadrature rules based on hybrid functions and Haar wavelets are used as supporting tools to handle the case of singularity of the meshless collocation method. Error bounds of the proposed methods are calculated and numerically verified by solving some benchmark test problems.