Meshless and wavelets based complex quadrature of highly oscillatory integrals and the integrals with stationary points

In this paper, we formulate and employ efficient and accurate methods for numerical evaluation of highly oscillatory integrals and integrals having stationary points. Two new approaches using radial basis function (RBF) and wavelets are discussed. The first approach is related to meshless method (MM) which is based on multiquadric (MQ) RBF, and is specially designed for integrands having oscillatory character. This approach stems from the Levinʹs method. In this procedure, the solution is obtained by solving the corresponding ODE or PDE instead of finding a numerical solution of the integration problem. In situations when the integrand has stationary points, MM fails to deliver. We opt for quadrature rules based on Haar wavelets and hybrid functions. The proposed methods are tested on a number of benchmark tests considered in available literature. The performance of the new methods is compared with the existing methods. Better accuracy of the proposed methods is reported for a variety of problems.