New quadrature rules for highly oscillatory integrals with stationary points

• A new framework for the numerical solution of highly oscillatory integrals is proposed.
• The integrals are bifurcated in the neighborhood of stationary point.
• The integral on the smaller subinterval is solved by hybrid functions and Haar wavelets.
• The integral on the longer subinterval is solved by the meshless method with MQ radial basis functions.
• Convergence analysis of the proposed methods is performed.

In this paper new algorithms are being proposed for evaluation of highly oscillatory integrals (HOIs) with stationary point(s). The algorithms are based on modified Levin quadrature (MLQ) with multiquadric radial basis functions (RBFs) coupled with quadrature rules based on hybrid functions of order 8 (HFQ8) and Haar wavelets quadrature (HWQ) (Aziz et al. 2011). Part of the new procedure presented in this paper is comprised of transplanting monomials (which are used in the conventional Levin method) by the RBFs. The linear and Hermite polynomials based quadratures (Xiang, 2007) are being replaced by the new methods based on HWQ and HFQ8 respectively. Both the methods are merged with MLQ to obtain the numerical solution of highly oscillatory integrals having stationary points. The accuracy of the new methods is neither dampened by presence of the stationary point(s) nor by the large value of frequency parameter ωω. Theoretical facts about the error analysis of the new methods are analyzed and proved. Numerical examples are included to show efficiency and accuracy of the new methods.