Approximation of highly oscillatory integrals containing special functions

Abstract

A modified form of Levin’s method based on multi-quadric radial basis function is presented for numerical evaluation of integrals having squared oscillatory Bessel function of the first kind of order μ and oscillatory Airy function. The method converts the oscillatory integrals into a system of coupled ordinary differential equations (ODEs) and subsequently, numerical solution of the coupled ODEs is obtained by meshless collocation procedure. A multi-resolution quadrature is used to tackle singularity of the proposed method. Theoretical error analysis of the proposed methods is performed in asymptotic sense. Numerical test problems are included to verify accuracy and efficiency of the new methods in the context of oscillatory Airy and Bessel integrals.