A well-conditioned Levin method for calculation of highly oscillatory integrals and its application

This paper is devoted to studying efficient calculation of generalized Fourier transform ∫−1xf(t)eiωg(t)dt. For the general phase function g(t), we develop a modified Levin method by the spectral coefficient approach. A sparse and well-conditioned linear system is constructed to help accelerate calculation of highly oscillatory integrals. Numerical examples are included to show the convergence properties of the new method with respect to both quantities of collocation points and the frequency ω. Furthermore, we apply this approach to solving oscillatory Volterra integral equations.