Reproducing kernel function-based Filon and Levin methods for solving highly oscillatory integral

•Based on the RKF theory, new Filon and Levin methods for highly oscillatory integrals.•The moments in RKF-based Filon method can be computed exactly.•The RKF-based Levin method is meshless.•The present method can give accurate approximation to the highly oscillatory integrals. The main theme of this paper is to develop new Filon and Levin methods for highly oscillatory integrals. The novel method is based on the spline reproducing kernel functions approximation in Sobolev reproducing kernel Hilbert space. The accuracy and efficiency of the present method is illustrated through some numerical experiments compared with some effective methods appeared in the literature.