Galerkin-Levin method for highly oscillatory integrals

In this paper, a new method, namely Galerkin-Levin method, is proposed to investigate the highly oscillatory integrals. Indeed, we construct an algorithm based on the Levin and Galerkin methods. Using the Levin method, the given integral is transformed into a certain ordinary differential equation, then, a Galerkin method is applied for solving the obtained ODE. The efficiency of the approach will be shown by applying the procedure on some prototype examples.