An Accurate Computation of Highly Oscillatory Integrals with Critical Points

Abstract

New methods are proposed for evaluation of one-dimensional highly oscillatory integrals with and without critical points. These integrals are hard to approximate due to the existence of high oscillations of the integrand and to take care of the critical point in the domain interval. Levin procedure with Gaussian radial basis function is implemented to evaluate oscillatory integrals without critical point. The meshless method is coupled with multi-resolution analysis in a new shape to handle the critical points. Test problems verify accuracy and efficiency of the new methods.