On computation of finite-part integrals of highly oscillatory functions

Abstract

In this paper, we propose some methods to compute finite-part integral involving hypersingular and highly oscillatory factors. We first write the integral as the Cauchy principle value integral which is computed based on the variable upper limit integral and frequency parameterization. For computing the nonsingular integral, we use the integration by parts and interpolation. On this basis, we get an asymptotic method and a Filon-type method. In order to test the efficiency of the proposed methods and verify the correctness of the proposed theories, some numerical experiments are performed.