Efficient numerical methods for hypersingular finite-part integrals with highly oscillatory integrands

In this paper, we focus on the numerical evaluation of hypersingular finite-part integrals with two kinds of highly oscillatory integrands. Suppose that f is analytic in the first quadrant of the complex plane, based on complex integration theory, both of them are transformed into the problem of integrating two integrals on [0,+∞), such that the integrands do not oscillate and decay exponentially and thus can be computed efficiently by constructing the corresponding Gaussian quadrature rule for them. Moreover, error analyses are made for the proposed methods. Finally, several numerical examples are given to verify the theoretical results and illustrate the accuracy of the proposed methods.