Gauss-type quadrature for highly oscillatory integrals with algebraic singularities and applications. Issue 6 (3rd June 2017) - Xu, Zhenhua; Xiang, Shuhuang
ABSTRACT: In this paper, we study the numerical methods for the highly oscillatory integral of the type∫ a b f ( x ) ( x − a ) α ( b − x ) β e i ω g ( x ) d x, whereα > − 1, β > − 1, f is analytic in a sufficiently large complex region containing[ a, b ] . Based on substituting the original interval of integration by the paths of steepest descent, the integral can be rewritten as a sum of several line integrals, which can be efficiently computed by Gaussian quadrature rules with different weight functions. Also, we apply this method to the implementation of discontinuous Galerkin method for Volterra integral equation with the Fourier kernel. Numerical examples are used to illustrate the efficiency and accuracy of the proposed methods.