A sparse fractional Jacobi–Galerkin–Levin quadrature rule for highly oscillatory integrals

Abstract

Levin’s quadrature rule is well-known for its good performance on numerical treatment with highly oscillatory integrals. However, most existing work treats with sufficiently smooth integrands. In this paper, we study Levin’s quadrature rule for a class of singular and oscillatory integrals. Based on fractional Jacobi polynomials, a class of differential equation solvers for Levin’s equation are developed, which leads to the fractional Jacobi–Galerkin–Levin method. The discretized equation is turned into a sparse linear system by properly choosing Jacobi polynomials and the inner product. Furthermore, convergence analysis with respect to the oscillation is presented by studying coefficients of the fractional Jacobi expansion of the error function. Numerical experiments indicate that in contrast to existing quadrature rules, the new method is efficient for computing oscillatory integrals with stationary points and unknown singular parameters.