Simpson's rule to approximate Hilbert integral and its application

In this paper, the computation of Hilbert singular integral with generalized composite Simpson's rule for is discussed. When singular points coincide with some a priori known point, the convergence rate of Simpson's rule higher than global one, we obtain the pointwise superconvergence phenomenon. Which means the especial function equal zero, the superconvergence points are got. Then choosing the superconvergence point as the collocation points, we get a collocation scheme for solving the relevant Hilbert integral equation. At last, some numerical examples are presented to validate the theoretical analysis.