A collocation scheme for a certain Cauchy singular integral equation based on the superconvergence analysis

In this paper, we investigate the composite midpoint rule for the evaluation of Cauchy principal value integral in an interval and place the key point on its pointwise superconvergence phenomenon. The error expansion of the rule is obtained, which shows that the superconvergence phenomenon occurs at the points of each subinterval whose local coordinate is the zeros of some function. Then, by applying the midpoint rule to approximate the Cauchy principal value integral and choosing the superconvergence points as the collocation points, we obtain a collocation scheme for solving a certain Cauchy singular integral equation. The more interesting thing is that the coefficient matrix of the resulting linear system possesses some good properties, from which we obtain an optimal error estimate. Finally, some numerical examples are provided to validate the theoretical analysis.