Extrapolation methods for solving the hypersingular integral equation of the first kind

Hypersingular integral equations have garnered extensive attention in the context of boundary element methods, particularly within natural boundary element methods. The asymptotic expansion of the composite rectangular rule's error function in Hadamard finite-part integrals yields a hypersingular kernel of $1/{\sin }^2(x-s)$. An extrapolation algorithm was developed to address this issue. To solve the hypersingular integral equation, we employed superconvergence points as collocation points, thereby constructing an extrapolation algorithm for hypersingular integral equations and establishing its convergence rate. A numerical example was provided to validate the efficacy of the method, corroborated by theoretical results that demonstrate the algorithm's effectiveness.