Extrapolation methods for solving hypersingular integral equation of the first kind

Abstract

The hypersingular integral equation has been studied widely in boundary element methods, especially in natural boundary element methods. The asymptotic expansion of error function of composite rectangle rule for the computation of Hadamard finite-part integrals with the hypersingular kernel 1/ sin 2 (x − s) is obtained. Extrapolation algorithm is constructed. In order to solve the hypersin-gular integral equation, the superconvergence point is taken as the collocation point, then the extrapolation algorithm is presented and the convergence rate of extrapolation algorithm for hypersingular integral equation is presented. At last, some numerical results are also illustrated to confirm the theoretical results and show the efficiency of the algorithms. dt − 4f (s) cot ϵ 2 } (3) The equivalent of (2) and (3) can be similarly obtained in [32]. The composite middle rectangle rule is still valid when the singular point is located at the middle rectangle of each subintervals, in the words, the superconvergence phenomenon of the middle rectangle rule appears at this moment, we obtain the error expansion of the error functional. Moreover, we apply the asymptotic expansion results to construct a algorithm scheme for solving hypersingular integral equations. MSC Classification: 65D05 65M70.