Superconvergence of the Composite Rectangle Rule for Computing Hypersingular Integral on Interval

Abstract

The generalized middle rectangle rule for the computation of certain hypersingular integrals is discussed. A generalized middle rectangle rule with the density function approximated and the singular kernel analysis calculated is presented and the asymptotic expansion of error functional is obtained. When the special function in the error functional equals to zero, the superconvergence point is obtained and the superconvergence phenomenon which is one order higher than the general case is presented. At last, numerical examples are given to confirm the theoretical results.