Extrapolation Methods to Compute the Hypersingular Integral on Interval

The composite trapezoidal rule for the computation of Hadamard finite-part integral on interval with the hyper singular kernel 1/(t-s)² is discussed and the case of the mesh point coinciding with the singular point by generalized finite-part definition is considered. The asymptotic expansion is obtained and an extrapolation algorithm is presented to accelerate the convergence rate. Based on the Toeplitz matrix discrete, we prove of convergence rate the singular integral equation is O(h). At last, some numerical results are also presented to confirm the theoretical results and the efficiency of the algorithms is shown.