Numerical computation of hypersingular integrals on the real semiaxis

摘要

In this paper we propose some different strategies to approximate hypersingular integrals where p is a positive integer, t > 0 and the integral is understood in the Hadamard finite part sense. Hadamard Finite Part integrals (shortly FP integrals), regarded as pth derivative of Cauchy principal value integrals, are of interest in the solution of hypersingular BIE, which model many different kind of Physical and Engineering problems (see [1] and the references therein, [2], [3, 4]).The procedure we employ here is based on a simple tool like the “truncated” Gaussian rule (see [5]), conveniently modified to remove numerical cancellation. We will consider functions having different decays at infinity. The method is shown to be numerically stable and convergent and some error estimates in suitable Zygmund-type spaces are proved. Finally, some numerical tests which confirm the efficiency of the proposed procedures are presented.