Numerical quadratures for singular and hypersingular integrals

We present a procedure for the design of high-order quadrature rules for th e numerical evaluation of singular and hypersingular integrals; such integr als are frequently encountered in solution of integral equations of potenti al theory in two dimensions. Unlike integrals of both smooth and weakly sin gular functions, hypersingular integrals are pseudo-differential operators, being limits of certain integrals; as a result, standard quadrature formul ae fail for hypersingular integrals. On the other hand, such expressions ar e often encountered in mathematical physics (see, for example, [1]), and it is desirable to have simple and efficient "quadrature" formulae for them. The algorithm we present constructs high-order "quadratures" for the evalua tion of hypersingular integrals. The additional advantage of the scheme is the fact that each of the quadratures it produces can be used simultaneousl y for the efficient evaluation of hypersingular integrals, Hilbert transfor ms, and integrals involving both smooth and logarithmically singular functi ons; this results in significantly simplified implementations. The performa nce of the procedure is illustrated with several numerical examples. (C) 20 01 Elsevier Science Ltd. All rights reserved.