Accuracy optimal methods for evaluating hypersingular integrals

This paper describes asymptotically optimal and optimal in order algorithms for numerical evaluation of one-dimensional hypersingular integrals with fixed and variable singularities. The obtained results in the paper can be divided into two groups:(i)For the first time, some optimal in order quadrature formulas are constructed for a set of hypersingular integrals. In particular, some novel optimal in order quadrature formulas have been constructed for numerical evaluation of the hypersingular integrals with a variable singularity.(ii)Some known quadrature formulas are substantially modified to obtain optimal in order quadrature formulas. In particular, previously developed asymptotically optimal and optimal in order quadrature formulas for evaluating hypersingular integrals with a fixed singularity [I.V. Boykov, N.F. Dobrynina, L.N. Domnin, Approximate Methods of Evaluating Hadamard's Integrals and Solution of Hypersingular Integral Equations, The Penza Technical State University, Penza, 1996, 188 p. (in Russian)] required a considerable “pre-hand” treatment of data. In the paper, optimal in order algorithms are constructed which do not require any provisional treatment. Illustrative examples demonstrate the accuracy and efficiency of the developed algorithms.