Details

Autor(en) / Beteiligte

• WANG, TONGKE
• LIU, ZHIFANG
• ZHANG, ZHIYUE

Titel

The modified composite Gauss type rules for singular integrals using Puiseux expansions

Ist Teil von

• Mathematics of computation, 2017-01, Vol.86 (303), p.345-373

Ort / Verlag

American Mathematical Society

Erscheinungsjahr

2017

Link zum Volltext

Quelle

Alma/SFX Local Collection

Beschreibungen/Notizen

• This paper is devoted to designing some modified composite Gauss type rules for the integrals involving algebraic and logarithmic endpoint singularities, at which the integrands possess the Puiseux expansions in series. Firstly, the error asymptotic expansion of a general composite quadrature rule is obtained directly by using the asymptotic expansions of the partial sum of the Hurwitz zeta function and its higher derivatives. Secondly, the deduced error asymptotic expansion is applied to the composite Gauss-Legendre and Gauss-Kronrod rules. By simplifying the evaluations of the Hurwitz zeta function and its derivatives, two modified composite Gaussian rules and their error estimates are obtained. The methods can also effectively deal with infinite range singular integrals, the singular and oscillatory Fourier transforms and the Cauchy principal value integrals by simple variable transformations. The advantage of the practical algorithms is that three ways can be used to increase the accuracy of the algorithms, which are decreasing the step length of the composite rule, increasing the orders of the Puiseux expansions and increasing the number of nodes of the Gaussian quadrature rules. Finally, the excellent performance of the proposed methods is demonstrated through several typical numerical examples. It is shown that the algorithms can be used to automatically evaluate a wide range of singular integrals over finite or infinite intervals.