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Improvement of the asymptotic behaviour of the Euler–Maclaurin formula for Cauchy principal value and Hadamard finite‐part integrals
Author(s) -
Choi U. Jin,
Kim Shin Wook,
Yun Beong In
Publication year - 2004
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1077
Subject(s) - hadamard transform , mathematics , cauchy principal value , euler's formula , sigmoid function , parametric statistics , transformation (genetics) , principal part , cauchy distribution , mathematical analysis , cauchy boundary condition , statistics , computer science , biochemistry , chemistry , machine learning , artificial neural network , gene , boundary value problem , free boundary problem
In the recent works ( Commun. Numer. Meth. Engng 2001; 17 : 881; to appear), the superiority of the non‐linear transformations containing a real parameter b ≠ 0 has been demonstrated in numerical evaluation of weakly singular integrals. Based on these transformations, we define a so‐called parametric sigmoidal transformation and employ it to evaluate the Cauchy principal value and Hadamard finite‐part integrals by using the Euler–Maclaurin formula. Better approximation is expected due to the prominent properties of the parametric sigmoidal transformation of whose local behaviour near x = 0 is governed by parameter b . Through the asymptotic error analysis of the Euler–Maclaurin formula using the parametric sigmoidal transformation, we can observe that it provides a distinct improvement on its predecessors using traditional sigmoidal transformations. Numerical results of some examples show the availability of the present method. Copyright © 2004 John Wiley & Sons, Ltd.