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A systematic formula for the asymptotic expansion of singular integrals

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Abstract

Asymptotic theories like the lifting-line, the slender body or the slender ship lead to lineintegrals with singular kernels. Sometimes these integrals are “improper”, that is to say that they are defined only by their Finite Part. To find asymptotic expansions of these integrals, the Matched Asymptotic Expansion Method is widely used along with other more specific methods depending on the kernel type. The first method is laborious and not systematic, and the other methods are sometimes too much specific to treat general cases. Moreover, all of them are not well adapted to deal with Finite Part integrals.

Here, a new method is proposed to avoid the previous difficulties. This method is systematic for homogeneous kernels and gives approximations up to any order, as long as the derivative of the weight function exists at this given order. Moreover the occurrence of logarithmic terms in the expansion is explained and easily predictable. An elliptic integral and the classical lifting-line theory are treated to illustrate the ease of this method.

Résumé

Les théories asymptotiques telles que la ligne portante, le corps élancé ou le navire de grand allongement conduisent à des intégrales curvilignes à noyaux singuliers. Parfois, ces intégrales sont “impropres” c'est à dire qu'elles sont définies en Parties Finies. Différentes méthodes ont été mises au point pour trouver les développements asymptotiques de ces intégrales. Généralement elles dépendent fortement de la nature du noyau, et c'est finalement la méthode des développements raccordés qui est utilisées quand le noyau est trop compliqué. Cependant, cette méthode est laborieuse et comme les précèdentes non adaptée aux intégrales défines par leur Partie Finie.

Une nouvelle méthode est proposée pour surmonter ces difficultés. Cette méthode est systématique pour les noyaux homogènes et donne les approximations à tout ordre pourvu que les dérivées de la fonction poids existent jusqu'à cet ordre. De plus la présence de termes logarithmiques dans le développement est expliquée et aisément prédictible.

Une intégrale elliptique, ainsi que la fameuse théorie de la ligne portante sont traités pour illustrer les possibilités de la méthode.

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Abbreviations

D :

domain of integration

f(x):

weight function

FP :

Finite Part

h(x):

weight function

I ɛ,I o :

bounded intervals

j, J :

integers

K(x, ε):

singular kernel

L, L :

integers

M :

integer defining the approximation order

P k (x):

Legendre polynomial

R :

set of real numbers

R(ß):

equals 1 ifβ is an integer and 0 if not

R f, J ,R K, L :

remainders of Taylor developments

S (α):

equals either 1 or the sign function:sgn(α)

t, u, v, x :

variable of integration

α, λ:

real numbers

β :

homogeneity order of the kernel

F (α):

Euler's integral (gamma function)

ε:

“small” parameter

[.]:

integer part of

References

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Author notes

  1. Jean Luc Guermond

    Present address: Bassin d'Essais des Carènes de Paris, France

Authors and Affiliations

  1. Visiting scientist at the Department of Ocean Engineering, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Jean Luc Guermond

Authors
  1. Jean Luc Guermond
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Guermond, J.L. A systematic formula for the asymptotic expansion of singular integrals. Journal of Applied Mathematics and Physics (ZAMP) 38, 717–729 (1987). https://doi.org/10.1007/BF00948292

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  • DOI: https://doi.org/10.1007/BF00948292

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