On a biharmonic problem in a circular ring

Rogosin, Sergei; Vaitekhovich, Tatyana

doi:10.1524/anly.2010.0935

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“…Complex analytic methods are applied also in [33] and in Vaitekhovich [36], where explicit solutions of boundary value problems for complex differential equations are obtained in the case of 'standard' domains (in particular, for the circular ring domain).…”

Section: Introductionmentioning

confidence: 99%

Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole

Riva

Musolino

Rogosin

2015

Asymptotic Analysis

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We consider the Dirichlet problem for the Laplace equation in a planar domain with a small hole. The diameter of the hole is proportional to a real parameter ε and we denote by uε the corresponding solution. If p is a point of the domain, then for ε small we write uε(p) as a convergent power series of ε and of 1/(r 0 + (2π) −1 log |ε|), with r 0 ∈ R. The coefficients of such series are given in terms of solutions of recursive systems of integral equations. We obtain a simplified expression for the series expansion of uε(p) in the case of a ring domain.

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Section: Introductionmentioning

confidence: 99%