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Pré-Publication, Document De Travail Année : 2023

The Airfoil Integral Equation over Disjoint Intervals: Analytic Solutions and Asymptotic Expansions

Résumé

The airfoil integral equation over two disjoint intervals, separated by a distance $\kappa$, is considered. An analytic solution is obtained in terms of polynomials that satisfy a generalized three-term recurrence relation, particularly when utilizing a Chebyshev polynomial of the first kind as the input function. A new class of polynomials is defined on two disjoint intervals, demonstrating a generalization of a classical integral relationship previously established for Chebyshev polynomials on a single interval. The solutions are efficiently calculated, and comparisons with the original problem, defined on a continuous domain, are presented. Additionally, as $\kappa \rightarrow 0$, we derive approximate solutions for the Airfoil equation and exhibit the first terms of an asymptotic expansion of them in power series of $\kappa$ within weighted Sobolev spaces. The paper also outlines a spectral method for the general airfoil integral equation over two disjoint intervals.
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Dates et versions

hal-04237680 , version 1 (11-10-2023)

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  • HAL Id : hal-04237680 , version 1

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Leandro Farina, Marcos Ferreira, Victor Péron. The Airfoil Integral Equation over Disjoint Intervals: Analytic Solutions and Asymptotic Expansions. 2023. ⟨hal-04237680⟩
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