The airfoil integral equation over disjoint intervals: analytic solutions and asymptotic expansions

The airfoil integral equation over two disjoint intervals, separated by a distance \(\kappa \) $\kappa$ , is considered. An analytic solution is obtained in terms of a polynomial series that satisfies a generalized three-term recurrence relation, with the solution being particularly tailored for cases where the input function is a Chebyshev polynomial of the first kind. 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 \(\kappa \rightarrow 0\) $\kappa \to 0$ , we derive approximate solutions for the airfoil equation and exhibit the first terms of an asymptotic expansion of them in power series of \(\kappa \) $\kappa$ within weighted Sobolev spaces. The paper further elaborates on the implementation of spectral methods, specifically the collocation and Galerkin versions, for solving the general airfoil integral equation over two disjoint intervals.