The airfoil integral equation over disjoint intervals: analytic solutions and asymptotic expansions (Q6608141)

In the article the authors consider the airfoil integral equation over two disjoint intervals, separated by the set \(\{k\}\) is considered, where \(k\) is a real number. They obtain particular polynomial solutions of the airfoil equation defined over the domain \((-1, -k)\cup (k, 1)\) for an arbitrary \(k\), when the input function \(g\) is a Chebyshev polynomial of the first kind. This 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, demonstrate the generalization of a classical integral relationship previously established for Chebyshev polynomials on a single interval. The solutions are efficiently calculated, and are compared with the solutions of the original problem, which is continuous domain, are presented. Additionally, as \(k\to 0\), we derive approximate solutions for the airfoil equation and exhibit the first terms of an asymptotic expansion of them in power series of \(\kappa\) within weighted Sobolev spaces.