Quadrature methods for integro-differential equations of Prandtl's type in weighted spaces of continuous functions

TL;DR

This paper introduces quadrature methods using optimal Lagrange interpolation for solving Prandtl-type integro-differential equations, demonstrating stability, convergence, and practical effectiveness through numerical experiments and an application to airflow around wing profiles.

Contribution

It proposes new quadrature methods with proven stability and convergence for Prandtl-type equations in weighted spaces, including practical implementation and testing.

Findings

Methods are stable and convergent under certain conditions.

Numerical experiments show efficiency and accuracy.

Application to airflow around wing profiles demonstrates practical utility.

Abstract

The paper deals with the approximate solution of integro-differential equations of Prandtl's type. Quadrature methods involving ``optimal'' Lagrange interpolation processes are proposed and conditions under which they are stable and convergent in suitable weighted spaces of continuous functions are proved. The efficiency of the method has been tested by some numerical experiments, some of them including comparisons with other numerical procedures. In particular, as an application, we have implemented the method for solving Prandtl's equation governing the circulation air flow along the contour of a plane wing profile, in the case of elliptic or rectangular wing-shape.