Direct spectral Chebyshev method for solving the generalized wing equation

The generalized Prandtl integro-differential equation is considered, in which the singular integrals contain both the unknown solution and its derivative. Direct numerical methods for solving this problem are constructed, based on representing the solution and the variable coefficients of the equation in the form of Chebyshev interpolation polynomials. The developed approach, which utilizes known spectral relations, makes it possible to obtain analytical expressions for the singular component of the problem and reduces its solution to a system of linear algebraic equations for the coefficients of the interpolation polynomial. Two computational schemes for the solution are constructed using Chebyshev polynomials of the first and second kind, respectively. Using a model problem as an example, it is shown that both schemes ensure good conditioning of the algebraic problem and possess high accuracy, demonstrating an exponential convergence rate of the error.