Numerical solution of singular integro-differential Prandtl equation by the method of orthogonal polynomials

The paper is constructed and proved computational scheme for a solution of singular Prandtl of integro-differential equations with singular integral over the interval of the real axis, understood in the sense of the Cauchy principal value. This equation reduces to the equivalent Fredholm equation of the second kind by inversion of the singular integral in the class of functions unbounded at the ends and the application of spectral relations for the singular integral. At the same time we investigate the condition of solvability of a Fredholm integral equation of the second kind with a logarithmic kernel of a special kind. The new computational scheme is based on applying spectral relations for the singular integral to the integral entering into the equivalent equation. Uniform estimates of the errors of approximate solutions are obtained.