On the numerical solution of Prandtl's integral equation (Q1121665)

Prandtl's singular integrodifferential equation of the form \[ \Gamma (x)/\beta (x)-(1/2\pi)\int^{1}_{-1}-\Gamma '(t)dt/(t-x)=f(x),\quad - 1<x<1 \] with the conditions \(\Gamma (1)=\Gamma (-1)=0\) is considered. Here, \(\Gamma\) is the unknown function, and B and f are known functions. For a numerical solution of that equation the authors propose a method based on the Jacobi-Chebyshev quadrature rule and suggest a natural interpolation formula for the approximation of the unknown function. Numerical applications are given in 6 examples.