A fast algorithm for Prandtl's integro-differential equation (Q674422)

The authors obtain optimal convergence rates for collocation and quadrature methods for singular integral equations of the form \[ g(x)v(x)- {1\over \pi} \int^1_{-1} {v(t) \over (t-x)^2} dt+ {1\over \pi} \int^1_{-1} h(x,t) v(t)dt= f(x),\;-1<x<1, \tag{*} \] in weighted Sobolev norms. They develop the idea of a fast algorithm for the numerical solution of (*) based on the quadrature method. Some numerical examples are given to illustrate the results.