The solution of singular integral equations by approximate projection methods

The approximate solution of singular equations of the 1st and 2nd kinds, when the line of integration is a segment, is considered. By contraction of the domain of definition or range of values of the operator, the one-to-one property of the mapping is established. Versions of the Bubnov-Galerkin method are used for the approximation solution. Chebyshev and Jacobi polynomials are used as coordinate elements. It is shown that the algebraic system is uniquely solvable for fairly large $n$, and that the approximate solutions converge to the exact solution in spaces with a weight. The process is stable.