Justification of Galerkin and collocations methods for one class of singular integro-differential equations on interval

We justify the Galerkin and collocations methods for one class of singular integro-differential equations defined on the pair of the weighted Sobolev spaces. The exact solution of the considered equation is approximated by the linear combinations of the Chebyshev polynomials of the first kind. According to the Galerkin method, we equate the Fourier coefficients with respect to the Chebyshev polynomials of the second kind in the right-hand side and the left-hand side of the equation. According to collocations method, we equate the values of the right-hand side and the left-hand side of the equation at the nodes being the roots of the Chebyshev polynomials the second kind.
The choice of the first kind Chebyshev polynomials as coordinate functions is due to the possibility to calculate explicitly the singular integrals with Cauchy kernel of the products of these polynomials and corresponding weight functions. This allows us to construct simple well converging methods for the wide class of singular integro-differential equations on the interval $(-1,1)$.
The Galerkin method is justified by the Gabdulkhaev–Kantorovich technique. The convergence of collocations method is proved by the Arnold–Wendland technique as a consequence of convergence of the Galerkin method. Thus, the covergence of both methods is proved and effective estimates for the errors are obtained.