APPROXIMATE SOLUTION OF SINGULAR INTEGRAL EQUATIONS ON THE END OF UNLIMITED INTEGRATION OF CHEBYSHEV SERIES
A computational scheme is constructed for the approximate solution of a singular integral equation of the first kind of an unbounded integration segment at the ends [- 1, 1]. The solution of the equation is sought in the form of a series in Chebyshev polynomials of the first kind. The kernel and the right-hand side of the equation decompose into series using the Chebyshev polynomials of the first kind, whose coefficients are calculated approximately by Gaussian quadrature formulas. For the coefficients of the decomposition of Chebyshev polynomials of the second kind into series in Chebyshev polynomials of the first kind, exact values are found. The coefficients of the expansion of the unknown function, that is, the solution, are found from the solution of linear algebraic equations. To justify the computational scheme, methods of functional analysis and the theory of orthogonal polynomials are used. We introduce the space of Holder functions with the corresponding norms. In this space, we consider the given singular and corresponding approximate operators. Conditions for the existence of an inverse singular operator are given and the existence of an inverse approximate operator is proved. When the existence condition for the given functions having derivatives up to some order belonging to the Holder class is satisfied, the error in the computation is evaluated and the order of its tendency to zero is given.