| Summary: | Background. The work is devoted to the study of sets of functions in which the
condition for the unique solvability of degenerate singular integral equations is satisfied.
At present, the study of many sections of singular integral equations can be
considered completed. An exception is singular integral equations that vanish on
manifolds with a measure greater than zero. The theory of singular integral equations
in degenerate cases is constructed, from which it follows that, firstly, degenerate
singular integral equations have an infinite number of solutions; secondly, the
first and second Noether theorems are not valid for these equations. But specific algorithms
and approximate methods for solving singular integral equations in degenerate
cases are absent. Due to the fact that many processes in physics and technology
are modeled by degenerate singular integral equations, it becomes necessary to
develop approximate methods for solving them. In addition, since in the Holder
space and in the space L2 of functions summable in a square, degenerate singular
integral equations have an infinite number of solutions, the actual problem of distinguishing
the uniqueness sets of the solutions of these equations arises, as well as the
equally urgent problem of constructing approximate methods for solving them.
Materials and methods. To distinguish classes of functions in which degenerate
singular integral equations have a unique solution, methods of the theory of functions
of a complex variable, Riemann boundary value problems, and the theory of
singular integral equations are used. When constructing approximate methods, iterative-
projection methods are used.
Results. Classes of functions are constructed on which solutions, if they exist,
are uniquely determined. In this regard, a new formulation of the solution of degenerate
singular integral equations is proposed. Collocation and mechanical quadrature
methods for solving degenerate singular integral equations on the constructed classes
of functions are proposed and substantiated.
Conclusions. The proposed results can be directly used in solving many problems
of physics and technology, in particular, in the problems of integral geometry,
aerodynamics, and hydrodynamics. It is of interest to extend these results to degenerate
polysingular integral equations.
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