AN APPROXIMATE METHODS FOR SOLVING POLYSINGULAR INTEGRAL EQUATIONS IN DEGENERATE CASES

Abstract

Background. This work is devoted to the study of sets of functions in which the condition of unique solvability of degenerate polysingular integral equations is satisfied, and to the construction of approximate methods for solving polysingular integral equations in degenerate cases. Nowadays, the study of many sections of singular integral equations can be considered largely completed. Some of the exceptions are singular and polysingular integral equations that vanish on manifolds with measure greater than zero. The theory of singular integral equations in degenerate cases is constructed, from which it follows that degenerate singular integral equations have an infinite number of solutions and for these equations the first and second Noether theorems are not valid. For polysingular integral equations, a similar theory has not yet been constructed. Moreover, there are no specific algorithms and approximate methods for solving polysingular integral equations in degenerate cases. Due to the fact that many processes in physics and technology are modeled by degenerate polysingular integral equations, it becomes necessary to develop approximate methods for their solution. In addition, since in the Hölder space and in the space of functions summable in a square, the degenerate polysingular integral equations have an infinite number of solutions, the actual problem of identifying the sets of uniqueness of solutions of these equations arises. The problem of constructing approximate methods for solving degenerate polysingular integral equations is no less urgent. Materials and methods. To distinguish classes of functions in which degenerate polysingular 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, iterativeprojection methods are used. Results. Classes of functions are constructed on which solutions of degenerate polysingular integral equations, if they exist, are uniquely determined. In this regard, a new formulation of the problem of solving degenerate polysingular integral equations is proposed. Methods of collocation and mechanical quadratures for solving degenerate polysingular 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 problems of integral geometry, aerodynamics, hydrodynamics. It is of interest to extend these results to degenerate multidimensional singular integral equations.

Keywords

• polysingular integral equations
• degenerate symbol
• uniqueness
• projection- iterative method