Solutions of the boundary value problem of the Carleman type in classes of generalized meta-analytic functions in a unit circle

This paper researches the boundary value problems of the Carleman type in classes of generalized meta-analytic functions of a complex variable and develops effective numerical methods to solve these problems. The explicit solution is of substantial significance. In other words there is the possibility of solving these problems with formulas of the classical Carleman problem for analytical functions and a finite number of linear algebraic equations and/or linear differential equations when the matrix of the system can be written in quadratures. The paper considers one of the main boundary value problems of the Carleman type in classes of generalized meta-analytic functions in simply connected domains. Using a representation of generalized meta-analytic functions using a pair of analytic functions of a complex variable, a constructive algorithm for an explicit method for solving the problem is established in the case when the unit circle is the carrier of the boundary conditions.
The paper proves that the solution of the boundary value problem in the unit circle is reduced to the solution of two classical boundary value problems of the Carleman type for analytic functions and some systems of algebraic equations. In addition, the paper describes the solvability of the boundary value problem in the unit circle and obtains conditions for its Noetherian property.