О явном решении краевой задачи типа Неймана для обобщенных аналитических функций в единичном круге

Abstract

To research the boundary value problems in some classes of functions of a complex variable and to develop the effective numerical solving methods for these problems, the problem of the explicit solution is of substantial significance, i.e. the possibility of solving these problems with formulas of the classical Riemann problem and Hilbert problem for analytical functions and finite number of linear algebraic equations and/or linear differential equations when the matrix of the system can be written in quadratures. In this article, on the complex plane we consider one family of differential equations with second-order partial derivatives and a coefficient at the sought-for function, depending on a natural parameter n . Solutions of these equations are commonly called generalized analytic functions of n order. In addition, in this article we give general formulation of the boundary value problem of Neumann type for the generalized analytic functions of the arbitrary order n . We obtain a new method of solving of the formulated problem for the generalized analytic functions of the first order in case when boundary is the unit circle. It is established that in the case under consideration the solution of the Neumann type problem for the generalized analytic functions is reduced to the consecutive solution of simple scalar Riemann problem in the class of limited at infinity piecewise analytic functions and Euler linear differential equation of the second order. The obtained general results are given as applied to a specific example.