ON ONE METHOD OF RESEARCH OF MULTIPERIODIC SOLUTION OF BLOCK-MATRIX TYPE SYSTEM WITH VARIOUS DIFFERENTIATION OPERATORS

Аннотация

There is researched the problem of existence and integral representation of a unique multiperiodic solution in all independent variables of a linear system with constant coefficients and with various differentiation operators in the direction of a vector field. Based on the Cauchy characteristics method, a methodology is developed for constructing solutions of initial problem for a linear system with constant coefficients and various special differentiation operators along two straight lines of the independent variables space, where integration characteristics are determined using a projector. It is given a methodology for constructing a matrix of homogeneous block-triangular system, as well as a matricant of a homogeneous linear system in the general case when a Jordan block is split into the sum of two sub-blocks. The Cauchy problems for linear homogeneous and nonhomogeneous systems with integral representation are solved using this methodology. At the same time, the introduced projectors for determining characteristics were of significant importance. Along with the construction of general solutions of linear systems with two differentiation operators, a theorem on the conditions of multiperiodicity of their solutions is proved. On their basis, in noncritical case, the theorem on existence and uniqueness of a multiperiodic solution of linear nonhomogeneous system is proved and its integral representation is given. The developed methodology has the perspective of extending the results obtained to the quasilinear case of system under consideration, as well as to the cases of a system with n various differentiation operators and multiperiodic matrices with partial derivatives of the desired vector function.