Integration of a Linear Equation with Differential Operator, Corresponding to the Main Diagonal in the Space of Independent Variables, and Coefficients, Constant on the Diagonal

On One Method of Research of Multiperiodic Solution of Block-Matrix Type System With Various Differentiation Operators

“…Therefore, the development of methods for solving multiperiodic solutions problems of such systems is at the initial stage of its development. Some ideas of these works methods, based on researches [14][15][16][17], are extended in [18,19] to study multiperiodic solutions problems of quasilinear equations systems with various differentiation operators along their characteristics. The questions of multiperiodic solutions of quasilinear equations systems with various differentiation operators are studied in [5] in terms of the matricant when the matrix of coefficients is block-diagonal.…”

“…Note that the developed method can be generalized to the quasilinear case when the coefficients of linear part are multiperiodic. Meanwhile, it will be necessary to use generalizations of methods of works [4][5][6][7][8][9][10], [14][15][16] for the case under consideration. уақыт және кеңістік тәуелсіз айнымалылары бойынша көппериодты шешімінің бар болатындығын зерттеудің жаңа тәсілдемесін құруға әкелді.…”

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10] and applied to the study of solutions different problems in the partial differential equations [11,12]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12], in particular, problems on multi-frequency solutions of equations from control theory [13]. The methods of research for multiperiodic solutions are successfully combined by methods for studying solutions of boundary value problems for equations of mathematical physics.…”

Research of Multiperiodic Solutions of Perturbed Linear Autonomous Systems With Differentiation Operator on the Vector Field

“…The foundations of the method used in this note were laid in [1,2], which were further developed in [3][4][5][6][7][8][9][10][11][12][13][14] and applied to the study of solutions different problems in the partial differential equations [15,16]. These methods with simple modifications extend to the study solutions of problems of the differential and integro-differential equations of different types [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16], in particular, problems on multi-frequency solutions of equations from control theory [17]. Many oscillatory phenomena are described by systems with a differentiation operator with respect to toroidal vector fields, and new methods based on the ideas of the Fourier [18], Poincaré-Lyapunov and Hamilton-Jacobi methods [19,20] appear to establish their periodic oscillatory solutions.…”