Geodesic
On multiperiodic integrals of a linear system with the differentiation operator in the direction of the main diagonal in the space of independent variables
Eurasian mathematical journal, Tome 8 (2017) no. 1, pp. 67-75 
A. A. Kulzhumiyeva; Zh. A. Sartabanov. On multiperiodic integrals of a linear system with the differentiation operator in the direction of the main diagonal in the space of independent variables. Eurasian mathematical journal, Tome 8 (2017) no. 1, pp. 67-75. http://geodesic.mathdoc.fr/item/EMJ_2017_8_1_a6/
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Voir la notice de l'article provenant de la source Math-Net.Ru

In the general theory of first order partial differential equations one of the effective ways of integration is a common method of finding solutions by the complete integral [4–5]. In this note we propose a method of research of problems of multiperiodic solutions of linear systems of equations with the same differentiation operator in the direction of a vector field, which is based on the results of the study multiperiodicity of their complete integral. Such an approach is not found in earlier studies based on the methods of fundamental works [8–9] on multiperiodic solutions of such systems. The elements of the proposed method are used in [1–3, 6–7]. In this note in order to establish the multiperiodicity of a complete integral Green's function is introduced in the absence of non-trivial integral multiperiodic solutions of homogeneous system.

[1] A. A. Kulzhumiyeva, Zh. A. Sartabanov, “Periodic on multivariate time of solutions of the systems equations with differentiation operator on direction of the vector field”, Eurasian Math. J., 2007, no. 1, 62–72 (in Russian) | MR

[2] A. A. Kulzhumiyeva, “Study of periodic solutions is rectangulared to canonical type of the systems with linear differential operator on multivariate time”, Eurasian Math. J., 2008, no. 2, 69–73 (in Russian)

[3] A. A. Kulzhumiyeva, Zh. A. Sartabanov, Periodic solutions of systems of differential equations with multivariate time, Uralsk, 2013 (in Russian)

[4] R. Courant, Partial differential equations, M., 1964 (in Russian) | MR

[5] P. S. Nazimov, On integration of differential equations with partial derivatives (physico-mathematical heritage), M., 2015 (in Russian)

[6] Zh. A. Sartabanov, “About one way of the study of periodic solutions of the partial differential equation special type”, Izvestiya AN KazSSR, Seriya Physico-mathematical, 1989, no. 1, 42–48 (in Russian) | MR | Zbl

[7] Zh. A. Sartabanov, A. A. Kulzhumiyeva, B. Zh. Muhambetova, “The structure of the general integral and multiperiodic solution of one quasilinear system with differential operator in the partial derivatives”, Vestnik of Karaganda University, 78:2 (2015), 112–117 (in Russian)

[8] D. U. Umbetzhanov, Almost multiperiodic solutions of the partial differential equations, Alma-Ata, 1979 (in Russian) | MR

[9] V. H. Harasahal, Almost periodic solutions of the ordinary differential equations, Alma-Ata, 1970 (in Russian) | MR