Geodesic
On some overdetermined systems of three differential equations in second-order partial derivatives
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 4, pp. 544-556 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper considers carefully a linear system of differential equations in partial derivatives from the deformable solid body mechanics and a class of systems of three equations including three second-order derivatives of one unknown function on the right-hand side in a non-linear manner. Using replacement of the first and second orders in the right parts with new unknown functions, transition to systems with a larger number of unknown functions has been performed and links with systems in total differentials have been established. For the studied overdetermined systems, the explicit compatibility conditions have been found and the variety of solutions containing not more than seven arbitrary constants have been determined. The obtained results, though mainly theoretical, can be used for further development of the theory of systems of differential equations with partial derivatives to solve specific problems of hydrodynamics (i.e., for construction of exact (“test”) solutions of the plane problems of hydrodynamics), gas dynamics, theory of elasticity, as well as a number of branches of mechanics, physics, and other basic sciences.
Keywords: compatibility conditions, variety of solutions, overdetermined systems. 
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R. Pirov. On some overdetermined systems of three differential equations in second-order partial derivatives. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 158 (2016) no. 4, pp. 544-556. http://geodesic.mathdoc.fr/item/UZKU_2016_158_4_a6/

[1] Goursat E., Leçonq sur l'intéqration des équations aux dérivées partielles du premier ordre, Paris, 1921, 454 pp. (In French) | MR

[2] Mikhailov L. G., Some Overdetermined Systems of Partial Differential Equations with Two Unknown Functions, Donish, Dushanbe, 1986, 116 pp. (In Russian) | MR

[3] Pirov R., “The study of certain nonlinear systems of second-order partial differential equations with one unknown function on the plane”, Kraiovy Zadachi Difer. Rivnyan', 14, Prut, Chernivtsi, 2006, 313–320 (In Russian)

[4] Hartman F., Ordinary Differential Equations, Mir, Moscow, 1970, 720 pp. (In Russian) | MR

[5] Il'yushin G. L., Plasticity, GITTL, Moscow, 1948, 479 pp. (In Russian)

[6] Brovko G. L., “The necessary and sufficient conditions for simple uniform-deformation”, Prikl. Mat. Mekh., 42 (1978), 701–710 (In Russian) | MR

[7] Lenskaya S. E., “On inhomogeneously simple processes”, Vestn. Mosk. Univ., Ser. 1: Mat., Mekh., 1988, no. 1, 100–103 (In Russian) | Zbl

[8] Pirov R., On an Overdetermined System of Second-Order Partial Differential Equations, Dep. Tadzh. NIINTI: June 19, 1989, no. 22(622), Dushanbe, 1989, 15 pp. (In Russian)

[9] Pirov R., “On the compatibility conditions and manifold solutions for one class of nonlinear overdetermined systems of differential equations in second-order partial derivatives in space”, Vestn. Inst. Predprinimatel'stva Servisa (Dushanbe), 2010, no. 20, 89–95 (In Russian)

[10] Pirov R., “On the compatibility of some overdetermined systems of equations in second-order partial derivatives with two unknown functions”, Dokl. Akad. Nauk Resp. Tadzh., 54:5 (2011), 359–366 (In Russian)