Abstract
We consider nonlinear systems of five partial differential equations of the second order with one and two unknown functions. We obtain explicit conditions of compatibility which provide the unique solvability of problems with initial data.
References
Goursat, É. Leçons sur l’Intégration des équations aux Dérivés Partielles du Premier Ordre (J.Hermann, Paris, 1921).
Mikhailov, L. G. Certain Overdetermined Systems of Partial Differential Equations with Two Unknown Functions (Donish, Dushanbe, 1986) [in Russian].
Pirov, R. “Investigations of Certain Overdetermined Nonlinear Systems of Partial Differential Equations of the FirstOrder with Two Unknown Functions on a Plane”, Candidate’s Dissertation inMathematics and Physics (Dushanbe, 1984).
Pirov, R. “Investigation of Certain Nonlinear Systems of Partial Differential Equations of the Second Order with One Unknown Function on a Plane” Kraiovi Zadachi dlya Differentsialnykh Rivnyan’, No. 14, 313–320 (Prut, Chernivtsi, 2006).
Pirov, R. and Rakhimov, F. Sh. “On Conditions of Compatibility and Diversity of Solutions of Certain Systems of Five Partial Differential Equations of the Second Order on a Plane”, Vestnik Ped. Univ., 6–7 (Dushanbe, 2007).
Pirov, R. “On the Theory of Nonlinear Systems of Three and Four Partial Differential Equations of the Second Order with One Unknown Function in a Space”, in Proceedings of International Conference ‘Functional Spaces and Theory of Approximations of Functions’, dedicated to 110th Anniversary of Acadamician S.M. Nikol’ckii, Mathem. Inst. of RAS, 199–201 (Moscow, 2015).
Pirov, R. “Conditions of Compatibility and Diversity of Solutions of One Nonlinear System of Five Partial Differential Equations of Second Order in a Space”, in Transactions of N. I. Lobachevskii Math. Center ‘Theory of Function, its Applications and Related Questions’, 51, 348–352 (Kazan, 2015).
Pirov, R. “On the Theory of Nonlinear Overdetermined Systems Consisting of Three and Four Partial Differential Equations of the Second Order with One Unknown Function in a Space”, Izvestiya AN RT. Otd. fiz-mat., khim., geog. tekhn. n., No. 1 (158), 32–41 (2015).
Hartman, P. Ordinary Differential Equations (New York, Wiley, 1964; Mir, Moscow, 1970).
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Original Russian Text © R. Pirov, 2017, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2017, No. 1, pp. 97–100.
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Pirov, R. On certain nonlinear systems of partial differential equations of the second order. Russ Math. 61, 85–87 (2017). https://doi.org/10.3103/S1066369X1701011X
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DOI: https://doi.org/10.3103/S1066369X1701011X