Multi-Dimensional Non-Autonomous Evolutionary Equation of Monge-Ampere Type

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/o3604-7902-1000-g Multi-Dimensional Non-Autonomous Evolutionary Equation of Monge-Ampere Type Rakhmelevich, I. V. Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 1. Abstract: A multi-dimensional non-autonomous evolutionary equation of Monge-Ampere type is investigated. The left side of the equation contains the first time derivative with the coefficient depending on time, spatial variables and unknown function. The right side of the equation contains the determinant of Hessian matrix. The solutions with additive and multiplicative separation of variables are found. It is shown that a sufficient condition for the existence of such solutions is the representability of the coefficient of the time derivative as a product of functions in time and spatial variables. In the case when the time derivative coefficient is a function inverse to linear combination of spatial variables with coefficients depending on time, the solutions in the form of the quadratic polynomials in spatial variables is also found. The set of solutions in the form of the linear combination of functions of spatial variables with coefficients depending on time is obtained. Some reductions of the given equation to ordinary differential equations (ODE) in the cases when unknown function depends on sum of functions of spatial variables (in particular, sum of their squares) and function of the time are considered; in this case the functional separation of variables is used. Some reductions of the given equation to PDE of lower dimension are also found. In particular, the solutions in the form of function of the time and sum of squares of spatial variables as well as the solutions in the form of sum of such functions are obtained. Keywords: evolutionary equation, Monge-Ampere equation, separation of variables, reduction, ordinary differential equation, partial differential equation Language: Russian Download the full text For citation: Rakhmelevich, I. V. Multi-Dimensional Non-Autonomous Evolutionary Equation of Monge-Ampere Type, Vladikavkaz Math. J., 2023, vol. 25, no. 1, pp. 64-80 (in Russian). DOI 10.46698/o3604-7902-1000-g + References 1. Polyanin, A. D. and Zaytsev, V. F. Handbook of Nonlinear Partial Differential Equations, 2nd Ed., Boca Raton, London, Chapman and Hall-CRC Press, 2012. 2. Rozhdestvenskii, B. L. and Yanenko, N. N. Sistemy kvazilineynykh uravneniy i ih prilozheniya k gazovoy dinamike [Systems of Quasylinear Equations and Their Applications to Gas Dynamics], Moscow, Nauka Publ., 1988 (in Russian). 3. Khabirov, S. V. Nonisentropic One-Dimensional Gas Motions Constructed with by Means of Contact Group of the Nonhomogeneous Monge-Ampere Equation, Mathematics of the USSR-Sbornik, 1992, vol. 71, no. 2, pp. 447-462. DOI: 10.1070/SM1992v071n02ABEH001405. 4. Shablovskii, O. N. Parametric Solution for the Monge-Ampere Equation and Gas Flow with Variable Entropy, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk State University Journal of Mathematics and Mechanics], 2015, no. 1(33), pp. 105-118 (in Russian). DOI: 10.17223/19988621/33/11. 5. Pogorelov, A. V. Mnogomernoe uravnenie Monga-Ampera [Multi-Dimensional Monge-Ampere Equation], Moscow, Nauka Publ., 1988 (in Russian). 6. Guberman, I. Ya. On Existence of Many Solutions of Dirichlet Problem for Multi-Dimensional Equation of Monge-Ampere Type, Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 1965, no. 4, pp. 54-63 (in Russian). 7. Rakhmelevich, I. V. Multi-Dimensional Monge-Ampere Equation with Power-Law Non-Linearities on the First Derivatives, Vestnik Voronezhskogo gosudarstvennogo universiteta. Seriya Fizika. Matematika [Proceedings of Voronezh State University. Ser. Physics. Mathematics], 2020, no. 2, pp. 86-98. 8. Ibragimov, N. H. CRC Handbook of Lie Groups to Differential Equations. Vol. 1. Symmetries, Exact Solutions and Conservation Laws, Boca Raton, CRC Press, 1994, 429 p. 9. Rakhmelevich, I. V. On the Solutions of Two-Dimensional Monge-Ampere Equation with Power-Law Non-Linearity on the First Derivatives, Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mekhanika [Tomsk State University Journal of Mathematics and Mechanics], 2016, no. 4(42), pp. 33-43 (in Russian). DOI: 10.17223/19988621/42/4. 10. Galaktionov, V. A. and Svirshchevskii, S. R. Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Boca Raton, Chapman and Hall/CRC Press, 2006. 11. Gutierres, C. E. The Monge-Ampere Equation, Boston, Birkhauser, 2001. 12. Taylor, M. E. Partial Differential Equations III. Nonlinear Equations, New York, Springer-Verlag, 1996. 13. Polyanin, A. D. and Zhurov, A. I. Metody razdeleniya peremennykh i tochnye resheniya nelineynykh uravneniy matematicheskoy fiziki [Variables Separation Methods and Exact Solutions of Nonlinear Equations in Mathematical Physics], Moscow, IPMech RAN Publ., 2020 (in Russian). 14. Faddeev, D. K. and Sominsky, I. S. Sbornik zadach po vyshey algebre [Collection of Problems on Highest Algebra], Moscow, 1977 (in Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). 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