 +7 (3412) 91 60 92 imi@udsu.ru

## Archive of Issues

Uzbekistan Tashkent
Year
2019
Volume
29
Issue
1
Pages
52-60
 Section Mathematics Title Invariant solutions of the two-dimensional heat equation Author(-s) Narmanov O.A.a Affiliations Tashkent University of Information Technologya Abstract The symmetry group of a given differential equation is the group of transformations that translate the solutions of the equation into solutions. If the infinitesimal generators of symmetry groups are known, then we can find solutions that are invariant under this group. For systems of partial differential equations, the symmetry group can be used to explicitly find particular types of solutions that are themselves invariant under a certain subgroup of the full symmetry group of the system. For example, solutions of an equation with partial derivatives of two independent variables, invariant under a given one-parameter symmetry group, are found by solving a system of ordinary differential equations. The class of solutions that are invariant with respect to a group includes many exact solutions that have immediate mathematical or physical meaning. In this paper, using the well-known infinitesimal generators of some symmetry groups of the two-dimensional heat conduction equation, solutions are found that are invariant with respect to these groups. First we consider the two-dimensional heat conduction equation with a source that describes the process of heat propagation in a flat region. For this case, a family of exact solutions was found, depending on an arbitrary constant. Then invariant solutions of the two-dimensional heat conduction equation without source are found. Keywords symmetry group, heat equation, infinitesimal generator, vector field UDC 517.958 MSC 35Q79, 35K05 DOI 10.20537/vm190105 Received 20 December 2018 Language Russian Citation Narmanov O.A. Invariant solutions of the two-dimensional heat equation, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 1, pp. 52-60. References Lie S., Sheffers G. Simmetrii differentsial'nykh uravnenii. Tom 1. Lektsii o differentsial'nykh uravneniyakh s izvestnymi infinitezimal'nymi preobrazovaniyami (Symmetries of differential equations. Vol. 1. Lectures on differential equations with known infinitesimal transformations), Moscow-Izhevsk: Regular and Chaotic Dynamics, 2011. Lie S., Sheffers G. Simmetrii differentsial'nykh uravnenii. Tom 3. Geometriya kontaktnykh preobrazovanii (Symmetries of differential equations. Vol. 3. Geometry of contact transformations), Izhevsk: Regular and Chaotic Dynamics, 2011. Gainetdinova A.A. Integration of systems of ordinary differential equations with a small parameter which admit approximate Lie algebras, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 143-160 (in Russian). https://doi.org/10.20537/vm180202 Ayub M., Khan M., Mahomed F.M. Second-order systems of ODEs admitting three-dimensional Lie algebras and integrability, Journal of Applied Mathematics, 2013, vol. 2013, article ID 147921, 15 p. https://doi.org/10.1155/2013/147921 Gainetdinova A.A., Gazizov R.K. Integrability of systems of two second-order ordinary differential equations admitting four-dimensional Lie algebras, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 2017, vol. 473, issue 2197, 20160461. https://doi.org/10.1098/rspa.2016.0461 Wafo Soh C., Mahomed F.M. Reduction of order for systems of ordinary differential equations, Journal of Nonlinear Mathematical Physics, 2004, vol. 11, issue 1, pp. 13-20. https://doi.org/10.2991/jnmp.2004.11.1.3 Dorodnitsyn V.A., Knyazeva I.V., Svirshchevskij S.R. Group properties of the heat-conduction equation with a source in the two- and three-dimensional cases, Differential Equations, 1983, vol. 19, pp. 901-908. https://zbmath.org/?q=an:0541.35036 Olver P.J. Applications of Lie groups to differential equations, Springer, 1986, 513 p. Ovsiannikov L.V. Group analysis of differential equations, Academic Press, 1982, 432 p. https://doi.org/10.1016/C2013-0-07470-1 Narmanov O.A. Lie algebra of infinitesimal generators of the symmetry group of the heat equation, Journal of Applied Mathematics and Physics, 2018, vol. 6, no. 2, pp. 373-381. https://doi.org/10.4236/jamp.2018.62035 Samarskii А.А., Galaktionov V.А., Kurdyumov S.P., Mikhailov А.P. Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Moscow: Nauka, 1987, 481 p. Full text