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 oneparameter 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 wellknown infinitesimal generators of some symmetry groups of the twodimensional heat conduction equation, solutions are found that are invariant with respect to these groups. First we consider the twodimensional 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 twodimensional heat conduction equation without source are found.

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), MoscowIzhevsk: 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. 143160 (in Russian). https://doi.org/10.20537/vm180202
 Ayub M., Khan M., Mahomed F.M. Secondorder systems of ODEs admitting threedimensional 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 secondorder ordinary differential equations admitting fourdimensional 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. 1320. https://doi.org/10.2991/jnmp.2004.11.1.3
 Dorodnitsyn V.A., Knyazeva I.V., Svirshchevskij S.R. Group properties of the heatconduction equation with a source in the two and threedimensional cases, Differential Equations, 1983, vol. 19, pp. 901908. 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/C20130074701
 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. 373381. 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.
