Geodesic
Invariant solutions of the two-dimensional heat equation
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 1, pp. 52-60
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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. 
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O. A. Narmanov. Invariant solutions of the two-dimensional heat equation. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 29 (2019) no. 1, pp. 52-60. http://geodesic.mathdoc.fr/item/VUU_2019_29_1_a4/

[1] Lie S., Sheffers G., Symmetries of differential equations, v. 1, Lectures on differential equations with known infinitesimal transformations, Regular and Chaotic Dynamics, M.–Izhevsk, 2011

[2] Lie S., Sheffers G., Symmetries of differential equations, v. 3, Geometry of contact transformations, Regular and Chaotic Dynamics, Izhevsk, 2011

[3] 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, 28:2 (2018), 143–160 (in Russian) | DOI | MR | Zbl

[4] Ayub M., Khan M., Mahomed F. M., “Sesond-order systems of ODEs admitting three-dimensional Lie algebras and integrability”, Journal of Applied Mathematiss, 2013 (2013), 147921, 15 pp. | DOI | MR | Zbl

[5] 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, 473:2197 (2017), 20160461 | DOI | MR | Zbl

[6] Wafo Soh C., Mahomed F. M., “Reduction of order for systems of ordinary differential equations”, Journal of Nonlinear Mathematical Physics, 11:1 (2004), 13–20 | DOI | MR | Zbl

[7] 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, 19 (1983), 901–908 | MR | Zbl

[8] Olver P. J., Applications of Lie groups to differential equations, Springer, 1986, 513 pp. | MR | Zbl

[9] Ovsiannikov L. V., Group analysis of differential equations, Academic Press, 1982, 432 pp. | DOI | MR | Zbl

[10] Narmanov O. A., “Lie algebra of infinitesimal generators of the symmetry group of the heat equation”, Journal of Applied Mathematics and Physics, 6:2 (2018), 373–381 | DOI

[11] Samarskii A. A., Galaktionov V. A., Kurdyumov S. P., Mikhailov A. P., Rezhimy s obostreniem v zadachakh dlya kvazilineinykh parabolicheskikh uravnenii, Nauka, M., 1987, 481 pp. | MR