Title: Some Invariant Solutions of Two Dimensional Heat Equation
Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814
Article ID: MTJPAM-D-20-00060; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 334-343
Document Type: Research Paper
aDepartment of Applied mathematics and computer analysis, National University of Uzbekistan, Tashkent, 100174, Tashkent, Uzbekistan
bDepartment of Mathematical Modelling and algorithms,Tashkent university of information technologies, Tashkent, 100200, Tashkent, Uzbekistan
Received: 3 January 2021, Accepted: 6 April 2021, Published: 25 April 2021.
Corresponding Author: Mersaid Aripov (Email address: email@example.com)
Full Text: PDF
The symmetry group of a differential equation is the group of transformations
which transform solutions of the differential equation to solutions.
For systems of partial differential equations, the symmetry group can be used to explicitly find particular types of solutions that are themselves invariant with respect to some subgroup of the full group of symmetries of the system.
Group analysis methods are widely used to study partial differential equations and to integrate ordinary differential equations.
The heat equation is a certain partial differential equation and the most widely studied equation in pure mathematics, and its analysis is regarded as fundamental to the broader field of partial differential equations.
In the paper we find solutions of the two-dimensional heat equation which are invariant with respect some symmetry groups and show that some solutions can be found using the well-known Bessel functions.
Keywords: Heat equation, Symmetry group, Invariant solution, Lie groupReferences:
- M. Ayub, M. Khan and F. M. Mahomed, Second-order systems of ODEs admitting three-dimensional Lie algebras and integrability, Journal of Applied Mathematics Article ID 147921, 15 p, 2013. DOI: 10.1155/2013/147921
- V. A. Dorodnitsyn, I. V. Knyazeva and S. R. Svirshchevskii, Group properties of the heat equation with source in the two-dimensional and three-dimensional cases, Differ. Equations 19 (7), 1215-1223, 1983.
- A. A. Gainetdinova, Integration of ordinary differential systems equations with a small parameter that admit approximate Lie algebras, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki 28 (2), 143-160, 2018. DOI: 10.20537/vm180202
- A. A. Gainetdinova and R. K. Gazizov, 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. DOI: 10.1098/rspa.2016.0461
- E. Kamke, Differentialgleichungen: Losungsmethoden und Losungen, Leipzig, 1977.
- S. Lie and G. Sheffers, Symmetries of differential equations. Vol. 1. Lectures on differential equations with known infinitesimal transformations, Moscow- Izhevsk, Regular and Chaotic Dynamimics, 2011.
- S. Lie and G. Sheffers, Symmetries of differential equations. Vol. 3. Geometry of contact transformations, Izhevsk, Udmurtskiy gosudarstvenniy universitet, 2011.
- O. A. Narmanov, Lie algebra of infinitesimal generators of the symmetry group of the heat equation, Journal of Applied Mathematics and Physics 6, 373-381, 2018. DOI: 10.4236/jamp.2018.62035
- O. A. Narmanov, Invariant solutions of two dimensional heat equation, Journal of Applied Mathematics and Physics 7, 1488-1497, 2019. https: //doi.org/10.4236/ jamp.2019.77100
- O. A. Narmanov, Invariant solutions of the two-dimensional heat equation, Bulletin of Udmurt University. Mathematics, Mechanics, Computer Science, 29 (1), 52-60, 2019.
- P. Olver, Applications of Lie Groups to Differential Equations, Springer 1986.
- L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, 1982, DOI: 10.1016/C2013-0-07470-1
- A. A. Samarskiy, V. A. Galaktionov, S. P. Kurdumov and A. P. Mikhaylov, Rejimy s obostreniyem v zadachax dlya kvazilineynyx parabolicheskix uravneniy, Nauka, Moscow, in Russian, 1987.
- S. C. Wafo and F. M. Mahomed, Reduction of order for systems of ordinary differential equations, Journal of Nonlinear Mathematical Physics, 11 (1), 13-20, 2004. DOI: 10.2991/jnmp.2004.11.1.3