**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

**Author(s):** Mersaid Aripov ^{a} , Otabek Narmanov ^{b}

^{a}Department of Applied mathematics and computer analysis, National University of Uzbekistan, Tashkent, 100174, Tashkent, Uzbekistan

^{b}Department 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: mirsaidaripov@mail.ru)

**Full Text:** PDF

**Abstract**

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 group

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