Lie point symmetries of a general class of PDEs: The heat equation

TL;DR

This paper explores the Lie point symmetries of a broad class of PDEs, including the heat equation, revealing their connection to geometric symmetries like conformal and homothetic algebras in various spaces.

Contribution

It generalizes the relationship between PDE symmetries and geometric algebras, extending known results to more complex equations like the heat equation in Riemannian spaces.

Findings

Lie symmetries of PDEs relate to conformal and homothetic algebras.

Symmetry analysis applied to wave and heat equations in different geometries.

Results confirmed with known cases like de Sitter and flat spaces.

Abstract

We give two theorems which show that the Lie point and the Noether symmetries of a second-order ordinary differential equation of the form (D/(Ds))(((Dx^{i}(s))/(Ds)))=F(x^{i}(s),x^{j}(s)) are subalgebras of the special projective and the homothetic algebra of the space respectively. We examine the possible extension of this result to partial differential equations (PDE) of the form A^{ij}u_{ij}-F(x^{i},u,u_{i})=0 where u(x^{i}) and u_{ij} stands for the second partial derivative. We find that if the coefficients A^{ij} are independent of u(x^{i}) then the Lie point symmetries of the PDE form a subgroup of the conformal symmetries of the metric defined by the coefficients A^{ij}. We specialize the study to linear forms of F(x^{i},u,u_{i}) and write the Lie symmetry conditions for this case. We apply this result to two cases. The wave equation in an inhomogeneous medium for which we derive the Lie symmetry vectors and check our results with those in the literature. Subsequently we consider the heat equation with a flux in an n-dimensional Riemannian space and show that the Lie symmetry algebra is a subalgebra of the homothetic algebra of the space. We discuss this result in the case of de Sitter space time and in flat space.