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

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.

References in corpus (1)

• Lie and Noether symmetries of geodesic equations and collineations

Cited by in corpus (20)

• Dynamical symmetries and observational constraints in scalar field cosmology
• The geometric origin of Lie point symmetries of the Schrödinger and the Klein Gordon equations
• Symmetry analysis of the Klein-Gordon equation in Bianchi I spacetimes
• Group invariant transformations for the Klein-Gordon equation in three dimensional flat spaces
• Lie Symmetry Analysis of the Black-Scholes-Merton Model for European Options with Stochastic Volatility
• Lie and Noether point symmetries of a class of quasilinear systems of second-order differential equations
• Variational Contact Symmetries of Constraint Lagrangians
• Quadratic first integrals of autonomous conservative dynamical systems
• Noether symmetries and isometries of the minimal surface Lagrangian under constant volume in a Riemannian space
• Quadratic conservation laws and collineations: a discussion
• Lie symmetries for systems of evolution equations
• Symmetries of second-order PDEs and conformal Killing vectors
• The reduction of Laplace equation in certain Riemannian spaces and the resulting Type II hidden symmetries
• Quadratic first integrals of time-dependent dynamical systems of the form ${\ddot{q}}^a=-{\Gamma }_{bc}^a{\dot{q}}^b{\dot{q}}^c-\omega (t)Q^a(q)$
• Constructing the CKVs of Bianchi III and V spacetimes
• Geometrization of Lie and Noether symmetries with applications in Cosmology
• The algebraic properties of the space- and time-dependent one-factor model of commodities
• Type II hidden symmetries for the homogeneous heat equation in some general classes of Riemannian spaces
• Lie symmetries of (1+2) nonautonomous evolution equations in Financial Mathematics
• Extended Theories of Gravity in cosmological and astrophysical applications