Lie Symmetries of Differential Equations: Classical Results and Recent Contributions

  title={Lie Symmetries of Differential Equations: Classical Results and Recent Contributions},
  author={Francesco Oliveri},
  • F. Oliveri
  • Published 8 April 2010
  • Mathematics
  • Symmetry
Lie symmetry analysis of differential equations provides a powerful and fundamental framework to the exploitation of systematic procedures leading to the integration by quadrature (or at least to lowering the order) of ordinary differential equations, to the determination of invariant solutions of initial and boundary value problems, to the derivation of conservation laws, to the construction of links between different differential equations that turn out to be equivalent. This paper reviews… 

Figures from this paper

Lie symmetries of first order neutral differential equations
In this paper we extend the method of obtaining symmetries of ordinary differential equations to first order non-homogeneous neutral differential equations with variable coefficients. The existing
A consistent approach to approximate Lie symmetries of differential equations
Lie theory of continuous transformations provides a unified and powerful approach for handling differential equations. Unfortunately, any small perturbation of an equation usually destroys some
Approximate Q-conditional symmetries of partial differential equations
Following a recently introduced approach to approximate Lie symmetries of differential equations which is consistent with the principles of perturbative analysis of differential equations containing
Consistent approximate Q-conditional symmetries of PDEs: application to a hyperbolic reaction-diffusion-convection equation
Within the theoretical framework of a recently introduced approach to approximate Lie symmetries of differential equations containing small terms, which is consistent with the principles of
Symmetry and Lie–Frobenius reduction of differential equations
Twisted symmetries, widely studied in the last decade, have proved to be as effective as standard ones in the analysis and reduction of nonlinear equations. We explain this effectiveness in terms of
Exact and Numerical Solutions of Some Nonlinear Partial Differential Equations
v for coupled short plus equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry generator. In all types, we find
Lie Point Symmetries, Traveling Wave Solutions and Conservation Laws of a Non-linear Viscoelastic Wave Equation
This paper studies a non-linear viscoelastic wave equation, with non-linear damping and source terms, from the point of view of the Lie groups theory. Firstly, we apply Lie’s symmetries method to the
The Invariance and Conservation Laws of fourth-order Difference Equations
We consider difference equations of order four and determine the one parameter Lie group of transformations (Lie symmetries) that leave them invariant. We introduce a technique for finding their
Symmetries and conservation laws of difference and iterative equations
We construct, using first principles, a number of non-trivial conservation laws of some partial difference equations, viz, the discrete Liouville equation and the discrete Sine-Gordon equation.


Complete symmetry groups of ordinary differential equations and their integrals : Some basic considerations
Abstract The concept of the complete symmetry group of a differential equation introduced by J. Krause (1994, J. Math. Phys. 35 , 5734–5748) is extended to integrals of such equations. This paper is
Symmetries and differential equations
The knowledge of the maximal Lie group or abstract monoid of symmetries of an ordinary non-singular differential equation (or system of equations) allows us to obtain solutions of them. Traditional
Symmetries of Differential Equations: From Sophus Lie to Computer Algebra
The topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it. The
New methods of reduction for ordinary differential equations
We introduce a new class of symmetries, that strictly includes Lie symmetries, for which there exists an algorithm that lets us reduce the order of an ordinary differential equation. Many of the
Exact solutions of Euler equations of ideal gasdynamics via Lie group analysis
Abstract.In this paper, we explicitly characterize a class of solutions to the first order quasilinear system of partial differential equations (PDEs), governing one dimensional unsteady planar and
Lie symmetries of differential equations:direct and inverse problems
This paper reviews some relevant problems arising within the context of Lie group analysis of dieren tial equations either in the direct approach or in the inverse one. For what concerns the direct
Hidden symmetries associated with the projective group of nonlinear first-order ordinary differential equations
Hidden symmetries, those not found by the classical Lie group method for point symmetries, are reported for nonlinear first-order ordinary differential equations (ODEs) which arise frequently in
Direct construction method for conservation laws of partial differential equations Part I: Examples of conservation law classifications
An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not