Discrete derivatives and symmetries of difference equations

@article{Levi2001DiscreteDA,
  title={Discrete derivatives and symmetries of difference equations},
  author={Decio Levi and J. Negro and Mariano A. del Olmo},
  journal={Journal of Physics A},
  year={2001},
  volume={34},
  pages={2023-2030}
}
We show with an example of the discrete heat equation that for any given discrete derivative we can construct a nontrivial Leibniz rule suitable for finding the symmetries of discrete equations. In this way we obtain a symmetry Lie algebra, defined in terms of shift operators, isomorphic to that of the continuous heat equation. 
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References

SHOWING 1-10 OF 12 REFERENCES
Continuous symmetries of discrete equations
Abstract Lie group techniques for solving differential equations are extended to differential-difference equations. As an application, it is shown that the two-dimensional Toda lattice has anExpand
Symmetries of the wave equation in a uniform lattice
The symmetries of discrete versions of a class of equations (that includes Klein - Gordon and wave equations) on a two-dimensional grid are studied. They close the same Lie algebra as theExpand
Symmetries of the heat equation on the lattice
Discrete versions of the heat equation on two-dimensional uniform lattices are shown to possess the same symmetry algebra as their continuum limits. Solutions with definite symmetry properties areExpand
Symmetries and conditional symmetries of differential difference equations
Two different methods of finding Lie point symmetries of differential‐difference equations are presented and applied to the two‐dimensional Toda lattice. Continuous symmetries are combined withExpand
Lie group formalism for difference equations
The methods of Lie group analysis of differential equations are generalized so as to provide an infinitesimal formalism for calculating symmetries of difference equations. Several examples areExpand
Continuous symmetries of differential-difference equations: the Kac-van Moerbeke equation and Painlevé reduction
Abstract A method is given to derive the point symmetries of partial differential-difference equations. Applying the method to the Kacvan Moerbeke equation we find its symmetries form aExpand
Transformation groups in net spaces
We consider formal groups of transformations on the space of differential and net (finite-difference) variables. We show that preservation of meaning of difference derivatives under transformationsExpand
Representation of Lie groups and special functions
At first only elementary functions were studied in mathematical analysis. Then new functions were introduced to evaluate integrals. They were named special functions: integral sine, logarithms, theExpand
The Similarity Method for Difference Equations
On applique la methode de similitude aux equations aux differences. Quand une equation aux differences a une dimension admet une symetrie, l'equation devient lineaire et on peut obtenir uneExpand
Applications of lie groups to differential equations
1 Introduction to Lie Groups.- 1.1. Manifolds.- Change of Coordinates.- Maps Between Manifolds.- The Maximal Rank Condition.- Submanifolds.- Regular Submanifolds.- Implicit Submanifolds.- Curves andExpand
...
1
2
...