Difference Equations by Differential Equation Methods

Abstract

Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.

Symmetries, Conservation Laws, and Noether's Theorem for Differential‐Difference Equations

“…In this situation, is understood as a vector of independent variables, while is a vector of dependent variables. A first construction of Noether's theorem for ordinary difference equations was studied by Maeda ; a general extension has now been well studied, see, for instance, . In particular, Kupershmidt's book includes fundamental analysis for variational principles with respect to difference equations as well as DDEs.…”

“…Here, P is a p ‐tuple and the operator id is the identity operator. For a difference system of the Kovalevskaya form (see, e.g., ), there exist functions such that In the difference case, integration by parts is replaced by the formula of summation by parts for any functions and . Here, the operator is the inverse (or adjoint) of S ; it is also called the backward shift operator.…”

“…Infinitesimally, it is equivalent to the existence of a p ‐tuple R such that where X is the infinitesimal generator of a symmetry group. A discrete version of Noether's theorem exists (see, e.g., ); it also relates symmetries for difference variational problems and conservation laws of the associated Euler–Lagrange equations via their characteristics. Namely, there exists a p ‐tuple P such that the characteristic of a variational symmetry satisfies …”

“…Example The difference equation admits the following characteristics of symmetries (e.g., ) In , the equation was solved by obtaining all of its invariant first integrals with respect to the first two generators. It was also noted therein that through a change of variables , the resulting equation is governed by a Lagrangian The Euler–Lagrange equation is .…”

“…For the interest of the broader community, other excellent textbooks are also available, see, for example, [8][9][10][11][12][13][14]. For (finite) difference equations, the methods of continuous symmetries and conservation laws, as well as their applications have been well adapted during the last few decades, see, for instance, [15][16][17][18][19][20][21][22][23][24]. The first effort to understand Noether's theorems was made by Maeda [25], who considered symmetries of discrete mechanical systems.…”