# Applications of Lie Groups to Difference Equations

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## Book Description

Intended for researchers, numerical analysts, and graduate students in various fields of applied mathematics, physics, mechanics, and engineering sciences, **Applications of Lie Groups to Difference Equations** is the first book to provide a systematic construction of invariant difference schemes for nonlinear differential equations. A guide to methods and results in a new area of application of Lie groups to difference equations, difference meshes (lattices), and difference functionals, this book focuses on the preservation of complete symmetry of original differential equations in numerical schemes. This symmetry preservation results in symmetry reduction of the difference model along with that of the original partial differential equations and in order reduction for ordinary difference equations.

A substantial part of the book is concerned with conservation laws and first integrals for difference models. The variational approach and Noether type theorems for difference equations are presented in the framework of the Lagrangian and Hamiltonian formalism for difference equations.

In addition, the book develops difference mesh geometry based on a symmetry group, because different symmetries are shown to require different geometric mesh structures. The method of finite-difference invariants provides the mesh generating equation, any special case of which guarantees the mesh invariance. A number of examples of invariant meshes is presented. In particular, and with numerous applications in numerics for continuous media, that most evolution PDEs need to be approximated on moving meshes.

Based on the developed method of finite-difference invariants, the practical sections of the book present dozens of examples of invariant schemes and meshes for physics and mechanics. In particular, there are new examples of invariant schemes for second-order ODEs, for the linear and nonlinear heat equation with a source, and for well-known equations including Burgers equation, the KdV equation, and the Schrödinger equation.

## Table of Contents

**Preface Introduction **Brief introduction to Lie group analysis of differential equations

Preliminaries: Heuristic approach in examples

**Finite Differences and Transformation Groups in Space of Discrete Variables**

The Taylor group and finite-difference derivatives

Difference analog of the Leibniz rule

Invariant difference meshes

Transformations preserving the geometric meaning of finite-difference derivatives

Newton’s group and Lagrange’s formula

Commutation properties and factorization of group operators on uniform difference meshes

Finite-difference integration and prolongation of the mesh space to nonlocal variables

Change of variables in the mesh space

**Invariance of Finite-Difference Models**

An invariance criterion for finite-difference equations on the difference mesh

Symmetry preservation in difference modeling: Method of finite-difference invariants

Examples of construction of difference models preserving the symmetry of the original continuous models

**Invariant Difference Models of Ordinary Differential Equations**

First-order invariant difference equations and lattices

Invariant second-order difference equations and lattices

**Invariant Difference Models of Partial Differential Equations**

Symmetry preserving difference schemes for the nonlinear heat equation with a source

Symmetry preserving difference schemes for the linear heat equation

Invariant difference models for the Burgers equation

Invariant difference model of the heat equation with heat flux relaxation

Invariant difference model of the Korteweg–de Vries equation

Invariant difference model of the nonlinear Shrödinger equation

**Combined Mathematical Models and Some Generalizations**

Second-order ordinary delay differential equations

Partial delay differential equations

Symmetry of differential-difference equations

Lagrangian Formalism for Difference Equations

Discrete representation of Euler’s operator

Criterion for the invariance of difference functionals

Invariance of difference Euler equations

Variation of difference functional and quasi-extremal equations

Invariance of global extremal equations and properties of quasiextremal equations

Conservation laws for difference equations

Noether-type identities and difference analog of Noether’s theorem

Necessary and sufficient conditions for global extremal equations to be invariant

Applications of Lagrangian formalism to second-order difference equations

Moving mesh schemes for the nonlinear Shrödinger equation

**Hamiltonian Formalism for Difference Equations: Symmetries and First Integrals**

Discrete Legendre transform

Variational statement of the difference Hamiltonian equations

Symplecticity of difference Hamiltonian equations

Invariance of the Hamiltonian action

Difference Hamiltonian identity and Noether-type theorem for difference Hamiltonian equations

Invariance of difference Hamiltonian equations

Examples

**Discrete Representation of Ordinary Differential Equations with Symmetries**

The discrete representation of ODE as a series

Three-point exact schemes for nonlinear ODE

**Bibliography**

Index

Index

## Reviews

The book provides a systematic application of Lie groups to difference equations, difference meshes, and difference functionals. Besides the well-explained theoretical background and motivations, there is also a large number of concrete examples discussed in reasonable details. Due to the fairly broad introductory part, the book is indeed self-contained. The main ideas and concepts appear understandable not only to experts.

—Vojtech Zadnik,Zentralblatt MATH1236In recent years "difference geometry" and its applications to integrable systems and mathematical physics have attracted significant attention and this monograph will contribute to the ongoing developments in this general area. It is clearly written and largely self-contained …

—Peter J. Vassiliou,Mathematical Reviews, 2012e