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Applications of Lie Groups to Difference Equations

Vladimir Dorodnitsyn
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Differential and Integral Equations and Their Applications
We do not plan to review this book.

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
Discrete Representation of Ordinary Differential Equations with Symmetries
The discrete representation of ODE as a series
Three-point exact schemes for nonlinear ODE