Applications of Lie Groups to Difference Equations (Differential & Integral Equations & Their Applications v. 8)
By: Vladimir Dorodnitsyn (author), Andrei D. Polyanin (series_editor)Hardback
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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 Schrodinger equation.
Keldysh Institute of Applied Mathematics, Moscow, Russia Russian Academy of Sciences, Moscow
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 Shrodinger 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 Shrodinger 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
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- ID: 9781420083095
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