Read e-book online Applications of Lie Groups to Difference Equations PDF

By Vladimir Dorodnitsyn

ISBN-10: 1420083090

ISBN-13: 9781420083095

Intended for researchers, numerical analysts, and graduate scholars in quite a few fields of utilized arithmetic, physics, mechanics, and engineering sciences, Applications of Lie teams to distinction Equations is the 1st publication to supply a scientific development of invariant distinction schemes for nonlinear differential equations. A consultant to equipment and ends up in a brand new quarter of software of Lie teams to distinction equations, distinction meshes (lattices), and distinction functionals, this e-book makes a speciality of the maintenance of whole symmetry of unique differential equations in numerical schemes. This symmetry upkeep leads to symmetry aid of the variation version in addition to that of the unique partial differential equations and so as aid for usual distinction equations.

A giant a part of the publication is worried with conservation legislation and primary integrals for distinction versions. The variational technique and Noether kind theorems for distinction equations are awarded within the framework of the Lagrangian and Hamiltonian formalism for distinction equations.

In addition, the e-book develops distinction mesh geometry in keeping with a symmetry workforce, simply because diversified symmetries are proven to require diverse geometric mesh buildings. the strategy of finite-difference invariants offers the mesh producing equation, any precise case of which promises the mesh invariance. a couple of examples of invariant meshes is gifted. specifically, and with various functions in numerics for non-stop media, that the majority evolution PDEs must be approximated on relocating meshes.

Based at the constructed approach to finite-difference invariants, the sensible sections of the ebook current dozens of examples of invariant schemes and meshes for physics and mechanics. specifically, there are new examples of invariant schemes for second-order ODEs, for the linear and nonlinear warmth equation with a resource, and for famous equations together with Burgers equation, the KdV equation, and the Schrödinger equation.

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Extra resources for Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications)

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And by z i we denote its ith coordinate. In the space Z, we introduce a mapping D (differentiation) by the rule D(x) = 1, D(u) = u1 , . . , D(us ) = us+1 , s = 1, 2, . . Let A be the space of analytic functions F(z) of finitely many variables z. (Different functions F(z) contained in A can depend on different variables in the sequence (x, u, u1 , u2 . . ) Identifying D with the action of the first-order linear differential operator D= ∂ ∂ ∂ ∂ + u1 + u2 + · · · + us+1 + ··· , ∂x ∂u ∂u1 ∂us xlviii I NTRODUCTION we generalize the differentiation D to functions in A, and D(F (z)) ∈ A in this case.

44) is called an invariant solution if it is an invariant manifold of the subgroup H. We restrict our consideration to solutions that form a nonsingular invariant manifold. Moreover, the manifolds given by Eqs. 44) are also assumed to be nonsingular. The nonsingular manifold u = Φ(x) has some rank ρ, which is called the rank of the invariant solution. Let the subgroup H be generated by the subalgebra of operators ∂ ∂ + ηαk k , α = 1, 2, . . , r. 45); then the following assertion holds. P ROPOSITION .

Moreover, to reduce a given system of equations, one can use differential rather than finite invariants. In this case, differential-invariant solutions are constructed with the use of relations between differential invariants. 13. Contact symmetries of differential equations Contact (tangent) transformations have been widely used in mechanics and the theory of differential equations for a long time. Sophus Lie used the group of contact transformations of the form x∗i = f i (x, u, u1 , a), u∗ = g(x, u, u1 , a), i = 1, 2, .

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Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications) by Vladimir Dorodnitsyn


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