Applications of Lie Groups to Difference Equations by Vladimir Dorodnitsyn PDF

By Vladimir Dorodnitsyn

ISBN-10: 1420083090

ISBN-13: 9781420083095

Intended for researchers, numerical analysts, and graduate scholars in a number of fields of utilized arithmetic, physics, mechanics, and engineering sciences, Applications of Lie teams to distinction Equations is the 1st booklet to supply a scientific building of invariant distinction schemes for nonlinear differential equations. A advisor to tools and leads to a brand new sector of software of Lie teams to distinction equations, distinction meshes (lattices), and distinction functionals, this booklet specializes in the upkeep of whole symmetry of unique differential equations in numerical schemes. This symmetry renovation ends up in symmetry relief of the variation version in addition to that of the unique partial differential equations and so as aid for usual distinction equations.

A enormous a part of the booklet is worried with conservation legislation and primary integrals for distinction versions. The variational strategy and Noether style theorems for distinction equations are offered within the framework of the Lagrangian and Hamiltonian formalism for distinction equations.

In addition, the publication develops distinction mesh geometry in accordance with a symmetry workforce, simply because assorted symmetries are proven to require varied geometric mesh buildings. the tactic 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. particularly, and with a number of functions in numerics for non-stop media, that almost all evolution PDEs have to be approximated on relocating meshes.

Based at the built approach to finite-difference invariants, the sensible sections of the booklet 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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Additional resources for Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications)

Example text

The Legendre transformation relates the Hamiltonian and Lagrange functions, ˙ = pi q˙i − H(t, q, p), L(t, q, q) lviii I NTRODUCTION where ∂H ∂L , q˙ = . ∂ q˙ ∂p This permits one to establish the equivalence of the Euler–Lagrange and Hamiltonian equations [6]. Indeed, from the Euler–Lagrange equations in the form (m = 1) p= δL ∂L = −D δq k ∂q k ∂L ∂ q˙k = 0, k = 1, . . 70) by using the Legendre transformation. It should be noted that the Legendre transformation is not a point transformation. Hence, there is no conservation of Lie group properties of the corresponding Euler–Lagrange equations and Hamiltonian equations within the class of point transformations.

44) where x ∈ Rn , u ∈ Rm , and us is the set of sth partial derivatives. We assume that Eqs. 44) admit a transformation group Gr N , and H is a subgroup of Gr N . D EFINITION . 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.

See [77]). We assume that the symmetry of a given equation contains a two-dimensional subalgebra of operators X1 = ξ1 (x, y) ∂ ∂ + η1 (x, y) , ∂x ∂y X2 = ξ2 (x, y) ∂ ∂ + η2 (x, y) . 1. B RIEF INTRODUCTION TO L IE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS xli of the operators. 3. 3: Standard forms of L2 and canonical forms of second-order ODE Structure of L2 Standard form of L2 Canonical second-order ODE [X1 , X2 ] = 0, ξ1 η2 − ξ2 η1 = 0 X1 = ∂ ∂ , X2 = ∂t ∂u u = F (u ) [X1 , X2 ] = 0, ξ1 η2 − ξ2 η1 = 0 X1 = ∂ ∂ , X2 = t ∂u ∂u u = F (t) ∂ , ∂u ∂ ∂ X2 = t + u ∂t ∂u X1 = [X1 , X2 ] = X1 , ξ1 η2 − ξ2 η1 = 0 [X1 , X2 ] = X1 , ξ1 η2 − ξ2 η1 = 0 X1 = 1 u = F (u ) t ∂ ∂ , X2 = u u = u F (t) ∂u ∂u Thus, to integrate a second-order ordinary differential equation, we should • Find the transformation group admitted by the equation and single out a twodimensional subalgebra L2 if any.

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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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