Applications of Lie Groups to Difference Equations by Vladimir Dorodnitsyn

By Vladimir Dorodnitsyn

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

A large a part of the e-book is worried with conservation legislation and primary integrals for distinction versions. The variational technique and Noether variety theorems for distinction equations are offered within the framework of the Lagrangian and Hamiltonian formalism for distinction equations.

In addition, the e-book develops distinction mesh geometry in accordance with a symmetry crew, simply because assorted symmetries are proven to require various geometric mesh buildings. the strategy of finite-difference invariants presents the mesh producing equation, any designated case of which promises the mesh invariance. a few examples of invariant meshes is gifted. specifically, and with quite a few functions in numerics for non-stop media, that almost all evolution PDEs must be approximated on relocating meshes.

Based at the built approach to finite-difference invariants, the sensible sections of the publication 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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This operator identity was called the Noether identity in [73]: ξα i ∂L ∂L ∂L + ηα k k + [Di (ηα k ) − uj k Di (ξα j )] + LDi (ξα i ) ∂xi ∂u ∂ui k δL ∂L + Lξα i , ≡ (ηα k − ui k ξα i ) k + Di (ηα k − ui k ξα i ) δu ∂ui k α = 1, 2, . . , r. This identity permits stating a necessary and sufficient condition for the existence of conservation laws. liv I NTRODUCTION T HEOREM ( [73]). 64). The Noether identity gives a relationship between three types of differentiation: the total differentiation Di , the variational differentiation δuδ k , and the Lie derivative Xα .

The central theorem in this approach, the so-called π-theorem, is a special case of the theorem on the invariant representation of the solution determined by the dilation group. This approach is not invariant under the change of the coordinate system; it suffices to pass to another coordinate system, and the group ceases to be the dilation group. In our last example, the change of variables x¯ = ln x, t¯ = ln t, u¯ = ln u transforms the operator X4 into the translation operator ∂ σ ∂ + . 2 ∂ x¯ ∂ u¯ For the translation operator, dimension analysis does not work any more, but the theorem on the invariant representation of the solution remains valid.

50). This transformation takes a sequence z i to the sequence z i∗ . 1. B RIEF INTRODUCTION TO L IE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS (or formal power series) of finitely many variables z i . 51) is well defined: ∞ z i∗∗ = f i (z ∗ , b) = ∞ Aik (z ∗ )bk = k=0 Aik (f (z, a))bk , i = 1, 2 . . 50) out of Z. 51) are closed in Z: ∞ z i∗∗ i ∗ i Aik (z)(a + b)k , = f (z , b) = f (z, a + b) = i = 1, 2, . . 51) form a formal one-parameter group in Z. 50) (see [107]): ∞ z i∗ = f i (z, a) = eaX (z i ) ≡ s=0 as (s) i X (z ), s!

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