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Applications of Lie Groups to Difference Equations by Vladimir Dorodnitsyn

By Vladimir Dorodnitsyn

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 ebook to supply a scientific building of invariant distinction schemes for nonlinear differential equations. A advisor to equipment and ends up in a brand new region of software of Lie teams to distinction equations, distinction meshes (lattices), and distinction functionals, this publication specializes in the renovation of whole symmetry of unique differential equations in numerical schemes. This symmetry upkeep ends up in symmetry relief of the variation version besides that of the unique partial differential equations and so as relief for usual distinction equations.

A great a part of the booklet is worried with conservation legislation and primary integrals for distinction types. The variational process and Noether sort theorems for distinction equations are awarded within the framework of the Lagrangian and Hamiltonian formalism for distinction equations.

In addition, the ebook develops distinction mesh geometry in keeping with a symmetry staff, simply because various symmetries are proven to require varied geometric mesh constructions. the tactic of finite-difference invariants offers the mesh producing equation, any targeted case of which promises the mesh invariance. a couple of examples of invariant meshes is gifted. specifically, and with various purposes 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 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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See [60,98]). The variations δq i and δpi are arbitrary and satisfy δq i (t1 ) = δq i (t2 ) = 0, i = 1, . . , n. Then we have t2 t2 i t1 t1 t2 q˙i − = t1 ∂H i ∂H δq − δpi dt ∂q i ∂pi ∂H t δpi − p˙i + i δq i dt + pi δq i t21 . ∂q δpi q˙i + pi δ q˙i − pi q˙ − H(t, q, p) dt = δ ∂H ∂pi The last term vanishes, because δq i = 0 at the endpoints. 70). 70) can be obtained by applying the variational operators and ∂ ∂ δ = −D , δpi ∂pi ∂ p˙i i = 1, . . 72) δ ∂ ∂ = − D , δq i ∂q i ∂ q˙i i = 1, . .

50) k=0 in a single symbol (parameter) a, where Aik (z) ∈ A and Ai 0 ≡ z i . Here z i is the ith coordinate of a vector in Z. We denote the space of sequences (f 1 (z, a), f 2 (z, a), . . , f s (z, a), . 50) by Z. The sequences (x, u, u1 , u2 , . . ) are a special case of such sequences, Z ⊂ Z. In Z, by definition, we introduce the operations of addition and multiplication by a number and the product of formal series as follows (these operations coincide with the operations for converging series): ∞ ∞ Aik ak α +β k=0 ∞ ∞ Bki ak k=0 ∞ Aik1 ak1 k1 =0 αAik + βBki ak , = k=0 ∞ Bki 2 ak2 ∞ Aik1 Bki 2 ak , = k2 =0 k=0 k1 +k2 =k where i = 1, 2, .

37). Thus, the knowledge of a nonzero solution of the determining equation is equivalent to the knowledge of an integrating factor, which permits integrating the firstorder equation. Unfortunately, the problem of finding a nonzero solution of the determining equation is not at all simpler than the problem of integration of the original equation. 41) gives an efficient integration method for the equation. This property underlies all elementary integration methods for first-order ordinary differential equations presented in numerous manuals.

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