By Vladimir Dorodnitsyn
Meant for researchers, numerical analysts, and graduate scholars in numerous fields of utilized arithmetic, physics, mechanics, and engineering sciences, functions of Lie teams to distinction Equations is the 1st ebook to supply a scientific building of invariant distinction schemes for nonlinear differential equations. A consultant to tools and leads to a brand new quarter of program of Lie teams to distinction equations, distinction meshes (lattices), and distinction functionals, this ebook specializes in the renovation of entire symmetry of unique differential equations in numerical schemes. This symmetry renovation ends up in symmetry relief of the adaptation version besides that of the unique partial differential equations and so as aid for usual distinction equations. a considerable a part of the ebook is anxious with conservation legislation and primary integrals for distinction versions. The variational technique and Noether style theorems for distinction equations are offered within the framework of the Lagrangian and Hamiltonian formalism for distinction equations. moreover, the e-book develops distinction mesh geometry according to a symmetry staff, simply because assorted symmetries are proven to require varied geometric mesh constructions. the tactic of finite-difference invariants presents the mesh producing equation, any detailed case of which promises the mesh invariance. a few examples of invariant meshes is gifted. particularly, and with quite a few purposes in numerics for non-stop media, that the majority evolution PDEs must be approximated on relocating meshes. in response to 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. particularly, 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 Schr?dinger equation.
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Meant for researchers, numerical analysts, and graduate scholars in a variety of fields of utilized arithmetic, physics, mechanics, and engineering sciences, functions of Lie teams to distinction Equations is the 1st e-book to supply a scientific building of invariant distinction schemes for nonlinear differential equations.
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Extra resources for Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications, Volume 8)
1. B RIEF INTRODUCTION TO L IE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS xxxv E XAMPLE . We return to our example of the three-dimensional algebra admitted by Eq. 19). The only differential invariant in the space (x, u, u , u ) is I = u3 u . Any sufficiently smooth function of this invariant determines an invariant of the second-order differential equation F (u3 u ) = 0. A special case of this equation coincides with Eq. 19), I = u3 u = 1. The equation thus obtained is an invariant representation of Eq.
In this case, the invariants are J1 = t, J2 = ux−2/σ . We seek the solution as a relation between the two invariants, ux−2/σ = V (t). We reduce the original equation to an ordinary differential equation for the unknown function V (t) by substituting the invariant representation u = V (t)x2/σ of the solution into the heat equation, V = 2(σ + 2) σ+1 V . σ2 By integrating this equation, we obtain the self-similar solution u= x2 C − 2(σ + 2)t/σ 1/σ , C = const. 1. B RIEF INTRODUCTION TO L IE GROUP ANALYSIS OF DIFFERENTIAL EQUATIONS xlv C.
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 pi q˙ − H(t, q, p) dt = δ 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 − i ∂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, . . 74) is the operator of total differentiation with respect to time, to the function pi q˙i − H(t, q, p).
Applications of Lie Groups to Difference Equations (Differential and Integral Equations and Their Applications, Volume 8) by Vladimir Dorodnitsyn