Related papers: Discrete derivatives and symmetries of difference …
The present work is devoted to the extension of some general properties of automorphisms and derivations which are known for Lie algebras to finite dimensional complex Leibniz algebras. The analogues of the Jordan-Chevalley decomposition…
In this article, we study the set of all solutions of linear differential equations using Hurwitz series. We first obtain explicit recursive expressions for solutions of such equations and study the group of differential automorphisms of…
Classical and nonclassical symmetries of the nonlinear heat equation $$u_t=u_{xx}+f(u),\eqno(1)$$ are considered. The method of differential Gr\"obner bases is used both to find the conditions on $f(u)$ under which symmetries other than the…
It is known that many equations of interest in Mathematical Physics display solutions which are only asymptotically invariant under transformations (e.g. scaling and/or translations) which are not symmetries of the considered equation. In…
A generalization of the classical Leibniz rule for the covariant derivative on a vector bundle is obtained.
We provide a general theoretical framework allowing us to extend the classical Lie theory for partial differential equations to the case of equations of fractional order. We propose a general prolongation formula for the study of Lie…
I will sketchily illustrate how the theory of symmetry helps in determining solutions of (deterministic) differential equations, both ODEs and PDEs, staying within the classical theory. I will then present a quick discussion of some more…
A powerful method for solving non-linear first-order ordinary differential equations, which is based on geometrical understanding of the corresponding dynamics of the so called Lie systems, is developed. This method allows us not only to…
We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding…
We give a characterization of symplectic quadratic Lie algebras that their Lie algebra of inner derivations has an invertible derivation. A family of symplectic quadratic Lie algebras is introduced to illustrate this situation. Finally, we…
We study the low-energy approximation for calculation of the heat kernel which is determined by the strong slowly varying background fields in strongly curved quasi-homogeneous manifolds. A new covariant algebraic approach, based on taking…
In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, call them Lie-central…
The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations.
We show that nonlocal reductions of systems of integrable nonlinear partial differential equations are the special discrete symmetry transformations.
In this paper, we consider the use of discrete gradients for differential-algebraic equations (DAEs) with a conservation/dissipation law. As one of the most popular numerical methods for conservative/dissipative ordinary differential…
We study the difference discrete variational principle in the framework of multi-parameter differential approach by regarding the forward difference as an entire geometric object in view of noncomutative differential geometry. By virtue of…
We develop a geometric version of the inverse problem of the calculus of variations for discrete mechanics and constrained discrete mechanics. The geometric approach consists of using suitable Lagrangian and isotropic submanifolds. We also…
We briefly review two different methods of applying Lie group theory in the numerical solution of ordinary differential equations. On specific examples we show how the symmetry preserving discretization provides difference schemes for which…
This paper deals with the classification of Leibniz central extensions of a naturally graded filiform Lie algebra. We choose a basis with respect to that the table of multiplication has a simple form. In low dimensional cases isomorphism…
We apply symmetry and invariance methods to analyse systems of difference equations. Non trivial symmetries are derived and their exact solutions obtained.