Related papers: On certain sixth order difference equations

This paper deals with the solution, stability character and asymptotic behavior of the rational difference equation \begin{equation*} x_{n+1}=\frac{\alpha x_{n-1}+\beta}{ \gamma x_{n}x_{n-1}},\qquad n \in \mathbb{N}_{0}, \end{equation*}…

Essentially generalizing Lie's results, we prove that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein-Gordon equations is the first-order prolongation of its point equivalence groupoid, and then…

Lie group theory states that knowledge of a $m$-parameters solvable group of symmetries of a system of ordinary differential equations allows to reduce by $m$ the number of equations. We apply this principle by finding some \emph{affine…

We discuss the basic properties of Lie groupoids, Lie algebroids and Lie pseudo-groups in view of applying these techniques to the analysis of Jordan-H\"older resolutions and, subsequently, to the integration of partial differential…

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three…

We study a Lie algebra $\mathcal A_{a_1,\ldots,a_{n-1}}$ of deformed skew-symmetric $n \times n$ matrices endowed with a Lie bracket given by a choice of deformed symmetric matrix. The deformations are parametrized by a sequence of real…

Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as…

This paper investigates the Jordan--Kronecker invariant of finite dimensional complex Lie algebras. We present an explicit algorithm for determining the type of a given Lie algebra from its Jordan--Kronecker invariant. The algorithm is…

Let $B_n$ be the $n$-th balancing number. In this paper, we give some explicit expressions of $\sum_{l=0}^{2 r-3}(-1)^l\binom{2 r-3}{l}\sum_{j_1+\cdots+j_r=n-2 l\atop j_1,\dots,j_r\ge 1}B_{j_1}\cdots B_{j_r}$ and…

We use the notion of the principal three-dimensional subgroup of a simple Lie group to identify certain special subspaces of the Lie algebra and address the question of whether these are calibrated for invariant forms on the group.

Through the following, we establish the conditions which allow us to express recursive sequences of real numbers, enumerated through the recurrence relation a_{n+1} = Aa_n + Ba_{n-1}, by means of algebraic equations in two variables of…

Lie group method provides an efficient tool to solve a differential equation. This paper suggests a fractional partner for fractional partial differential equations using a fractional characteristic method. A space-time fractional diffusion…

The scalar difference equation $x_{n+1}=f_{n}(x_{n},x_{n-1},...,x_{n-k})$ may exhibit symmetries in its form that allow for reduction of order through substitution or a change of variables. Such form symmetries can be defined generally…

We study a class of nonlinear evolution systems of time fractional partial differential equations using Lie symmetry analysis. We obtain not only infinitesimal symmetries but also a complete group classification and a classification of…

We consider finite-dimensional complex Lie algebras admitting a periodic derivation, i.e., a nonsingular derivation which has finite multiplicative order. We show that such Lie algebras are at most two-step nilpotent and give several…

A heat equation with non-constant diffusivity depending as a power law on the spatial variable is analysed using Lie's method to identify classical point symmetries. It is shown that the group invariant solutions of a four-dimensional…

Lie symmetry analysis is one of the powerful tools to analyze nonlinear ordinary differential equations. We review the effectiveness of this method in terms of various symmetries. We present the method of deriving Lie point symmetries,…

A definition of invariance in Lie's sense for a boundary value problem (BVP) with the basic evolution differential equations is proposed. A problem of group classification at a wide class of BVPs parameterized by arbitrary elements is…

The symmetry algebra of the differential--difference equation $$\dot u_n = [P(u_n)u_{n+1}u_{n-1} + Q(u_n)(u_{n+1}+u_{n-1})+ R(u_n)]/(u_{n+1}-u_{n-1}),$$ where $P$, $Q$ and $R$ are arbitrary analytic functions is shown to have the dimension…

This paper studies non-autonomous Lyness type recurrences of the form $x_{n+2}=(a_n+x_{n+1})/x_{n}$, where $\{a_n\}$ is a $k$-periodic sequence of positive numbers with primitive period $k$. We show that for the cases $k\in\{1,2,3,6\}$ the…