Invariance analysis and exact solutions of some sixth-order difference equations

Abstract

We perform a full Lie point symmetry analysis of difference equations of the formwhere the initial conditions are non-zero real numbers. Consequently, we obtain four non-trivial symmetries. Eventually, we get solutions of the difference equation for random sequences (A n ) and (B n ). This work is a generalization of a recent result

On certain rational recursive sequences of order four

“…after clearing fractions. By thrice differentiating (17) with respect to u n , keeping u n+1 fixed, we obtain…”

On certain sixth order difference equations

“…for some arbitrary functions α n and β n that depend on n. Substituting (15) and its shifts in (11), and then replacing the expression of u n+6 given in (10) in the resulting equation yields…”

“…The method consists of finding a group of transformations that maps solutions onto themselves. Symmetry method is a valuable tool and it has been used to solve several difference equations [8,9,14,15]. In this paper, our objective is to obtain the symmetry operators of…”

On a system of second-order difference equations

“…However, calculations can be cumbersome and to the best of our knowledge, there are no computer software packages that generate symmetries for difference equations. For ideas on how to derive solutions via the symmetry approach, the reader is referred to [4,5,7,11]. Our interest is in rational ordinary difference equations, which have been researched widely using different approaches, see [1-3, 9, 14-16].…”