Note on theoretical and practical solvability of a class of discrete equations generalizing the hyperbolic-cotangent class

AbstractThere has been some recent interest in investigating the hyperbolic-cotangent types of difference equations and systems of difference equations. Among other things their solvability has been studied. We show that there is a class of theoretically solvable difference equations generalizing the hyperbolic-cotangent one. Our analysis shows a bit unexpected fact, namely that the solvability of the class is based on some algebraic relations, not closely related to some trigonometric ones, which enable us to solve them in an elegant way. Some examples of the difference equations belonging to the class which are practically solvable are presented, as well as some interesting comments on connections of the equations with some iteration processes.

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Analytical Study of the Difference Equation xn+1 = xnxn-6 / xn-5 (±1 – xnxn-6)

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v12i230081 ◽

2019 ◽

pp. 1-12

Author(s):

Abdualrazaq Sanbo ◽

Elsayed M. Elsayed ◽

Faris Alzahrani

Keyword(s):

Difference Equation ◽

Difference Equations ◽

Analytical Study ◽

Initial Conditions ◽

Specific Form ◽

Positive Real ◽

Rational Difference Equations ◽

Special Cases ◽

The Difference

This paper is devoted to find the form of the solutions of a rational difference equations with arbitrary positive real initial conditions. Specific form of the solutions of two special cases of this equation are given.

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Solvability of the difference equations for the dynamics of cumulative sums

Issues of Analysis ◽

10.15393/j3.art.2013.2384 ◽

2013 ◽

Vol 20 (2) ◽

pp. 68-81

Author(s):

N N Nikitina ◽

I A Chernov

Keyword(s):

Difference Equations ◽

The Difference ◽

Cumulative Sums

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Stability of Solutions of Differential-operator and Operator-difference Equations in the Sense of Perturbation of Operators

Computational Methods in Applied Mathematics ◽

10.2478/cmam-2006-0015 ◽

2006 ◽

Vol 6 (3) ◽

pp. 269-290 ◽

Cited By ~ 3

Author(s):

B. S. Jovanović ◽

S. V. Lemeshevsky ◽

P. P. Matus ◽

P. N. Vabishchevich

Keyword(s):

Differential Operator ◽

Difference Equations ◽

Hyperbolic Equations ◽

Second Order ◽

Difference Schemes ◽

Order Differential Operator ◽

Stability Of Solutions ◽

Operator Equations ◽

The Difference ◽

Coefficient Stability

Abstract Estimates of stability in the sense perturbation of the operator for solving first- and second-order differential-operator equations have been obtained. For two- and three-level operator-difference schemes with weights similar estimates hold. Using the results obtained, we construct estimates of the coefficient stability for onedimensional parabolic and hyperbolic equations as well as for the difference schemes approximating the corresponding differential problems.

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Limiting Values and Functional and Difference Equations

Mathematics ◽

10.3390/math8030407 ◽

2020 ◽

Vol 8 (3) ◽

pp. 407 ◽

Cited By ~ 1

Author(s):

N.-L. Wang ◽

Praveen Agarwal ◽

S. Kanemitsu

Keyword(s):

Difference Equations ◽

Functional Equations ◽

Boundary Behavior ◽

Special Functions ◽

Asymptotic Formulas ◽

Limit Values ◽

Summatory Function ◽

The Difference ◽

Limiting Values ◽

Mathematics And Science

Boundary behavior of a given important function or its limit values are essential in the whole spectrum of mathematics and science. We consider some tractable cases of limit values in which either a difference of two ingredients or a difference equation is used coupled with the relevant functional equations to give rise to unexpected results. As main results, this involves the expression for the Laurent coefficients including the residue, the Kronecker limit formulas and higher order coefficients as well as the difference formed to cancel the inaccessible part, typically the Clausen functions. We establish these by the relation between bases of the Kubert space of functions. Then these expressions are equated with other expressions in terms of special functions introduced by some difference equations, giving rise to analogues of the Lerch-Chowla-Selberg formula. We also state Abelian results which not only yield asymptotic formulas for weighted summatory function from that for the original summatory function but assures the existence of the limit expression for Laurent coefficients.

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Finiteness Properties of Affine Difference Algebraic Groups

International Mathematics Research Notices ◽

10.1093/imrn/rnaa177 ◽

2020 ◽

Cited By ~ 1

Author(s):

Michael Wibmer

Keyword(s):

Differential Equation ◽

Linear Differential Equation ◽

Difference Equations ◽

General Linear Group ◽

Algebraic Groups ◽

The Matrix ◽

Algebraic Difference ◽

The Difference ◽

Linear Differential ◽

Finiteness Properties

Abstract We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.

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On a Theory for Analysing Second-Order Systems of Ordinary Discrete Equations

Journal of Mathematics ◽

10.1155/2019/8256867 ◽

2019 ◽

Vol 2019 ◽

pp. 1-14

Author(s):

J. J. H. Bashingwa ◽

A. H. Kara

Keyword(s):

Conservation Laws ◽

Difference Equations ◽

Second Order ◽

Discrete Equations ◽

Second Order Systems

We present geometric based methods for solving systems of discrete or difference equations and introduce a technique for finding conservation laws for such systems.

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Existence of a Period-Two Solution in Linearizable Difference Equations

Discrete Dynamics in Nature and Society ◽

10.1155/2013/421545 ◽

2013 ◽

Vol 2013 ◽

pp. 1-9

Author(s):

E. J. Janowski ◽

M. R. S. Kulenović

Keyword(s):

Difference Equation ◽

Linear Equation ◽

Difference Equations ◽

Initial Conditions ◽

Sufficient Conditions ◽

Minimal Period ◽

Real Numbers ◽

Existence And Nonexistence ◽

The Difference ◽

Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

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Dynamics of a Higher Order Rational Difference Equation

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.216.50 ◽

2011 ◽

Vol 216 ◽

pp. 50-55 ◽

Cited By ~ 1

Author(s):

Yi Yang ◽

Fei Bao Lv

Keyword(s):

Difference Equation ◽

Difference Equations ◽

Initial Conditions ◽

Higher Order ◽

Rational Difference Equation ◽

The Difference

In this paper, we address the difference equation xn=pxn-s+xn-t/q+xn-t n=0,1,... with positive initial conditions where s, t are distinct nonnegative integers, p, q > 0. Our results not only include some previously known results, but apply to some difference equations that have not been investigated so far.

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