On a recursive sequence

Kybernetes ◽  
2007 ◽  
Vol 36 (1) ◽  
pp. 98-115
Author(s):  
Mehdi Dehghan ◽  
Reza Mazrooei‐Sebdani
Keyword(s):  
Difference Equation ◽  
Difference Equations ◽  
Open Problem ◽  
Design Methodology ◽  
Initial Conditions ◽  
Open Problems ◽  
Content Type ◽  
Rational Difference Equations ◽  
Global Behaviour ◽  
Recursive Sequence

PurposeThe aim in this paper is to investigate the dynamics of difference equation yn+1=(pyn+yn−k)/(qyn+yn−k), n=0,1,2,… where k∈{1,2,3,…}, the initial conditions y−k, … ,y−1,y0 and the parameters p and q are non‐negative.Design/methodology/approachThe paper studies characteristics such as the character of semicycles, periodicity and the global stability of the above mentioned difference equation.FindingsIn particular, the results solve the open problem introduced by Kulenovic and Ladas in their monograph, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures.Originality/valueThe global behaviour of the solutions of equation yn+1=(pyn+yn−k)/(qyn+yn−k), n=0,1,2,… were investigated providing valuable conclusions on practical data.

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

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.

scholarly journals On the Recursive Sequence

2010 ◽  
Vol 2010 ◽  
pp. 1-13 ◽  
Author(s):  
Ibrahim Yalcinkaya ◽  
Cengiz Cinar ◽  
Ali Gelisken
Keyword(s):  
Difference Equation ◽  
Open Problem ◽  
Initial Conditions ◽  
Rational Numbers ◽  
Recursive Sequence

We investigate the periodic nature of the solution of the max-type difference equation , , where the initial conditions are and for , and that and are positive rational numbers. The results in this paper solve the Open Problem proposed by Grove and Ladas (2005).

Periodicity of the solutions of general system of rational difference equations related to fibonacci numbers

2021 ◽  
pp. 2250087
Author(s):  
Ibtissam Talha ◽  
Salim Badidja
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
General System ◽  
Real Numbers ◽  
Rational Difference Equations

In this paper, we deal with the periodicity of solutions of the following general system rational of difference equations: [Formula: see text] where [Formula: see text] [Formula: see text] and the initial conditions are arbitrary nonzero real numbers.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

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.

Dynamics of a Higher Order Rational Difference Equation

2011 ◽  
Vol 216 ◽  
pp. 50-55 ◽  
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.

Discrete Verhulst model based on a linear time-varying

2014 ◽  
Vol 4 (2) ◽  
pp. 299-310 ◽  
Author(s):  
Xia Long ◽  
Yong Wei ◽  
Zhao Long
Keyword(s):  
Differential Equations ◽  
Difference Equations ◽  
Design Methodology ◽  
Linear Time ◽  
Original Data ◽  
Time Varying ◽  
Sequential Simulation ◽  
Content Type ◽  
Verhulst Model ◽  
Practical Implications

Purpose – The purpose of this paper is to build a linear time-varying discrete Verhulst model (LTDVM), to realise the convert from continuous forms to discrete forms, and to eliminate traditional grey Verhulst model's error caused by difference equations directly jumping to differential equations. Design/methodology/approach – The methodology of the paper is by the light of discrete thoughts and countdown to the original data sequence. Findings – The research of this model manifests that LTDVM is unbiased on the “s” sequential simulation. Practical implications – The example analysis shows that LTDVM embodies simulation and prediction with high precision. Originality/value – This paper is to realise the convert from continuous forms to discrete forms, and to eliminate traditional grey Verhulst model's error caused by difference equations directly jumping to differential equations. Meanwhile, the research of this model manifests that LTDVM is unbiased on the “s” sequential simulation.

On the global attractivity and the solution of recursive sequence

2010 ◽  
Vol 47 (3) ◽  
pp. 401-418 ◽  
Author(s):  
Elsayed Elsayed
Keyword(s):  
Difference Equation ◽  
Global Attractivity ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
Special Cases ◽  
Recursive Sequence ◽  
The Difference

In this paper we study the behavior of the difference equation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x_{n + 1} = ax_{n - 2} + \frac{{bx_n x_{n - 2} }}{{cx_n + dx_{n - 3} }},n = 0,1,...$$ \end{document} where the initial conditions x−3 , x−2 , x−1 , x0 are arbitrary positive real numbers and a, b, c, d are positive constants. Also, we give the solution of some special cases of this equation.

scholarly journals Solutions of Some Difference Equations Systems and Periodicity

2019 ◽  
Vol 16 ◽  
pp. 8247-8261
Author(s):  
Elsayed M. Elsayed ◽  
Faris Alzahrani ◽  
Ibrahim Abbas ◽  
N. H. Alotaibi
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
Rational Difference Equations ◽  
Article Analysis

In this article, analysis and investigation have been conducted on the periodic nature as well as the type of the solutions of the subsequent schemes of rational difference equations with a nonzero real numbers initial conditions.

Numerical simulation of the fractional Lienard’s equation

2019 ◽  
Vol 30 (3) ◽  
pp. 1223-1232 ◽  
Author(s):  
Razan Alchikh ◽  
Suheil Khuri
Keyword(s):  
Decomposition Method ◽  
Design Methodology ◽  
Nonlinear Term ◽  
Initial Conditions ◽  
Recursive Algorithm ◽  
Content Type ◽  
Fractional Differential ◽  
Laplace Decomposition Method ◽  
First Time

Purpose The purpose of this paper is to apply an efficient semi-analytical method for the approximate solution of Lienard’s equation of fractional order. Design/methodology/approach A Laplace decomposition method (LDM) is implemented for the nonlinear fractional Lienard’s equation that is complemented with initial conditions. The nonlinear term is decomposed and then a recursive algorithm is constructed for the determination of the proposed infinite series solution. Findings A number of examples are tested to explicate the efficiency of the proposed technique. The results confirm that this approach is convergent and highly accurate by using only few iterations of the proposed scheme. Originality/value The approach is original and is of value because it is the first time that this approach is used successfully to tackle fractional differential equations, which are of great interest for authors in the recent years.

Robustness of convergence demonstrated byparametric-guiding andcomplex-root-tunneling algorithms for Bratu’s problem

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Zhi Liu ◽  
Tienmo Shih ◽  
Zhong Chen
Keyword(s):  
Fluid Flow ◽  
Boundary Value Problem ◽  
Design Methodology ◽  
Initial Conditions ◽  
Complex Root ◽  
Content Type ◽  
Related Literature ◽  
Bratu’S Problem ◽  
Newton Raphson ◽  
Heat And Fluid Flow

Purpose This study aims to propose the parametric-guiding algorithm, the complex-root (CR) tunneling algorithm and the method that integrates both algorithms for the heat and fluid flow (HFF) community, and apply them to nonlinear Bratu’s boundary-value problem (BVP) and Blasius BVP. Design/methodology/approach In the first algorithm, iterations are primarily guided by a diminishing parameter that is introduced to reduce magnitudes of fictitious source terms. In the second algorithm, when iteration-related barriers are encountered, CRs are generated to tunnel through the barrier. At the exit of the tunnel, imaginary parts of CRs are trimmed. Findings In terms of the robustness of convergence, the proposed method outperforms the traditional Newton–Raphson (NR) method. For most pulsed initial guesses that resemble pulsed initial conditions for the transient Bratu BVP, the proposed method has not failed to converge. Originality/value To the best of the authors’ knowledge, the parametric-guiding algorithm, the CR tunneling algorithm and the method that integrates both have not been reported in the computational-HFF-related literature.

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