On the Asymptotic Behaviour of the Solutions of the linear Homogeneous Differential-Difference Equation with Real Constant Coefficients in a Critical Case

1969 ◽  
Vol 40 (4-6) ◽  
pp. 201-223
Author(s):  
D. S. Subramanyam
Keyword(s):  
Difference Equation ◽  
Asymptotic Behaviour ◽  
Critical Case ◽  
Real Constant ◽  
Constant Coefficients ◽  
Differential Difference Equation

A Singular Perturbation Problem and A Neutral Differential-Difference Equation

1974 ◽  
Vol 17 (1) ◽  
pp. 77-83
Author(s):  
Edward Moore
Keyword(s):  
Difference Equation ◽  
Taylor Series Expansion ◽  
Linear Difference Equation ◽  
Constant Matrix ◽  
Perturbation Problem ◽  
Constant Coefficients ◽  
Explicit Formulae ◽  
Differential Difference Equation ◽  
The Difference ◽  
Image Position

Vasil’eva, [2], demonstrates a close connection between the explicit formulae for solutions to the linear difference equation with constant coefficients(1.1)where z is an n-vector, A an n×n constant matrix, τ>0, and a corresponding differential equation with constant coefficients(1.2)(1.2) is obtained from (1.1) by replacing the difference z(t—τ) by the first two terms of its Taylor Series expansion, combined with a suitable rearrangement of the terms.

The differential-difference equation w'(s) = g(s)[w(s-l)-w(s)]

1977 ◽  
Vol 78 (1-2) ◽  
pp. 41-55 ◽  
Author(s):  
Gillian L. Slater
Keyword(s):  
Difference Equation ◽  
Asymptotic Behaviour ◽  
Asymptotic Behaviour Of Solutions ◽  
Differential Difference Equation ◽  
Asymptotic Behaviours

This paper is concerned with the asymptotic behaviour of solutions of the differential-difference equation w'(s) = g(s)[w(s— 1)—w(s)], where g(s) is a continuous real-valued function. g(s) is assumed to have one of the following asymptotic behaviours: algebraic, exponential algebraic, periodic or zero.

scholarly journals Note on some representations of general solutions to homogeneous linear difference equations

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Stevo Stević ◽  
Bratislav Iričanin ◽  
Witold Kosmala ◽  
Zdeněk Šmarda
Keyword(s):  
General Solution ◽  
Difference Equation ◽  
Difference Equations ◽  
Fibonacci Sequence ◽  
Linear Difference Equation ◽  
Linear Difference Equations ◽  
Second Order Difference Equation ◽  
Constant Coefficients ◽  
Order Difference Equation ◽  
Homogeneous Linear

Abstract It is known that every solution to the second-order difference equation $x_{n}=x_{n-1}+x_{n-2}=0$ x n = x n − 1 + x n − 2 = 0 , $n\ge 2$ n ≥ 2 , can be written in the following form $x_{n}=x_{0}f_{n-1}+x_{1}f_{n}$ x n = x 0 f n − 1 + x 1 f n , where $f_{n}$ f n is the Fibonacci sequence. Here we find all the homogeneous linear difference equations with constant coefficients of any order whose general solution have a representation of a related form. We also present an interesting elementary procedure for finding a representation of general solution to any homogeneous linear difference equation with constant coefficients in terms of the coefficients of the equation, initial values, and an extension of the Fibonacci sequence. This is done for the case when all the roots of the characteristic polynomial associated with the equation are mutually different, and then it is shown that such obtained representation also holds in other cases. It is also shown that during application of the procedure the extension of the Fibonacci sequence appears naturally.

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