On the non-linear difference equation Δxn = kΦ (xn)

1932 ◽  
Vol 28 (2) ◽  
pp. 234-243 ◽  
Author(s):  
J. B. S. Haldane
Keyword(s):  
Differential Equation ◽  
Natural Selection ◽  
Finite Difference ◽  
Difference Equation ◽  
Linear Difference Equation ◽  
Finite Difference Equation ◽  
Non Linear ◽  
Image Position

1. Among the equations arising in the theory of natural selection is the finite difference equationwhere k is a constant which may have any value between 1 and − ∞ inclusive, but is often small. Its solution is discussed in the accompanying paper(1). It is a particular case of the equationwhere Φ(x) is a known one-valued function of x. When k is small this may obviously be solved approximately by treating it as a differential equationwhence

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.

scholarly journals On the Relation between Pincherle's Polynomials and the Hypergeometric Function

1920 ◽  
Vol 39 ◽  
pp. 58-62 ◽  
Author(s):  
Bevan B. Baker
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Hypergeometric Function ◽  
The Difference ◽  
Image Position

1. The Pincherle polynomials are defined as the coefficients in the expansion of {1 − 3 tx + t3}−½ in ascending powers of t. If the coefficient of tn be denoted by Pn(x), then the polynomials satisfy the difference equationand Pn(x) satisfies the differential equation

Sur l'existence d'une longueur élémentaire et d'un intervalle de temps élémentaire

1978 ◽  
Vol 56 (8) ◽  
pp. 1109-1115 ◽  
Author(s):  
Robert Lacroix
Keyword(s):  
Finite Difference ◽  
Difference Equation ◽  
One Dimension ◽  
Time Interval ◽  
Finite Difference Equation ◽  
Elementary Length

We have briefly examined several studies which have been made concerning the introduction of an elementary length l0 and an elementary time interval t0 into physical theories. We have discussed the arguments which we have found, arguments formulated by other authors, and which support the hypotheses concerning the existence of l0 and of t0. A finite difference equation is proposed and the solutions of some problems of movement in one dimension are given.

Solutions of a non-linear differential equation. I

1967 ◽  
Vol 63 (3) ◽  
pp. 743-754 ◽  
Author(s):  
C. E. Billigheimer
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Image Position

We consider in this paper solutions of the equationwhere the primes indicate differentiation with respect to s, and a, b, c are constants.

Dynamics of a non-linear difference equation

2006 ◽  
Vol 178 (2) ◽  
pp. 250-261 ◽  
Author(s):  
Reza Mazrooei-Sebdani ◽  
Mehdi Dehghan
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Non Linear

scholarly journals Solution of a general homogeneous linear difference equation

1980 ◽  
Vol 22 (2) ◽  
pp. 254-254
Author(s):  
V. N. Singh
Keyword(s):  
Difference Equation ◽  
Linear Difference Equation ◽  
Image Position ◽  
Homogeneous Linear

In Section 3, since relation (6) is valid only for n ≥ 2r, the condition n ≥ r in relation (9) should be replaced by n ≥ 2r. When r ≤ n ≤ 2r − 1, relation (9) still holds but now relations (6) and (7) should be replaced, respectively, by the relationsandthe asterisk again denoting omission of the last column. The proof of (9) for r ≤ n ≤ 2r − 1 is exactly similar to its proof for n ≥ 2r.

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