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0
votes
1answer
30 views

Solving the difference equation $h(\vec x)\cdot A(\vec x)=\sum_{i=1}^m A(\vec x - \vec e_i)$

I am trying to solve the following difference equation: $$h(x_1,x_2,x_3,x_4) \cdot A(x_1,x_2,x_3,x_4)=A(x_1-1,x_2,x_3,x_4)+A(x_1,x_2-1,x_3,x_4)+A(x_1,x_2,x_3-1,x_4)+A(x_1,x_2,x_3,x_4-1)$$ with ...
1
vote
0answers
12 views

Bifurcations in flows on two dimensional torus

I want to have a research about bifurcations which are appeared in flows on two dimensional torus. Especially bifurcations that can not be seen in flows of $\mathbb{R}^2$. Can anyone introduce me ...
4
votes
1answer
229 views

for which values of $\theta$ does this equation $x_{n+1}=\cos(\theta)x^2_{n}-\sin(\theta)x^2_{n-1}$ have bounded solutions?

I would like to investigate the global behavior of the following equation : $$x_{n+1}=Ax^2_{n}-Bx^2_{n-1}$$ where $A(\theta)= \cos(\theta)$ and $B(\theta) =\sin(\theta)$ are nonnegative parameters ...
0
votes
0answers
28 views

Multivariate recurrence relation

consider the recurrence relation \begin{equation} \mathbb{I}(m<M)[- k_{\rm on} c_{m,n} + (m+1) k_{\rm off} c_{m+1,n}] + \mathbb{I}(m>0)[-k_{\rm off} m \, c_{m,n} + k_{\rm on} c_{m-1,n} ] + ...
2
votes
0answers
50 views

Discrete “difference” equations that involve changes in both shift and scale

A standard use of the Z-transform ($F(z) = \sum_n (f[n] \cdot z^{-n} )$) is to understand the effect of a difference equation on a signal. For instance: $y[n] = x[n] + y[n-1]$ $Y(z) = X(z) + Y(z) ...
7
votes
1answer
816 views

Boundedness of solutions of a difference equation

Is there someone who can show me how I can prove this conjecture? Or at least show me how to do the first implication ? Conjecture: Assume $\alpha,\beta, \lambda \in [0,\infty)$. Then every ...
0
votes
0answers
99 views

Help solving a recurrence relation

For solving a related probability problem, I need to solve the following recurrence relation:- $q(r,b,L)=\frac{b}{r+b}q(r,b-1,L)+\sum_{k=1}^{L}\frac{r}{r+b}\times\cdots \times ...
0
votes
0answers
75 views

Rational dynamical system with nonnegative paramaters

let $A$ be a rational system of the form :$\begin{cases} x_{n+1}=\frac{\alpha_{1}}{y_{n}} \\ y_{n+1}=\frac{\alpha_{2}}{z_{n}} \\ ...
2
votes
1answer
133 views

a second order difference equation related to a real polynomials which seems to have only real roots

I am seeking solutions to the following difference equation: $$2c_k-c_{k-1}-c_{k+1}=\ln(k+A)-\ln(k+B)$$ where $A>B>0$. This equation is related to a real polynomial (see here) which I want to ...
-1
votes
1answer
163 views

Generic way to solve f(x+1) - f(x) = g(x) when g(x) is given [closed]

All I have been looking around for a general way to solve the problem of $f(x+1) - f(x) = g(x)$, where $g(x)$ is given. Has this problem been studied before? If there does not exist such a general ...
2
votes
1answer
244 views

Non-linear 1st order difference equation

I have been trying to solve the following difference equation for some time now : $$u^3(n+1) = a - b\cdot u^2(n) + u^3(n), \qquad a \ne 0 \ne b$$ I have tried various substitutions, simplifications ...
2
votes
2answers
206 views

Resource on Infinite Systems of Difference Equations

I have asked this question previously at Math.stackexchange, but it seems to receive little attention there. In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer ...
3
votes
0answers
227 views

Differential Equations vs Difference Equations

My question is: Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain? I mean if I know a solution to the difference ...
1
vote
1answer
161 views

Vortex equations on cylinder

Solutions to the vortex equations for a closed Riemann surface are well known (moduli space is a symmetric power). What do we know about solutions on surfaces with boundary or non compact surfaces? In ...
3
votes
1answer
192 views

nonlinear delay differential equation

Consider the delay differential equation: $ y_x(x) = \sqrt{y(x-\bar{x})} $ where $y$ is the unknown function of $x$, and where $\bar{x}$ is a fixed parameter. This equation does not seem to have a ...
3
votes
2answers
310 views

delay differential equation

I'm looking for exact solutions, if such exist, for the following non-linear delay differential equation (DDE): $ y_x(x) = A y(x-1)^a $ where $ 0 < a < 1 $ and $ A > 0 $ are given ...
11
votes
1answer
480 views

Is exponent of discrete-analytic function also discrete-analytic?

Lets define a discrete analytic function such a function that is equal to its Newton series: $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$ Is function $g(x)=e^{f(x)}$ also ...
0
votes
4answers
756 views

What are other applications of difference equations in other branches of mathematics ?

What are some of interesting results that arise from using difference equations in number theory , Combinatorics or any other field ?
3
votes
0answers
119 views

Are there existing resources on modular-esque recurrence relations?

Does anyone know where I would be able to get information on analyzing a class of polynomial recurrence relations of a form like this? $\begin{align*} f_{n,k}(x) & ...
12
votes
1answer
698 views

Why is there no formula for partial sums of some simple series?

I'm pretty sure that the sequences like $F_n=\sum_{k=1}^n \frac 1k$ are not traces of elementary functions on positive integers (take any reasonable definition of "elementary" you want, just make sure ...
2
votes
1answer
279 views

Difference equation $A(n,x)=p(x)A(n-1,x-1)+q(x)A(n-1,x)$

I asked this question on MSE, but didn't get enough information. If it is a violation of some norms, let me know, I'll delete it. I'm having problem solving this difference equation. Initially I ...
1
vote
1answer
254 views

Extension of polynomial functions

Let $P$ be an interval in $\mathbf{R}$, $n \in \mathbf{N}$. Assume that a function $f: P \rightarrow \mathbf{R}$ satisfies $\Delta^{n+1}_h f(x)=0$ for every $x \in P$ and every $h>0$ such that ...
8
votes
0answers
934 views

Are these two functions equal?

The question here is sparked by the discussion inside this question about indefinite sum(antidifference) of tan(x). A proposed solution was a function $$f_1(x)=ix-\psi _{e^{2 ...
2
votes
2answers
547 views

Discrete-analytic functions

I do not know if such concept already exists but lets consider functions which are equal to its Newton series. We know that functions which are equal to their Taylor series are called analytic, so ...
5
votes
2answers
169 views

Is there a solution/approximation for the non-linear difference equation $c_n = c_{n-1}+c_{\lceil \alpha n \rceil}$, where $0 < \alpha < 1$?

Is there a solution/approximation for the non-linear difference equation $c_n = c_{n-1}+c_{\lceil \alpha n \rceil}$, where $0 < \alpha < 1$?
12
votes
4answers
1k views

Shannon's communication paper and finite differences

In Shannon's 1948 paper "A Mathematical Theory of Communication", early on he derives the equation $$N(t)=N(t-t_1)+N(t-t_2)+\ldots+N(t-t_n).$$ He then says "according to a well-known result in finite ...
6
votes
1answer
344 views

Smooth and analytic structures on low dimensional euclidian spaces

So it is relatively easy to show that there exists only one smooth structure on the real line $\mathbb{R}$. So here are 2 natural questions: Q1: Up to equivalence, is there only one real analytic ...
5
votes
0answers
266 views

When does a triangle of numbers have a zero row sum?

Suppose we have a triangle of numbers defined by the recurrence relation $$\left| n \atop k \right| = f(n,k) \left| n-1 \atop k \right| +g(n,k) \left| n-1 \atop k-1 \right| + [n=k=0],$$ for some ...
0
votes
0answers
145 views

What is known about fractions in difference equations?

I'm using the following book as reference: ** Walter G. Kelly and Allan C. Peterson Difference Equations Second Edition, Harcourt Academic Press, 2001. ** I gather from the book and from research ...
0
votes
0answers
236 views

A second order non-linear difference equations

I am trying to explicitly solve [ if possible and if the solution exists ] a second order non-linear difference equation of the form : $ a _ {n+2} ^2 + a_{n} ^2 + K = \lambda a_{n+1} $, where K ...
6
votes
4answers
1k views

Laplace's summation formula

I recently came across the following formula, which is apparently known as Laplace's summation formula: $$\int_a^b f(x) dx = \sum_{k=a}^{b-1} f(k) + \frac{1}{2} \left(f(b) - f(a)\right) - ...
2
votes
0answers
355 views

Can any antidifference (indefinite sum) of a function be expressed in elementary functions and generalized polygamma function if its integral can be expressed in elementary functions?

If the integral or multiplicative integral of a function can be expressed with elementary functions, does it mean its indefinite sum (antidifference) or indefinite product respectively can be ...
9
votes
3answers
1k views

Solving a general two-term combinatorial recurrence relation

What is known about explicit (not necessarily closed-form) solutions to the recurrence $$R^n_k= (\alpha n) R^{n-1}_k + (\alpha' n + \beta' k) R^{n-1} _{k-1},$$ with initial condition $R_0^0 = 1$ and ...
2
votes
1answer
186 views

Elementary proof of bounds on discrete derivative applied to $(1+n)^s$

I would like to show that for $s \in \mathbb{R}$ and a nonnegative integer $k$ $$ \triangle^k ((1+n)^s) \lesssim (1+|n|)^{s-k} $$ where $\triangle$ is the discrete derivative, i.e. $\triangle^1 ...
2
votes
1answer
2k views

Solving partial difference equation

I am trying to solve the following partial difference equation: $$A_k^{n+1}=(k+1)A_{k+1}^n+(n+2-k)A_{k-1}^n $$ with initial condition: $$\begin{cases} A_0^0&=1\\ A_1^0&=1 \end{cases}$$ I ...
1
vote
1answer
312 views

[Numerical Mathemtics] How to solve hexagonal central differences

I want to simulate a 2d linear wave equation on a circle ($\displaystyle\frac{\partial^2 z(x,y,t)}{\partial t^2}=v^2\cdot\left(\displaystyle\frac{\partial^2 z(x,y,t)}{\partial ...
6
votes
5answers
2k views

Discrete version of Ito's lemma

Could anyone give me some references where I could find (a) discrete version(s) of Ito's lemma (b) a proof how it converges to the continuous form in the limit (c) its usage within stochastic ...
12
votes
3answers
847 views

Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$?

Are there any nonlinear solutions to $f(x+1) - f(x) = f'(x)$? (Asked by bcross at math.iuiui.edu on the Q&A board at JMM.)