Questions tagged [difference-equations]

Difference equations, including linear and nonlinear equations, discrete version of topics in analysis, partial difference equations, oscillation theory, periodic solutions, almost periodic solutions, bifurcation theory, stability theory.

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What are the limits of what the theory of time-scale calculus can capture?

Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article 1 and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
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1 vote
0 answers
32 views

Time-scale calculus (an similar approaches - measure chains) on more general "time" sets

Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article [1] and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
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2 votes
1 answer
249 views

Difference equation satisfied by discrete harmonic functions on square lattice

A function $f:\mathbb Z^2 \rightarrow \mathbb R$ is said to be discrete harmonic if it satisfies the discrete Laplacian equation $$ \Delta f(m,n) = f(m+1,n)+f(m-1,n)+f(m,n+1)+f(m,n-1)-4f(m,n) = 0~. $$ ...
1 vote
0 answers
40 views

Help with a surface of delay differential equations

This question is difficult for me to phrase, as it's very much outside of my mathematical purview. This is a question which intersects directly with my research, but as I work predominantly in ...
2 votes
1 answer
51 views

High order difference operator applied to 1/u

I need a formula for $\Delta^k \frac 1 u$, where $u(x)$ is a strictly positive function, $\Delta^k$ is the difference operator defined recursively as $\Delta^k=\Delta^1 \Delta^{k-1}$ and $\Delta^1 u(x)...
0 votes
1 answer
82 views

Unique zero solution to a difference equation via Laplace transform

We want to prove that the unique solution to the following difference equation is the null one: $$ au(x)+b\mathbf{1}_{(0,\frac{1}{2})}(x)u(x+\frac{1}{2})+c\mathbf{1}_{(\frac{1% }{2},1)}(x)u(x-\frac{1}{...
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1 vote
0 answers
198 views

Eigenvalues of an (almost) pentadiagonal symmetric Toeplitz matrix

I am looking for analytic expressions for the eigenvalues of matrices of the form $$A = \begin{bmatrix} 6 & -4 & 1 & 0 & 0 & 0 & 0 \\ -4 & 6 & -4 & 1 & 0 &...
1 vote
0 answers
172 views

Uniform lower bound for the distance between terms of a linear recurrence sequence

Let $(u_k)_{k=0}^\infty$ be a non-degenerate linear recurrence sequence of algebraic numbers. Denote by $\alpha_1,\dots,\alpha_m$ its characteristic roots, and suppose that $\alpha := \underset{i}{\...
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1 vote
0 answers
27 views

Prove monotonicity of a system of difference equations

I have the following system of difference equations. Fix any $(\alpha,K,p)\in\mathbb R_+\times\mathbb R_+\times (0,1)$. Let $u_0=1, \nu_0=0$. The sequence $(a_t,w_t,\nu_t,u_t)_{t\geq 1}$ is defined as ...
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2 votes
0 answers
83 views

Rational zeta series and differential-difference equations

In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$ A variation of the above identity arises by ...
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-1 votes
1 answer
149 views

Is this recurrent sequence decreasing?

Let $S_n$ be defined as $\frac{1}{n}\sum_{t=1}^{t=n} [px_t^2 - (p+q)x_t]$ where $x_t = 1-(1-p-q)^t$. We want to find the conditions on $p$ and $q$ such that $S_n$ is monotonically decreasing for all $...
0 votes
1 answer
323 views

How to solve this conditional recurrence relation?(two variable and conditions)

I am trying to solve the following recurrence relation $4 \leq n \;\; \; \; \; \;$ and $\; \; \; \; 2\leq i \leq \lfloor{\frac{n}{2}}\rfloor$ $F(2i,n)=$ $\begin{cases} \frac{1}{2(2i)-5}F(2i-2,...
2 votes
0 answers
84 views

discrete Fourier transform for composition of differential operators on a grid

This question pertains to stability analysis of finite difference methods using the discrete Fourier transform. Suppose I have a convection diffusion equation of the form: (1) $\hspace{.5in}u_t + \...
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2 votes
3 answers
455 views

Does this deceptively simple nonlinear recurrence relation have a closed form solution?

Given the base case $a_0 = 1$, does $a_n = a_{n-1} + \frac{1}{\left\lfloor{a_{n-1}}\right \rfloor}$ have a closed form solution? The sequence itself is divergent and simply goes {$1, 2, 2+\frac{1}{2}, ...
1 vote
0 answers
89 views

General solution for first-order difference equation

I have the following first-order difference equation $$y_{t} = \frac{x_{t}}{1-\rho L} + \epsilon_{t}$$ where $L$ denotes the backshift operator, i.e., $L(x_{t}) = x_{t-1}$. I can obtain a solution ...
1 vote
1 answer
248 views

On difference identities and $[K:F]$

Let $(K,\sigma)$ be a difference field of characteristic $0$, i.e. equiped with field morphism $\sigma:K\rightarrow K$. Assume that $K$ satisfy a non-trivial univariate difference polynomial identity $...
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1 vote
0 answers
27 views

How to change difference equation time steps when rearranging?

I am using difference equations to solve SDOF systems. I have the system $$m\ddot{y_i}+c\dot{y_i} + ky_i = x_i$$ Using the difference equation results for the derivatives, I am meant to end up with ...
5 votes
1 answer
913 views

A strange two-variable recursion

In some work I was doing with a colleague the following function of two natural number variables, defined by a recursion, came up and we have no clue how to solve it. Any suggestions or improvements ...
6 votes
0 answers
346 views

Hrushovski's proof of the Manin-Mumford Conjecture

For my master's thesis, I am studying Hrushovski's model-theoretic proof of the Manin-Mumford Conjecture. Among the references I have used are the following: Lecture notes 'Model Theory of Difference ...
2 votes
3 answers
211 views

Difference equation and formal series

For a given formal series $g(x)=\sum_{k=0}^\infty g_k x^k$ I would like to find a formal series $f(x)=\sum_{k=0}^\infty f_k x^k$ such that they satisfy the difference equation $$ f(x+1)-f(x)=g(x). $$ ...
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7 votes
2 answers
487 views

Combinatorial Identity with Connection Coefficients and Falling Factorial $\langle i x\rangle_n$

Let $j, k ,n$ be nonnegative integers such that $0 \leq j, k \leq n \leq k +j $. Pick integer $m$ such that $0 \leq m \leq k + j - n$. Let $\langle x \rangle_m$ denote the falling factorial $x(x-1)\...
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2 votes
1 answer
146 views

A problem from linear algebra and difference equations

Let $A$ be a linear second-order difference operator acting on the space of complex sequences as $$(Af)_{n}=f_{n-1}+a_{n}f_{n}+f_{n+1}, \quad n\in\mathbb{Z},$$ where $a_{n}\in\mathbb{C}$. Further, let ...
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1 vote
0 answers
183 views

Bounds for Discrete Poisson Kernel of a Square

I am having difficulty in proving the lower bound of the discrete Poisson kernel of a square denoted as $H$ described below. It is stated in Gregory F. Lawler's Randomm Walk and the Heat Equation as ...
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1 vote
1 answer
168 views

boundedness of a nonlinear recursive sequence

Consider a real sequence $(x_k)$ for $k=0,1,2,\dots,N$ as $x_0=1$ and for $k>0$ $$ x_k=x_{k-1}+\frac{\gamma}{N}x_{k-1}^2,\qquad (\gamma>0).$$ I wonder to show that the sequence is bounded as $N\...
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2 votes
0 answers
369 views

A two variable recurrence relation with conditionals

I have arrived at the combinatorial problem of enumerating certain types of ballot paths. This has led to analyzing the sequence defined by the following recurrence $$ f(n,m) = \begin{cases} f(n, \...
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9 votes
3 answers
531 views

Finite realization of irrational transfer functions

In the field of digital signal processing, linear time-invariant systems play a distinguished role. These are the systems for which there exists an impulse response, a function $h:\mathbb{Z}\to\...
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1 vote
1 answer
423 views

Integer Polynomial solutions to functional equation

Recently I came across a functional equation which always has a polynomial with integer coefficients solution. Let $$ L_n(x)=(2 x+1)^2f(x+1)-4x(x+n+1)f(x)-((2 n+1)!!)^2\prod_{i=1}^n(x+i). $$ Problem: ...
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2 votes
1 answer
114 views

how to solve this equation? [closed]

$Q=-{\frac {q \left( {{\rm e}^{{\it nb}\,{\it nv}\,\theta\,{\it Td}}}-1 \right) }{1-{{\rm e}^{\theta\,{\it Td}}}} \left( 1-{{\rm e}^{{\frac { Q\theta\,{\it Td}}{p}}}} \right) \left( 1-{{\rm e}^{{\...
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3 votes
0 answers
158 views

Deriving Milne's predictor of order four from extrapolation polynomial [closed]

I am trying to derive the following Milne's predictor formula of order four for the differential equation $\frac{dy}{dx}=f(x,y)$ from an extrapolation polynomial of degree four. $$y_{n+1} = y_{n-3}+\...
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2 votes
1 answer
75 views

Sum of difference equation involving hypergeometric functions 1F0

I'm trying to prove the sum of a sequence given by $a_{n+1} = \frac{nb-x}{(n+1)b} a_n$ with $a_1 = 1$. This gives the solution $a_n = \frac{(-x/b)_n}{n!}$. When trying to work out what this sums to, ...
5 votes
2 answers
460 views

closed form solution of the following iterative equation?

is it possible to obtain a closed-form solution w.r.t. ${P_j:\forall j}$ (or in terms of special functions) for the following equations: $\alpha P_0=P_1$, $\alpha<1$ $\alpha P_j=P_{j+1}+P_{j+2}+\...
2 votes
0 answers
102 views

Geometric unfolding of a difference equation

Does anyone know a link to the (superb) slides of a talk given by Zeeman in 1996, I think (with same title as this question)? It used to be at www.math.utsa.edu/ecz/gu.html .
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5 votes
1 answer
427 views

Derivative in terms of finite differences

Consider expanding the differentiation operator in terms of the forward difference operator as $f' = \log(1 + \Delta)f = \displaystyle \sum_{n = 1}^{\infty} \frac{(-1)^{n + 1}\Delta^n f}{n}$. For some ...
0 votes
1 answer
89 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 ...
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1 vote
0 answers
29 views

Bifurcations in flows on 2-dimensional torus

I am doing research on bifurcations which appear in flows on the 2-dimensional torus, in particular on such which do not appear in flows on $\mathbb{R}^2$. Can anyone provide some references on ...
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6 votes
1 answer
383 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 ...
2 votes
0 answers
84 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
2 answers
976 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
0 answers
230 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 \frac{r-(k-1)}{r+b-(k-1)...
0 votes
0 answers
115 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}} \\ z_{n+1}=\frac{\alpha_{3+}+\beta_{3}x_{n}+\sigma_{3}y_{n}+\lambda_{3}z_{...
2 votes
1 answer
235 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 ...
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0 votes
1 answer
228 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 ...
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2 votes
1 answer
301 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 ...
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2 votes
2 answers
302 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 ...
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3 votes
0 answers
561 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 ...
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1 vote
1 answer
258 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
1 answer
493 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 ...
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4 votes
2 answers
547 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 constants....
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10 votes
1 answer
531 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 ...
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1 vote
4 answers
1k 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 ?