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.
28
questions with no upvoted or accepted answers
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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
$$f_1(x)=ix-\psi _{e^{2 i}}^{(0)}\left(x+\frac{\...
6
votes
0
answers
372
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Hrushovski's proof of the Manin-Mumford Conjecture
For my master's thesis, I am studying Hrushovski's model-theoretic proof of the Manin-Mumford Conjecture. Among the references I have used are the following:
Lecture notes 'Model Theory of Difference ...
5
votes
0
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344
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When does a triangle of numbers have a zero row sum?
Suppose we have a triangle of numbers defined by the recurrence relation
$$\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],$$
for some ...
3
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0
answers
586
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Differential Equations vs Difference Equations
My question is:
Is there a duality between a solution of an ODE,PDE,SDE or integral equations with their analog counterpart in the discrete domain?
I mean if I know a solution to the difference ...
3
votes
0
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146
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Are there existing resources on modular-esque recurrence relations?
Does anyone know where I would be able to get information on analyzing a class of polynomial recurrence relations of a form like this?
$\begin{align*}
f_{n,k}(x) & =a(x)f_{n-1,k}(x)+b(x)f_{n-2,k}(...
2
votes
0
answers
47
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Coefficient growth upper bound of a recurrence relation
Consider the recurrence relation one can obtain from the radial Schrödinger equation for the hydrogen atom, where $\psi(r)=\sum_{n=0}^{\infty} a_nr^n$:
$$(n+3)(n+2)a_{n+2}+2a_{n+1}=2|E|a_n, n\geq0$$
...
2
votes
0
answers
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What are the limits of what the theory of time-scale calculus can capture?
Time-scale calculus [0], also called calculus on measure chains (introduced in this widely cited article 1 and also here [2]) unified the concept of derivative of a functions $\mathbb{R}\rightarrow\...
2
votes
0
answers
87
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Rational zeta series and differential-difference equations
In an earlier question, I mentioned I was looking for generalizations of $$\sum_{n=k+2}^{\infty} \binom{n-1}{k} (\zeta(n) -1) =1. \qquad \qquad (1) $$
A variation of the above identity arises by ...
2
votes
0
answers
102
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discrete Fourier transform for composition of differential operators on a grid
This question pertains to stability analysis of finite difference methods using the discrete Fourier transform.
Suppose I have a convection diffusion equation of the form:
(1) $\hspace{.5in}u_t + \...
2
votes
0
answers
383
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A two variable recurrence relation with conditionals
I have arrived at the combinatorial problem of enumerating certain types of ballot paths. This has led to analyzing the sequence defined by the following recurrence
$$
f(n,m) = \begin{cases} f(n, \...
2
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0
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112
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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 .
2
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0
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91
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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) \...
2
votes
0
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456
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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 ...
1
vote
0
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127
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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\...
1
vote
0
answers
42
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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 ...
1
vote
0
answers
292
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Eigenvalues of an (almost) pentadiagonal symmetric Toeplitz matrix
I am looking for analytic expressions for the eigenvalues of matrices of the form
$$A = \begin{bmatrix}
6 & -4 & 1 & 0 & 0 & 0 & 0 \\
-4 & 6 & -4 & 1 & 0 &...
1
vote
0
answers
182
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Uniform lower bound for the distance between terms of a linear recurrence sequence
Let $(u_k)_{k=0}^\infty$ be a non-degenerate linear recurrence sequence of algebraic numbers. Denote by $\alpha_1,\dots,\alpha_m$ its characteristic roots, and suppose that $\alpha := \underset{i}{\...
1
vote
0
answers
31
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Prove monotonicity of a system of difference equations
I have the following system of difference equations. Fix any $(\alpha,K,p)\in\mathbb R_+\times\mathbb R_+\times (0,1)$. Let $u_0=1, \nu_0=0$. The sequence $(a_t,w_t,\nu_t,u_t)_{t\geq 1}$ is defined as ...
1
vote
0
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99
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General solution for first-order difference equation
I have the following first-order difference equation
$$y_{t} = \frac{x_{t}}{1-\rho L} + \epsilon_{t}$$
where $L$ denotes the backshift operator, i.e., $L(x_{t}) = x_{t-1}$. I can obtain a solution ...
1
vote
0
answers
33
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How to change difference equation time steps when rearranging?
I am using difference equations to solve SDOF systems.
I have the system $$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
...
1
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0
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224
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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 ...
1
vote
0
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35
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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 ...
0
votes
0
answers
78
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Renormalization group in condensed physics and field theory
Renormalization group in field theory is differenct from the one in condensed physics in that the former satisfies a differential equation, but the latter does not.
Does Renormalization group in ...
0
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0
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49
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How to solve with FEM a semilinear elliptic equation?
I searched in many books regarding FEM how to solve semilinear elliptic equation, but I did not find too many things. They mostly treat linear and simple problems. For example in P.Ciarlet-The finite ...
0
votes
0
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68
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Urn model and recursion
We have an urn with $n$ white balls. In each iteration we pick a ball at random. If it's white, we paint it red and return it to the urn. If it's already red, we discard it. We lose the game if (after ...
0
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0
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241
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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
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124
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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_{...
0
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0
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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 ...