2
votes
0answers
19 views

Solution of partial difference equation

I want to find the explicit solution of the following difference equation $e_{i,j+1}=re_{i-1,j}+(1-2r)e_{i,j}+re_{i+1,j}+km_{i,j}$ where $r>0$, $k>0$ and $m_{i,j}$ are known and $e_{i,0}=0$. ...
3
votes
1answer
196 views

How to solve this recurrence Relation - Varying Coefficient

Sir,I have two questions related to this recurrence relation. It has been messing with me for long. Because of this I couldn't proceed my work for some time .This contains a polynomial term n+2 in ...
0
votes
1answer
44 views

General solution of a system of linear differential equations with multiple generalized eigenvectors

I am looking for general solutions for the linear sODE's $$\textbf{x}'(t) = A\textbf{x}(t)$$ with $t \geq 0$ and $A \in \mathbb{R}^{n \times n}$ Let focus on just real eigenvalues and eigenvectors. ...
1
vote
0answers
35 views

Closed form for a sequence defined recursively

Let $a_k$ be a sequence such that $a_0=0, a_1=0, a_2=1, a_3=1$ and $$a_{k+4}=-\frac{a_{k}+ka_{k+2}}{(k+1)(k+2)}$$ for $k\ge 0$. My question is: Is a closed form formula for $a_n, n\ge 4$ possible? ...
0
votes
0answers
29 views

Questions on Difference operators

Please I really need help on the following short problems on difference operators that I need even some clues on how to go by them: 1) $\sum_{t=1}^{4}{\dfrac{1}{(t+1)(t+2)(t+3)}}= ...
0
votes
0answers
23 views

A word problem in difference equations

The following problem is from Difference Equations by W. G. Kelly and A. C. Peterson (2nd edition). I just couldn't figure out where to start. 'Suppose that $t$ points are chosen on the perimeter of ...
1
vote
1answer
67 views

How to find solutions to the differential equation created by setting the Schwarzian Derivative equal to zero?

I'm in a Masters level course on difference equations, and last week we discussed theorems which can be applied to show the stability characteristics of non-hyperbolic equilibria of first order ...
1
vote
1answer
57 views

How do I solve the following difference differential equation

While studying a particular physical system, I arrived at the following difference differential equation: $$\frac{dx_n(t)}{dt} = -g \left\{\sqrt{(n + 1)(n + 2)}x_{n+1}(t) - (2n +1)x_n(t)\right\},$$ ...
1
vote
0answers
54 views

Simplification of differential equation when definition interval becomes small?

Assuming the following differential equation on the interval $0<x<c$ with a rational function $f(x,c)$ $$\left(\frac{d^2}{dx^2}+f(x,c)\right)y(x,c)=0,$$ what kind of simplifications (if any) ...
2
votes
1answer
107 views

Solving $f_n=\exp(f_{n-1})$ : Where is my mistake?

I was trying to solve the recurrence $f_n=\exp(f_{n-1})$. My logic was this : $f_n -f_{n-1}=\exp(f_{n-1})-f_{n-1}$. The associated differential equation would then be $\dfrac{dg}{dn}=e^g-g$. if ...
2
votes
0answers
51 views

Finding the best real value for $C$.

Consider the recurrence $f_{n+1}=f_n + \ln(f_n)$ with $f_0=2$. Also consider differential equations of type $g(0)=2$ and $\dfrac{d g}{d x}=\ln(g(x)- C \cdot \ln(g(x)))$. Lets call the solution ...
1
vote
0answers
71 views

When is it justified to approximate a difference equation with its corresponding differential equation?

Consider the difference equation $f_{x+1}-f_x=a(f_x)$ and the differential equation $g'_x=a(g_x)$. When and Why is it justified to say "$f_x - g_x = o(1) $ hence we can solve the difference equation ...
0
votes
0answers
47 views

Reduce a set of $N$ coupled differential equations to $N/2$ differential equations given a periodic boundary condition?

Is it possible to reduce a set of $N$ coupled differential equations of the form $U_n''=U_{n+1}-U_{n-1}-2U_n$ to $N/2$ coupled differential equations, given the periodicity condition that $U_{N+1}$ is ...
1
vote
0answers
51 views

Finding the linear representation of an ARMA process

I have the following $ARMA(2,2)$ process: $X_t + 0.9X_{t-1} = Z_t + 1.3Z_{t-1}$ I'd like to write it in the form: $X_t = \sum\limits_{i=0}^\infty\psi_iZ_{t-i}$ I just want to compute the first few ...
0
votes
1answer
195 views

Time-delay differential-difference equation

Is it possible that the system $$ \begin{cases} 2\dot{q}(t) + \dot{q}(t-1) + \dot{q}(t+1) = k & \text{if} \hspace{5mm} 0 \leqslant t \leqslant 2 \\ \dot{q}(t) + \dot{q}(t-1) = c & \text{if} ...
0
votes
1answer
65 views

Determine growth rate

Having struggled with calculating the following, I turn to you: In a model for growth af a certain type of cancer, the quantity of cancer-cells $N$ (measured in millions), can be described as a ...
1
vote
2answers
182 views

Differential to Difference equation with two variables?

For the following information : $$\frac{dx}{dt} = -10x+3y$$ $$\frac{dy}{dt} = 2$$ How do I convert this to a difference equation ?? I want to use a simple discretisation technique (first order) ...
2
votes
5answers
1k views

Differential equation to Difference equation?

I have the following equation : $$\frac{dx}{dt} = -5(x-2)$$ $$\frac{dy}{dt} = 0$$ How do I change this differential equation to a difference equation ? Do I use Euler forward method ? I remember ...
3
votes
0answers
147 views

The simplest delay differential equation

I am trying to understand a bit about solutions of delay differential equations, so I tried analyzing one of the most simple ones: $$u'(t)=-\beta u(t-1), \text{and for } t\in [-1,0), u(t)=\phi(t), ...
2
votes
2answers
166 views

Laplace transform exercise

I found this on Priestley's Complex Analysis in the Laplace transforms bit. Suppose $f$ satisfies $f'(t)=f(kt)$ for $t>0$, where $0<k<1$ and $f(0)=1$. Prove that ...
1
vote
0answers
34 views

What is the definition of Cauchy function associated with the differential or difference equations?

What is the definition of Cauchy function associated with the differential or difference equations? Where can I find the details?
3
votes
1answer
209 views

Is this a correct way to convert an convolution equation into differential/difference equation?

For functions $f,g,h$ that are defined over $\mathbb{R}$, suppose we have a convolution equation: $$ f = g * h. $$ I would like to convert it into a differential equation. Is it correct that $$ ...
1
vote
1answer
74 views

Does a fundamental solution set exist for homogeneous first-order difference equations?

When $A$ is diagonalisable, $\vec{x}_{k+1}=A\vec{x}_k$ implies that $\vec{x}_k = c_1\lambda_1^k\vec{v}_1 +...+c_n\lambda_n^k\vec{v}_n$ because an eigenbasis exists and any $\vec{x}$ can be ...
3
votes
1answer
302 views

Why some differential equation can be solved while similar difference equations cannot?

Take an equation $$w'+w-w^2-1=0$$ Its solution is $$w(x)=\frac{\sqrt{3}}{2} \tan \left( \frac{\sqrt{3}}2 C+\frac{\sqrt{3}}2 x\right)+\frac12$$ I wonder why a similar difference equation ...
0
votes
1answer
90 views

The puzzling eigenvalues of a differential equation system

I have a question here, in order to have a stable solution of a differential equation, is that right we should have all eigenvalues less one? But I have seen a incomprehensible situation in ...
0
votes
0answers
60 views

properties about number of groups of linearly independent solutions in the general solutions of linear functional-differential equations

Are there any effective methods to determinate the number of groups of linearly independent solutions in the general solutions of linear functional-differential equations of the form ...
10
votes
3answers
4k views

Links between difference and differential equations?

Does there exist any correspondence between difference equations and differential equations? In particular, can one cast some classes of ODEs into difference equations or vice versa?
1
vote
0answers
214 views

non-linear delay differential equation

I'm looking for an explicit (not numeric) solution to the following non-linear delay differential equation (aka difference differential equation). It's a sort of Riccati type equation. ...
2
votes
1answer
456 views

Correspondence between ODE and difference equation

In Wikipedia about difference equations, there is some description about correspondence between ODE and difference equation: If you consider the Taylor series of the solution to a linear ...