Questions related to difference equations, which are discrete analogs of differential equations.

learn more… | top users | synonyms

1
vote
1answer
11 views

How to find the basic reproductive number of a discrete SIS epidemic model

I have been following a textbook called Mathematical Models in Population Biology and Epidemiology. The SIS model is given by the system \begin{aligned} S_{n+1} &= \Lambda + S_n e^{-\mu} ...
2
votes
0answers
31 views

What are the main similarities between difference equations and differential equations, specifically in methods for solving them?

I know that that difference equations can be used to represent discrete dynamical systems and differential equations can be used to represent continuous dynamical systems. Therefore both are ...
2
votes
0answers
64 views

A similar, but hopefully easier problem than Gilbreath's conjecture

Gilbreath's conjecture says that for every positive integer $n$, if we write out the first $n$ primes $2,3,5,7,11,13,\ldots,p_n$ take the differences between consecutive terms ...
0
votes
1answer
19 views

Using Difference Equations to Solve Word problems

While I was studying about finite differences I noticed a question in difference equations.Does anyone knows how to solve this using difference equations? WORD PROBLEM Imagine you are to jump from ...
0
votes
0answers
11 views

Let $g_n^k=p_{n+k}-p_n$, where $p_n$ is the $n$th prime. Does there exist $g_{k+1}^1=2$ such that $g_1^k,g_2^k,\ldots$ is a “Gilbreath sequence?”

Call $(S_i)_{i=1}^{\infty}$ a Gilbreath sequence if $1=\lvert S_2-S_1\rvert=\lvert \lvert S_3-S_2\rvert-\lvert S_2-S_1\rvert\rvert=\cdots$, i.e., if the sequence can be substituted for the primes in ...
0
votes
1answer
32 views

Probability of a population increasing from size $N$ to size $N + 1$ in a time interval $(t, t + dt)$?

Let $\lambda$ be the birth rate and consider a time interval $(t, t + dt)$. If we have a population of size $N$ the probability of it increasing to size $N + 1$ within the interval $(t, t + dt)$ is ...
1
vote
2answers
61 views

Solving $T(n)= 2T(n/2) + \sqrt{n}$ without master theorem (algebraically & recurrence tree)

$$T(n)= 2T(n/2) + \sqrt{n}$$ This recurrence was in a stackoverflow question, and I want to solve it without relying on the master method. The solution was given, but wolframAlpha gives a slightly ...
1
vote
0answers
39 views

Difference equation of second order system with zero

I saw from lecture notes that difference equation of a first order system is like this: (1) (2) (3) (4) 1. What happens between (3) and (4)? It looks like inverse Z-transform but according ...
1
vote
1answer
25 views

Difference Equation, verify expression is solution to the equation

I am reading a book on Probability, and do not know how to solve this example question. Consider the following difference equation and initial condition(s). In each case, verify that the expression ...
0
votes
2answers
33 views

Difference equation, special solution

I have the difference equation: $x_{n+2} - \frac{1}{2}x_{n+1} + \frac{1}{8}x_{n} = \cos(\frac{n\pi}{2})$ I am guessing the special solution is on the form: $A\cos(\frac{n\pi}{2}) + ...
2
votes
0answers
41 views

Validity of approximating a difference equation with a differential equation

Consider the following difference-differential equation defined for positive integer indices $k$ and $t$: $$ A_k(t+1)-A_k(t)=\beta \frac{ A_{k-1}(t)- A_k(t)}{\alpha+2 t} + \delta_{k \beta} . \tag{1} ...
0
votes
0answers
22 views

Recurrence relations and their solutions

I recently read an article about difference equations and found the solution of the fibonacci recurence there. It is this function: $f(n) = \frac{1}{\sqrt5}\left (\frac{1+\sqrt5}{2} \right )^{n}- ...
0
votes
0answers
24 views

Verifying solution of difference equation?

I have the following difference equation - $2h_{x+1} - 5h + 2h_{x-1} = 0$ for $x = 1, 2, ...., 19$ The boundary conditions are $h_0 = 1$ and $h_{20} = 0$ How would I go about verifying that $h_x = ...
0
votes
0answers
14 views

Elementary differential equations, difference equation

Find the effective annual yield of a bank account that pays interest at a rate of 7%, compounded daily; that is, divide the difference between the final and initial balances by the initial balance.
0
votes
1answer
30 views

Solving I. y[n+2]-(1/3)y[n+1]=sin(n) and II. y[n+2]+3y[n+1]-4y[n]=n-1 difference equations

I have two difference equations, which I just can't solve. I hardly even get the method, so if you could help me with the steps, I would be grateful. $y_{n+2}-\frac{1}{3}y_{n+1}=\sin(n)$ ...
0
votes
0answers
11 views

How to compute this Z-Transform?

The exercise is like this: $$y(k+1) - 3y(k) = 4^k$$ How do I compute $Z$ transform of $4^k$? I understand that I have to use the Z-Transform formula and the result after applying it is : $$sum ...
1
vote
2answers
30 views

How to analyze convergence of non-linear difference equation (recurrrence relations)

I've a couple of functions, such as: $Y(t+1)=2-\ln(Y(t))$ $Y(t+1)=(Y(t))^{-2}$ $Y(t+2)=e^{-Y(t)}$ and I need to analyze stability and convergence. No problem with stability, but I can't figure out ...
0
votes
0answers
11 views

Discrete Analoge Methods for solving difference equations

For solving non-linear first order differential equations we can use separation of variables (sometimes) or an integrating factor to convert a DE to an exact DE. Are there any analog methods for ...
2
votes
3answers
68 views

Why do we set $u_n=r^n$ to solve recurrence relations?

This is something I have never found a convincing answer to; maybe I haven't looked in the right places. When solving a linear difference equation we usually put $u_n=r^n$, solve the resulting ...
0
votes
1answer
50 views

Find the difference equation for {2, 4, 16, 256, …}

Write a difference equation to represent the change during the nth interval as a function of the previous term in the sequence. b. {2,4,16, 256,...} I know that an= 22n but I can't figure out how to ...
2
votes
0answers
25 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 $e_{i,0}=0$. ...
0
votes
1answer
34 views

$\Delta^kn^\alpha$ converges monotonically to zero when $\alpha<k$

I am trying to prove that the $k$-th finite difference of the series $n^\alpha$ converges to zero monotonically as $n\to\infty$ when $\alpha<k$. The differential analogue is ...
2
votes
1answer
68 views

Limit of a Discrete Dynamical System

For the system defined below, the point by point evolution remains bounded for all $t$ so I could see that some sort of limit exists. However, the question is what sort of limit is it -- a single ...
0
votes
2answers
131 views

Why do we use the term “equivalent” with Operators but “equal” with Functions?

Why do we speak in terms of "equality" when we deal with functions but "equivalence" when dealing with operators? To elaborate: Two functions, f and g are equal to each other (denoted: f=g) if: ...
2
votes
0answers
61 views

Boundedness of solutions of Difference equation

Consider a second order difference equation in complex plane, \begin{equation} z_{n+1}=\frac{\alpha + \beta z_{n}}{1+z_{n-1}},\qquad n=0,1,\ldots \end{equation} where the parameters $\alpha, ~\beta$ ...
4
votes
0answers
72 views

How to solve a non-homogeneous second-order linear difference equation with both a forward and a backward difference?

Quite a long title for this: I'm looking for the general solution of the following difference equation: $$ax_{t+1} -bx_t + x_{t-1} = c + u_t$$ where $a,b,c$ are real constants and $u_t$ is a bounded ...
3
votes
1answer
214 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 ...
2
votes
1answer
74 views

a system of finite difference equations

Let $a,b>0$ such that $ab<1$ consider the system$$x_{t+1}=x_ty_t+ay_t$$ $$y_{t+1}=x_ty_t+bx_t$$ I would like you to help me answer the following: find values $a$ and $b$ ​​for which the ...
1
vote
0answers
15 views

Literature on functional difference equations

dear community. I'm looking for books/guides on functional difference equations. Can you recommend some? Below I try to explain what kind of equations I have in mind. As an example, one of the ...
0
votes
2answers
50 views

Calculating a percentage between two numbers

I have two numbers, a minimum value, and a maximum value. I also have a percent. This percent helps me find a value between the two numbers, the minimum value and the maximum value. I cannot figure ...
1
vote
0answers
30 views

Is there a better notation for difference equations?

Difference equations are quite messy to deal with, esp. in constraint optimization with many time subscripts that invite mistakes. Is there a better notation? Something like Feynman diagrams for ...
1
vote
1answer
93 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. ...
0
votes
2answers
38 views

Is there a difference for discount per unit and discount per purchase total?

I can't find relevant tags for my question so I wonder if this is a good place to ask. I wanted to ask this a long time ago but keep forgetting. Let's suppose when shopping for 3 units of specific ...
0
votes
0answers
20 views

difference equation soling

Need to help sovling a differene quation :) $p_t = - \frac{p_{t-1} + \alpha + \gamma \beta}{\delta \beta}$ My Thoughts: $p_t = p_{CF} + p_p$ where $p_{CF}$ is the complementary function and $p_p$ ...
1
vote
1answer
58 views

Example for finite dimensional analog of integral transforms

I understand that integral transforms are generalisations of the dot product of functions that could be interpreted as infinite dimensional vectors. The most significant advantage then is that ...
4
votes
4answers
261 views

Finding the billionth number in the series: $2, 3, 4, 6, 9, 13, 19, 28, 42, \ldots $?

Series is defined as $$a_{n+1} = \lfloor\frac{3\cdot a_n}{2}\rfloor,\qquad a_0 = 2$$ It can be viewed as the number of animals starting from a single pair if any pair of animals can produce a single ...
1
vote
0answers
37 views

$\cos(2\arccos(\frac{a}{a+1})x$

I have trying to prove that this cosine map: $$\frac{r}{4}((a+1)\cos\left(2\arccos\left(\frac{a}{a+1}\right)\ \left(X_n-\frac12\right)-a\right)$$ is a logistic map. What I have done so far: Using ...
1
vote
0answers
41 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? ...
-2
votes
2answers
145 views

Using generation functions solve the following difference equation

Using generation functions solve the following difference equation $$ a_{n+1} - 3a_{n+2} + 2a_n = 7n ; n\geq0; a_0 = -1; a_1 = 3. $$
0
votes
1answer
32 views

Linear Difference Equations and how to solve for $y_n$

I am currently trying to study difference equations for my first year undergrad Calculus course. I am struggling to understand how they work. I am currently trying this question: ...
5
votes
5answers
318 views

What particular solution should I guess for this relation?

Just trying to solve a non-homogeneous recurrence relation: $$f(n) = 2f(n-1) + n2^n$$ $$f(0) = 3$$ Characteristic equation is: $$f(n) - 2f(n-1) = 0$$ $$a-2 = 0$$ $$a = 2$$ Homogeneous ...
0
votes
3answers
66 views

How to find the particular solution of a second order difference equation

I am trying to solve the second order difference equation, ...
1
vote
2answers
57 views

difference equation( recurrence relation)

Let $y_n$ satisfy the nonlinear difference equation: $$(n+1)y_n=(2n)y_{n-1}+n.$$ Let $u_n=(n+1) y_n$. Show that $$u_n= 2u_{n-1}+n.$$ Solve the linear difference equation for $u_n$. Hence find ...
0
votes
1answer
29 views

second order difference equation question

$y_{n+2} - 2y_{n+1} + 2y_n = 62^n$ sub $y_n= r^n$ then $y_{n+2}=r^{n+2}$, $y_{n+1}=r^{n+1}$ so $r^{n+2} - 2r^{n+1} + 2r^n = 0$ $r^n( r^2 - 2r + 2) = 0$ I got a problem here, I can solve for $r$, ...
0
votes
1answer
24 views

Help with difference/recursion equation change of variable

I am in a self study of Dynamic Systems and am reading through David Luenberger's book and cannot seem to figure the following question out. Solve the difference equation using a change of variables ...
2
votes
1answer
66 views

second-order difference equation

I have a second-order difference equation question. yn + 2 - 78yn = 23n^2 What is the value of root in auxiliary equation? I have tried searching for videos online but I don't really quite ...
1
vote
2answers
68 views

Reference Request: Difference Equations

I am taking a second course in calculus and came across sequences defined inductively, as in recursively. My teacher told the class that a general formula for the $n$th term can be obtained using a ...
0
votes
3answers
32 views

Is this possible? - Controlled system equation

Does this mathematical equation even make sense. This is taken From my controlled system book, where 4.12 doesn't make sense for.. How come is that true??
0
votes
0answers
50 views

Linear Constant Coefficient Different Equation

The question I have is about linear constant coefficient question but I don't really know for sure how to do it. The question is: Suppose that $N_{m+1}-N_m=f(N_m,N_{m-1})$.(a) How would you determine ...
3
votes
0answers
53 views

Conditions of a Monotonic Process?

$f$ is the output of a discrete time process described by $f(k)=\sum_{i=1}^{k-1}w_{ki}f(i)$ where $f(1)\geq0$ is a known initial condition and $w_{ki}\geq0$ are weights of previous states on the ...