Questions tagged [difference-equations]
Tag for the usage of RSolve, RecurrenceTable, DifferenceRoot, and other functions related to difference equations.
295
questions
3
votes
1
answer
115
views
How to replace the symbol n in DifferenceRoot
I'd like to employ another symbol (say alpha or k) instead of n in the following (type of) ...
- 2,205
7
votes
1
answer
239
views
Incorrect answer given by RSolve
Bug introduced in 9.0 or earlier and persisting through 11.0.1 or later
Also posted in Wolfram Community.
I tried to solve the following nonlinear recurrence relation using ...
- 173
0
votes
2
answers
387
views
Sampling a continuous function to create sequence of discrete values
My question is straightforward: How do I convert a continuous function into a discrete-valued function?
The function I want to convert is the JacobiAmplitude function:
...
0
votes
1
answer
132
views
solving a non linear (log-linear) recurrence relation
I'd like to solve the recurrence relation
$$\begin{align*} f(n+1) &= 2 f(n)^2 h(n) \\ h(n+1) &= h(n) f(n) \\ f(2) & = h(2) = 2 \end{align*}$$ for an explicit solution for $h$ and $f$. So ...
- 135
15
votes
2
answers
2k
views
Finite difference method not converging to correct steady state or conserving area?
I am working with the following PDE, which is an advection-diffusion type equation. It describes the movement of a fluid-fluid interface inside an annulus of inner radius $R_1$ and outer $R_2$ under ...
- 713
3
votes
0
answers
482
views
Lyapunov Exponent as a function of parameter
I am working at Rossler system and I would like to ask you for some help.
This system is described by equations:
\begin{array}{ll}
\dot{x} = -y - z \\
\dot{y} = x + 0.2y \\ \label{eq:UkladRosSP}
\dot{...
- 59
4
votes
4
answers
795
views
Solving Recurrence Equation $a_{n+1}=a_n(1-a_n)$
How to solve following recurrence equation
$$a_0=2$$
$$a_{n+1}=a_n(1-a_n)$$
I tried:
...
- 8,648
1
vote
0
answers
122
views
Solving recurrence equation depending on a parameter
I have a recurrence relation which depends on a parameter $t$ and I am trying to solve it using RSolve. This is what I typed:
...
- 111
1
vote
2
answers
418
views
Dynamical system problem x(t+1) = Ax(t)
I have a problem that have been trying to solve but it's not going so good. I would like some guidelines on how to work myself around this problem:
Two neighboring countries spy on each other and ...
- 199
13
votes
3
answers
483
views
Archimedes' Scheme to find $\pi$
Given an Archimedes' scheme
$$a_{n+1}=\frac{2a_nb_n}{a_n+b_n}$$
$$b_{n+1}=\sqrt{a_{n+1}b_n}$$
with initial values $a_0=2\sqrt{3},b_0=3.$
How do I prove using Mathematica that this converges to $\...
- 275
4
votes
2
answers
2k
views
Plotting logistic maps
I have a question about plotting in Mathematica in general, and this particular recurrence relation as an example:
$\qquad x _{n+1} = rx_{n}(1-x_{n})$
How can I plot the following plots as shown ...
- 43
1
vote
1
answer
130
views
System of polynomial equations with Solve does not converge
The Exp-method [1] is an analytical solution method for differential equations. Essentially, the method boils down to the solution of a system
of nonlinear polynomial algebraic equations. In [1] (and ...
- 13
0
votes
1
answer
175
views
Finding eigenvalues and eigenfunctions of difference operator?
Consider the following difference operator
For n=2 this operator action is given by:
...
- 11.5k
7
votes
1
answer
983
views
Calculating the Feigenbaum Constants
I would like to calculate the $δ$ and the $α$ Feigenbaum constants for the Logistic Map:
\begin{equation}
x_{n+1}=4μx_n(1-x_n)=f(x_n)
\end{equation}
as part of an undergraduate assigment of mine. I ...
- 73
3
votes
2
answers
309
views
DifferenceRoot question
I was doing the following sum:
$$\sum_{i=2}^k \frac{(-1)^i}{i-1} \binom{2k-i-1}{k-1}x^i$$
First, Mathematica simplifies it to some DifferenceRoot function:
...
- 617
0
votes
1
answer
69
views
How can I write a program implementing difference equations system with parameter [closed]
How can I make a program with this dynamical difference equations system
P(n + 1) = Pn - 0.1(Qn - 200)
Q(n + 1) = Qn + 0.5(Pn - 50)
But in instead of ...
user38513
4
votes
1
answer
320
views
How to approximate the solution of a system of quartic equations
In an attempt to evaluate the point at which the cycle 2 becomes unstable for the given map:
\begin{equation}
x_{n+1}=μ-x_n^4=f(x_n), \quad μ \in \mathbb{R}
\end{equation}
I have so far managed to ...
- 707
1
vote
1
answer
66
views
Incorrect difference equation answer
The difference equation 2.5(y(t)-y(t-1))=0.1y(t) has a general solution of y(t)=8(1.04)^t, thus with y(0)=8 the particular solution is y(t)=8(1.04)^t.
If I run the problem with Maple, I get exactly ...
- 425
1
vote
1
answer
349
views
Use Recurrence Table with Generating Function
I tried to use Generating Function to solve Fibonacci series using following function.
Input:
...
- 153
4
votes
0
answers
191
views
Generating Functions with multiple arguments
I want to solve the recurrence of Ackermann function using Generating Function in Mathematica.
...
- 153
7
votes
4
answers
718
views
Paying off an installment
How do I determine the weekly installment I have to pay, if my loan is, let's say, loan=5000, with interest rate of 2% every ...
- 473
7
votes
3
answers
452
views
Solving coupled recursion equations for a target variable
I have a set of coupled recursion equations and want to express the final solution in terms of one of the variables:
...
- 35.8k
3
votes
1
answer
206
views
Solve a set of recurrence relations with a constraint
I am trying to solve a set of recurrence equations as given below:
$p_{1, n}*(l + n*u1) = p_{1, n-1}*l + p_{1, n+1}*(n+1)*u1 + p_{2, n}*u2$,
$p_{2, n}*(l + n*u1 + u2) = p_{2, n-1}*l + p_{2, n+1}*(n+...
- 55
3
votes
1
answer
152
views
Making a table or expression based on previous values
I would like to illustrate rational expectations based on the following equation
p[[t+1]] == 1/(1 - ψ (β - 1)) m[[t+1]] - ψ (β - 1)/(1 - ψ (β - 1)) p[[t]]
by ...
- 425
0
votes
1
answer
244
views
8
votes
1
answer
488
views
Two dimensional recurrence
My actual problem is a bit more complicated, but let us first assume that we have a recurrence relation of the following type
$a_{n,l} = a_{n-1,l} + a_{n-1,l-1} + a_{n,l-1}$
The boundary is given, ...
- 111
4
votes
2
answers
166
views
How can I find the coefficients of the next recursive symbolic function?
I have the following function
$$
f(x+3k) = -mf(x+2k)+f(x)
$$.
If we get $x=x+k$, then
$$
f(x+4k) = -mf(x+3k)+f(x+k) = m^2f(x+2k)+f(x+k)-mf(x)
$$. Also we obtain the coefficients of the final functions,...
- 390
0
votes
0
answers
51
views
Recursions where one definition depends on another. How to Block or have local variables?
I have been using Mathematica to solve a second order differential equation using the second order Verlet method. My code looks like:
...
- 120
0
votes
2
answers
330
views
How to find negative numbers for my recurrence formula?
The generalized recurrence formula is
F[n_]:=F[n]=F[n-1]+F[n-p-1];
with the initial conditions
...
- 390
7
votes
5
answers
2k
views
Limit n->Infinity of recursive sequence
I have defined a recursive sequence
a[0] := 1
a[n_] := Sqrt[3] + 1/2 a[n - 1]
because I want to calculate the Limit for this ...
- 173
3
votes
1
answer
373
views
Finite Difference - heat generation in a square (square within a square)
I am trying to develop a code that will allow me to solve the following problem. Lets say that I have a square block, inside that square block I have a smaller square box thats at a certain ...
- 45
10
votes
2
answers
308
views
Evolution of a trait in an asexual population (performance tuning)
I just coded a simple simulation module that looks at the evolution of a continuous trait in a haploid asexually reproducing population under density dependent competition in discrete time (i.e. non-...
- 887
1
vote
1
answer
199
views
How do I use RSolve to solve the problem below?
RSolve can't compute the solution. Appears that Mathematica is having a convergence problem. I don't know if the boundary conditions are properly defined...
...
- 2,760
6
votes
6
answers
480
views
Recursion question
The first two terms of a sequence are $a_1 = 1$ and $a_2 = \frac {1}{\sqrt3}$. For $n\ge1$,
$$a_{n + 2} = \frac {a_n + a_{n + 1}}{1 - a_na_{n + 1}}.$$
What is $|a_{2009}|$?
So I tried (thanks to ...
- 14.5k
-2
votes
1
answer
86
views
3
votes
1
answer
174
views
Kernel crash when evaluating NDSolveValue
I had a code that was working fine and I tried to add an extra few lines to find the solution of the same equation with different conditions. The original NDSolveValue solutions work fine but when it ...
- 167
2
votes
1
answer
1k
views
Partial differential equation, Finite difference Method
For my research project I am trying to solve following partial differential equation in Mathematica
(V[r]*ψ[u, v])/4 + Derivative[1, 1][ψ][u, v] == 0
with ...
- 33
4
votes
2
answers
920
views
Show Factorial instead of Gamma in the result of RSolve
Now I wanted to solve for a recurring function with RSolve. Here's how I tried:
...
- 175
1
vote
0
answers
606
views
How to derive finite-difference scheme automatically on a quite general stencil
Info
This question is a generalization of the following one Derivation of numerical scheme for linear transport equation on a variable stencil.
Statement of a problem
Linear scalar hyperbolic (...
- 119
2
votes
0
answers
91
views
Why doesn't RSolve work here? [closed]
RSolve[
{θ[n + 1] == θ[n] + 0.25*ϕ[n],
ϕ[n + 1] == ϕ[n] - 2.5*Sin[θ[n]],
θ[0] == π/2, ϕ[0] == 0},
{θ[n], ϕ[n]}, n]
What's wrong with my code? I know ...
- 21
6
votes
1
answer
209
views
Random Variable in Recurrence Function
Following the previously published question, I'm looking for the solution of RecurrenceTable with explicit random variable. For example, something like
...
- 183
1
vote
1
answer
245
views
Finite Difference equations - slow computations
The following code I have written runs very slow, in fact, so slow that it does not finish computation.
...
- 273
6
votes
2
answers
232
views
Solution to Simultaneous Arithmetic/Geometric Mean Recursion Relations
I'm trying to solve the simultaneous convergent sequences of geometric/arithmetic means where $a_{n+1}=\frac{1}{2}(a_n+b_n)$ and $b_{n+1}=\sqrt{a_nb_n}$ and initial values are $a_0=1+x$ and $b_0=1-x$. ...
- 163
2
votes
2
answers
458
views
Why does Mathematica not evaluate the difference recurrence using `RSolve` or `RecurrenceTable`?
Mathematica does not evaluate the following recurrence of two variables using RSolve and only echoes the input:
...
- 808
1
vote
1
answer
109
views
2
votes
3
answers
142
views
Permutations and recurrence equation
First problem:
I need to show all Permutations with length 4 of this elements {x, y, z, w, t} but with this condition: the element z mustn't be after element t. I ...
- 23
6
votes
1
answer
2k
views
Discrete-time Lyapunov exponent plot
I am trying to create a Lyapunov exponent plot as a function of $\alpha$ for two functions:
$$
f(x) = (\alpha + 1)x - \alpha x^{3}
$$
and a piecewise function given by following Mathematica code:
<...
- 195
1
vote
1
answer
304
views
RSolve with integrations
I want to solve this question with Mathematica.
$$T_0(t)=c(t),\quad T_n(t)=a\int_0^t T_{n-1}(s)\, \mathrm{d}s+b(t).$$
Is it possible to solve $T_n(t)$ with RSolve?...
- 245
2
votes
1
answer
172
views
Speeding up a differential equation system with nested modulo periodic events
I have the following differential equation system with periodic Events :
...
- 887
2
votes
1
answer
230
views
Getting the first $n$ terms of a recursion relation
Given a recursion relation, I would like to generate the first $n$ terms (automatically) without having to go term by term.
Background
Each term is of the form $a_{n}(t)$, that is, $a_{n}$ is a ...
- 121