Tag for the usage of RSolve, ReccurenceTable, and implementation of finite difference method(FDM) in Mathematica

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21
votes
1answer
891 views

Has this implementation of FDM touched the speed limit of Mathematica?

Still, I'll use the implementation of the 1D FDTD method (you can simply understand it as a kind of explicit finite difference scheme for the Maxwell's equation) as the example. Just for completeness, ...
15
votes
1answer
287 views

Mathematica 9 cannot solve a difference equation that Mathematica 8 can

Consider RSolve[{a[n + 1] == 2 a[n] + n (1 + 2^n), a[1] == 1}, a[n], n] Mathematica 8 can solve this quickly, but Mathematica 9, even after being allowed to run ...
12
votes
3answers
287 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 $\...
10
votes
1answer
559 views

Is it possible to find a limit of a sequence given by its recurrence relation?

I need to calculate a limit of a sequence given by its recurrence relation. I tried the following: ...
8
votes
2answers
195 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-...
7
votes
1answer
556 views

Poisson PDE over a irregular region with FDM

The problem modeled by the Poisson PDE is related to the torsion of prismatic beams and I use the Finite Differences Method (FDM). I've managed to solve the equation over a rectangle region with ...
7
votes
1answer
241 views

How to find a recurrence relation for a sequence?

I have a sequence given by an explicit formula for n-th term: ...
6
votes
6answers
414 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 ...
6
votes
3answers
208 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: ...
6
votes
3answers
562 views

How can I solve a forward-backward-looking recurrence system?

I would like to (numerically) solve a forward-backward looking recurrence equation in Mathematica. The system is something like: ...
6
votes
1answer
193 views

RSolve[{y[0] == y0, y[t + 1] == (1 - (t + 1)/100) y[t]}, y[t], t]

When I evaluate this code: RSolve[{y[0] == y0, y[t + 1] == (1 - (t + 1)/100) y[t]}, y[t], t] I get an RSolve::bvnul error. I don't understand why. Yet, if I ...
5
votes
4answers
181 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: ...
5
votes
2answers
156 views

RSolve gives a Context::ssle error

Bug introduced in 9.0 or earlier and fixed in 10.0 I want to solve the following recurrence equation RSolve[{(k + 2) c[k + 2] q^(k + 1) - c[k] == 0}, c[k], k] ...
5
votes
2answers
83 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 $b_0=1-x$. ...
5
votes
1answer
141 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 ...
5
votes
1answer
79 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, ...
4
votes
1answer
730 views

RSolve Lotka-Volterra: A Closed Form Solution to this Difference Equation?

I would like to solve a difference equation, a Lotka-Volterra difference equation. I tried a model from http://www.stolaf.edu/people/mckelvey/envision.dir/nonDE.lotka-volt.html Using their ...
4
votes
5answers
298 views

Can I generate a “piecewise” list from a list in a fast and elegant way?

Suppose we have a number list a: n = 1000000; a = RandomReal[{0, 1}, {n}]; We want to generate a list ...
4
votes
1answer
352 views

Mathematica 10 issue: working with Entities slows down recurrence equations [duplicate]

I was looking at the Mathematica code given at http://community.wolfram.com/groups/-/m/t/326240 where example code is given to simulate the spreading of a supposed Ebola outbreak in Nigeria: ...
4
votes
1answer
152 views

How can I implement this recurrence equation in Mathematica?

I asked a similar question yesterday but today I realized that I was implementing the wrong function. I have tried changing the implementation but I can't get it to work. Problem: There is a river ...
4
votes
0answers
79 views
+50

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 ...
3
votes
2answers
92 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: ...
3
votes
3answers
164 views

Tangled up in sequences and recurrence relations

I took a trip through sequences and recurrence relations and I got tangled up since the documentation didn't help much. I'm sure you can. Given a sequence ...
3
votes
1answer
108 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,...
3
votes
2answers
173 views

Rook walk and RecurrenceTable in Mathematica

This appears in a combinatorics book: $a(m,n)=2 a(m,n-1)+2 a(m-1,n)-3 a(m-1,n-1)$ It is a recurrence equation for the number of rook walks from $(0,0)$ to $(m,n)$. The initial conditions are: $ a(...
3
votes
1answer
100 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 ...
3
votes
2answers
128 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: ...
3
votes
1answer
51 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 ...
3
votes
1answer
113 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 ...
3
votes
1answer
86 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 ...
3
votes
2answers
706 views

RecurrenceTable with a recurrence function of two variables

I have to calculate the cool down process of a regenerative heat exchanger. I solved the problem in Excel before, but now I want to do it with Mathematica. I want to create a table with the variables ...
3
votes
2answers
461 views

Nonlinear recurrence relation

I entered the following recurrence relation into RSolve and it just gave the question back to me. $$a_{n+1} = \frac{(1+a_n +(a_{n−1})^3)} 3$$ Is there another way to get a formula for the nth term?
3
votes
1answer
60 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+...
3
votes
2answers
281 views

RSolve with assumptions

I want to solve the following non-linear first order difference equation: $$ k_{t+1} = (1-a) k_t^{a} $$ with $0 < a < 1$, and $k_0 = \text{const.}$ . Of course, the solution is easily found, ...
3
votes
0answers
87 views

Generating Functions with multiple arguments

I want to solve the recurrence of Ackermann function using Generating Function in Mathematica. ...
3
votes
0answers
83 views

How to combine DifferenceRoot objects for odd and even-indexed terms

I'm trying to analyze a certain infinite sequence $S$, indexed by positive integers starting from $1$. It can be split into two subsequences: $S^{odd}$ by removing all even-indexed elements and ...
3
votes
0answers
130 views

An recurrence equation that can be solved in version 7 but not in version 8

Backslide introduced in 8.0, fixed at least in 9.0.1. A friend of mine showed me this code sample: ...
2
votes
2answers
218 views

How can I express this recurrence function in Mathematica?

\begin{equation} r_{i+1}=r_{i}+V_{i}{\Delta}t \end{equation} Where \begin{equation} V_{i}=(0.01, 0.40) \end{equation} and ${\Delta}t$ is the difference in time. My implementation (this is for ${\...
2
votes
3answers
134 views

Computing a sequence by recurrence

I am trying to create a list where each element of the list is calculated using the sum of all previous elements of the list and also uses the $N$th element of another list. The initial value of the ...
2
votes
2answers
130 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 ...
2
votes
1answer
104 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 ...
2
votes
1answer
123 views

How to solve coupled complex-valued equations faster and get simplified solution

I have six coupled equations. How to solve these equations symbolically faster in Mathematica. In these equations, variables p,q,x,y are ...
2
votes
3answers
106 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 ...
2
votes
2answers
164 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: ...
2
votes
1answer
106 views

Speeding up a differential equation system with nested modulo periodic events

I have the following differential equation system with periodic Events : ...
2
votes
1answer
96 views

RSolve and a recurrence relation

In the book An Introduction to the Analysis of Algorithms there is a recurrence relation for the number of compares used in Quicksort algorithm: $$ C_N=N+1+\sum_{0≤k≤N-1}\frac{1}{N}(C_k+C_{N−k−1}) $$ (...
2
votes
1answer
110 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 ...
2
votes
2answers
189 views

RSolve with conditional equation

I have the following conditional equation: $f(n) = 1+2 f(n/2)$ when n is even $f(n) = 1+f((n-1)/2)+f((n+1)/2)$ when n is odd I want to represent this as a conditional equation to be used in ...
2
votes
1answer
199 views

How can I get DSolve to solve my ODE?

I evaluated DSolve[{x[t] x''[t] + x'[t]^2 - g x[t] == 0, x[0] == 0, x'[0] == 0}, x, t] and the output was {}. But this ODE ...
2
votes
1answer
150 views

RSolve: Errors while solving Partial Difference Equation

I tried solving following partial difference equation in Mathematica by typing ...