27
votes
Nonlinear differential equation: numerical solution
Introduction
I think there are several questions on this site about ODEs of the form
$$(x-a)^2 u''(x) = F(x,u,u')$$
with an initial condition at $x=a$.
There is no general guarantee that solutions ...
- 216k
25
votes
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How do I use the new nonlinear finite element in Mathematica 12 for this equation?
OK, there are a few things going on here. Let me explain them in turn. First, as the message suggests, this should be written in Inactive form (we'll get to the why later). If you click on the three ...
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22
votes
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20
votes
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Plotting separatrices for nonlinear system
We can solve (approximately) for the initial conditions of solutions that approach an equilbrium by comparing the displacement vector from the equilibrium with the vector field of the ODE. Such ...
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17
votes
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How does Rescale[] handle infinities?
The answers of the original questions by Szabolcs:
What does Rescale do when infinities are present? What's the justification for this behaviour? Where is it documented?
were guessed correctly ...
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16
votes
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Is there any predictor-corrector method in Mathematica for solving nonlinear system of algebraic equations?
Since @hesam asked about a command, and to get a better understanding of @DanielLichtblau's approach, I tried to generalize it and package it in a function. Feedback would be appreciated!
...
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15
votes
Plotting a Bifurcation diagram
Please note that though cosmetically appealing this is not rigorous or the best insight into basin of attraction of system as pointed out by bbgodfrey in comment below. I leave it as, perhaps, a road ...
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15
votes
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Singularity error. What is actually causing the problem here?
Description of the issues
It is a bit unusual to discuss an ODE system as a function of a parameter ν but with a fixed initial condition ...
- 216k
15
votes
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Linearization of a nonlinear system
a) To find equilibria, use Solve:
eq = Solve[{A - B*x - x*y^2, A*(x*y^2 - y)} == {0, 0}, {x, y}]
b) Linearizing around the ...
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14
votes
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Nonlinear PDE solver
The error message is misleading. NDSolve fails, because not enough boundary conditions in t have been supplied. If, for ...
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13
votes
Is there any predictor-corrector method in Mathematica for solving nonlinear system of algebraic equations?
As noted by @ChrisK, this works better starting at the top. Reason being there are no real solutions below the parameter value of 48.
Using FoldList one can ...
- 55.5k
13
votes
What can one do with extremely stiff problem in NDSolve?
EDIT #2
My error was useful. It brought me to the conclusion that the difficulties in solving the PDE of the OP are due to the drift term
$$\frac{\partial (x u(x,t))}{\partial x}$$
If the drift ...
- 12.8k
13
votes
Conservation of area solving a PDE via finite difference scheme
Partly NDSolve-based solution
Use a higher even order spatial grid to discretize the PDE to an ODE set seems to be a good approach. The definition of ...
- 52.5k
13
votes
How to apply different equations to different parts of a geometry in PDE?
Denote the disk by $\varOmega$ and its boundary by $\varGamma = \partial \varOmega$. I'd prefer to denote the function residing on the boundary by $u \colon \varGamma \to \mathbb{R}$; the function on ...
- 95.6k
13
votes
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Finding the Period of a Limit Cycle
Although it's primarily designed for ecological models, my EcoEvo package can help. First, you need to install it with
...
- 17.7k
12
votes
Numerical solution of coupled ODEs with boundary conditions
This is the most difficult of the nearly two dozen nonlinear ODE separatrix computations that I have encountered on Mathematica.SE. Nonetheless, it can be can be solved by a systematically refined ...
- 58.4k
12
votes
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Heat convection differential equations from 1952 - Mathematica "fails to converge"
The problem is with the default starting initial conditions used by the shooting method in NDSolve. The shooting method is where ...
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12
votes
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DSolve doesn't work
Edit: Derivation of symbolic solution
As commented by Mariusz Iwaniuk), the initial conditions in the question are inconsistent. Setting ...
- 58.4k
12
votes
Routh-Hurwitz criterion not giving correct answer when done manually?
I can't say why your approach didn't work, but my RouthHurwitzCriteria function uses a simplified test for 3x3 matrices due to Fuller (1968), which I first learned about from Gandolfo (1997):
There ...
- 17.7k
11
votes
Accepted
How to apply different equations to different parts of a geometry in PDE?
Since I have the code to solve the original problem described in the article GDI-Mediated Cell Polarization in Yeast Provides Precise Spatial and Temporal Control of Cdc42 Signaling, I will give here ...
- 34.1k
11
votes
Finding the Period of a Limit Cycle
Here is a simple approach to get the period of the unknown limit cycle.
The idea is to approximate the limitcycle by a circle (1st harmonic) around the mean of the limitcycle:
solution ...
- 37k
11
votes
Accepted
Steffensen's Method Implementation in Mathematica
I think you did everything right. The problem is that {Vx[1,1],Vy[1,1]} equals {253, 292} which is very far from ...
- 95.6k
10
votes
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Solve stiff system by shooting method
The goal of this question, I believe, is selecting Φ[r0] so that the solution connects smoothly to the asymptotic solution at large ...
- 58.4k
10
votes
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Stiff second order ODE
The equation is not stiff, despite the claim by Mathematica. It can be solved by using the "Shooting" Method.
...
- 58.4k
10
votes
Solving a coupled nonlinear PDE using low level FEM programming
This not a complete answer, but you ReactionCoefficients do not have to correct shape I think:
...
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10
votes
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Nonlinear model fit not fitting with good parameters. I can find the coefficients manually and origin manages to fit anyway
You need to use a constraint for c to avoid complex numbers.
...
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10
votes
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Using Fourier Series to acquire Nonlinear ODE Periodic Solutions
Direct solution of the last equation in the question also is feasible, because the Fourier series converges very rapidly. as will be seen below. The equation for a three term expansion can be written ...
- 58.4k
10
votes
Time varying delay differential equations
As noted in my earlier comment, I am unaware of an existing Mathematica function that can solve the variable delay ODE in the question. Certainly, NDSolve objects, ...
- 58.4k
10
votes
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Schemes for nonlinear advection equation
First of all, if you just want to solve the equation, NDSolve is enough:
...
- 52.5k
10
votes
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Need to fit curve to 5 parameters: what's a problem with NonlinearModelFit?
Because there is a term x^d, I dropped the point {0, 0} from the data (see here).
Next, the terms ...
- 23.2k
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