28 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 ...
  • 225k
25 votes
Accepted

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 ...
  • 38k
22 votes
Accepted

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 ...
  • 225k
22 votes
Accepted

Plotting a Bifurcation diagram

An alternative representation is ...
  • 59.6k
17 votes
Accepted

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 ...
16 votes
Accepted

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! ...
  • 18.9k
16 votes
Accepted

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 ...
  • 18.9k
16 votes
Accepted

Curve tracing for a given data set

You have to fit implicitly, for example fit a conic section (or ellipse ) ...
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 ...
  • 55.8k
15 votes
Accepted

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 ...
  • 225k
14 votes
Accepted

Nonlinear PDE solver

The error message is misleading. NDSolve fails, because not enough boundary conditions in t have been supplied. If, for ...
  • 59.6k
14 votes

NonlinearModelFit's fit is atrocious

We need to select another fit function( shift the function Sin[a*x] to Sin[a (x + p)] + q) ...
  • 49.6k
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 ...
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 ...
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 ...
  • 58.6k
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 ...
13 votes
Accepted

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 ...
  • 18.9k
13 votes
Accepted

NonlinearModelFit's fit is atrocious

Nonlinear optimization problems almost never just magically work without a push in the right direction. This is especially true for problems with multiple local minima like this one. You need to give ...
  • 19.7k
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 ...
  • 59.6k
12 votes
Accepted

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 ...
  • 225k
12 votes
Accepted

DSolve doesn't work

Edit: Derivation of symbolic solution As commented by Mariusz Iwaniuk), the initial conditions in the question are inconsistent. Setting ...
  • 59.6k
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 ...
  • 18.9k
12 votes
Accepted

Are there practical Mathematica tools for detection of limit cycles in two dimensional dynamical systems?

In general, I'm not sure there's a good algorithm to find all limit cycles for a given set of equations. But if there's one in particular you want to find, then it's not too hard with a decent ...
  • 18.9k
11 votes
Accepted

How to stop DSolve from solving equations

It seems that Solve will be called during the DSolve calculation, so if you stop Solve, the <...
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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 ...
  • 37.7k
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 ...
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 ...
10 votes

Solving Non-linear System of Equations

These equations are solvable, but the process is exceptionally slow and the output huge. To help Solve, replace the approximate real numbers ...
  • 59.6k
10 votes
Accepted

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 ...
  • 59.6k
10 votes
Accepted

Stiff second order ODE

The equation is not stiff, despite the claim by Mathematica. It can be solved by using the "Shooting" Method. ...
  • 59.6k

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