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Questions tagged [differential-equations]

Questions on the symbolic (DSolve, DifferentialRoot) and numerical (NDSolve) solutions of differential equations in Mathematica.

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3answers
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Dynamic Euler–Bernoulli beam equation

I am trying to solve for the vibration of a Euler–Bernoulli beam. The equation is $\frac{\partial ^2u(t,x)}{\partial t^2}+\frac{\partial ^4u(t,x)}{\partial x^4}=0$ For the boundary conditions I ...
40
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1answer
3k views

Analogue for Maple's dchange - change of variables in differential expressions

Maple owns an interesting function called dchange which can change the variables of differential equations, but there seems to be no such function in Mathematica. ...
26
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2answers
6k views

How to fit 3 data sets to a model of 4 differential equations?

I'm a biologist and a newbie in Mathematica. I want to fit three data sets to a model consisting of four differential equations and 10 parameters. I want to find the parameters best fitting to my ...
12
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2answers
908 views

Boundary condition with spatial derivative is ignored by NDSolve

Consider the following differential equation: $$\begin{align*}&\rho C_p\left(\frac{\partial T}{\partial t}\right)=k\left[\frac{\partial^2 T}{\partial x^2}\right]+\dot{q}\\ &\text{at }x=0,\;\...
15
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2answers
5k views

Error entering equation in DSolve

I entered a command incorrectly as follows: DSolve[{y'[x]=y[x]},y[x],x] I am now experiencing: ...
36
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6answers
4k views

How to splice together several instances of InterpolatingFunction?

I have a set of InterpolatingFunction returned by NDSolve which are valid over different (but overall continuous) domains. How ...
21
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4answers
9k views

Find all roots of an interpolating function (solution to a differential equation)

I'm trying to find all the roots of the solution to a differential equation. Using NSolve or Reduce I don't get the roots, so I'm using an iterative method which I found in physicsforums.com. This ...
32
votes
3answers
46k views

Plotting a Phase Portrait

I'm trying to plot a phase portrait for the differential equation $$x'' - (1 - x^2) x' + x = 0.5 \cos(1.1 t)\,.$$ The primes are derivatives with respect to $t$. I've reduced this second order ODE to ...
25
votes
4answers
9k views

Solving an ODE in power series

How do I find a series solution to an ODE? I do not mean taking the Taylor series of an exact solution; I want to solve nasty nonlinear differential equations locally via plug and chug. Surely, that ...
13
votes
2answers
592 views

What boundary is added when boundary condition is not sufficient?

When insufficient boundary conditions are given to NDSolve for solving PDE, usually the warning NDSolve::bcart pops up: ...
67
votes
3answers
7k views

Numerically solving Helmholtz equation in 2D for arbitrary shapes

I would like to solve the Helmholtz equation with Dirichlet boundary conditions in two dimensions for an arbitrary shape (for a qualitative comparison of the eigenstates to periodic orbits in the ...
24
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4answers
2k views

Accessing Reduce from DSolve

When solving transcendental equations, Solve frequently warns us that inverse functions are being used so that some solutions may not be found. We also see that <...
21
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2answers
4k views

Basins of Attraction

How does one shade the basin(s) of attraction of a phase plot in Mathematica? I have been trying to do this using the system $$\begin{align*} \dot x &= y\\ \dot y &= -9\sin(x) - 0.20y \end{...
30
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1answer
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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, ...
9
votes
2answers
10k views

Solving a system of ODEs with the Runge-Kutta method

I´m trying to solve a system of ODEs using a fourth-order Runge-Kutta method. I have to recreate certain results to obtain my degree. But I'm a beginner at Mathematica programming and with the Runge-...
5
votes
3answers
2k views

Nonlinear differential equation: numerical solution

I have to find and plot a numerical solution for this second order differential equation: u''[x] + (u'[x]/x) - (u[x]/(x^2)) + u[x] - u[x]^3 = 0 where $0\leq x &...
24
votes
6answers
40k views

How can I plot the direction field for a differential equation?

I'd like to plot the graph of the direction field for a differential equation, to get a feel for it. I'm a novice, right now, when it comes to plotting in Mathematica, so I'm hoping that someone can ...
16
votes
3answers
10k views

How to solve ODE with boundary at infinity

y''[x] - x y[x] == 0 y[0] == AiryAi[0], y[∞] == 0 The analytic solution to this ODE is the Airy function y[x] == AiryAi[x] ...
2
votes
1answer
2k views

Why my differential equations become True? [duplicate]

I've been trying to solve a system of nonlinear differential equation, but the conditions are a bit weird. Two of the differentials equate to the same equation, but have different boundary ...
37
votes
2answers
5k views

Complex valued 2+1D PDE Schrödinger equation, numerical method for `NDSolve`?

Based on the heat equation of the Mathematica Manual tutorial, I wrote the complex counterpart (Schrödinger) equation, for the free particle propagation of an initial wavepacket. ...
58
votes
4answers
3k views

Can Mathematica solve Plateau's problem (finding a minimal surface with specified boundary)?

Given a closed curve $\mathcal C$ in three dimensions, is it possible to use Mathematica's built-in functionality to compute a minimal surface whose boundary is $\mathcal C$? For simplicity, let us ...
33
votes
4answers
7k views

Animation of double pendulum

Sadly, I am a completely newbie. I am studying Physics and in our theoretical physics class we got the task to solve the double pendulum using Mathematica. We just got the program, but no introduction ...
14
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2answers
5k views

Is it possible to do vector calculus in Mathematica?

I am trying to rearrange and manipulate some vector differential equations in Mathematica. As far as I understand you have to tell Mathematica that a variable is a vector by specifying the components ...
29
votes
3answers
3k views

Numerical solution of coupled ODEs with boundary conditions

I have to solve the following set of ODEs and just can't get good results using Mathematica $$ r\frac{d}{dr}\left(\frac{1}{r}\frac{d}{dr}A(r)\right)-\xi^2F(r)^2\left(A(r)-1\right)=0 $$ $$ \frac{1}{r}\...
16
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2answers
504 views

NDSolve uses different difference order for different spatial derivative when solving PDE

I found something this tutorial for method of line doesn't tell us. Consider the following toy example: ...
16
votes
1answer
544 views

Easy way to plot ODE solutions from NDSolve?

Inspired by the closed question Beautify a NDSolve Graph ! and a comment someone made to me not too long ago: Is there some quick way to plot NDSolve results ...
15
votes
1answer
2k views

Poisson solver using Mathematica

I am looking for some help with a Poisson solver I am writing in Mathematica. The code is quite long with Arrays plugged in, so the full details can be found at http://pastebin.com/uSrSDcW6 I am ...
6
votes
2answers
4k views

How to plot and solve the numerical solution of a integro-differential equation

I have a integro-differential equation of the form $y'(t) = - \int_0^t {y(t_1 )} e^{t_1 - t} dt_1, {\rm{ t}} \in {\rm{[0,10], y(0) = 1}}$ My code is: ...
6
votes
2answers
1k views

Why Can't `DSolve` Find a Solution for this ODE?

I wanted to find a basis for the set of solutions of the following ODE. $$y^{''}+\frac{1}{x^2+1}y^{'}(x)+\left[-1-\frac{1}{x^2+1}\right]y(x)=0$$ But when I try to use ...
17
votes
4answers
6k views

Lyapunov Exponent

Does anyone know a (simple) Mathematica code for computing the Lyuponov Exponent for the Rossler System? Thank you Rossler System: ...
9
votes
3answers
2k views

Can NDSolve handle discontinuous data?

Backslide introduced in 9.0, persisting through 11.0.1 It is possible to numerically solve a differential equation if not-smooth data are involved? For example the following instruction return the ...
11
votes
2answers
1k views

Finding a 3d curve from torsion and curvature with NDSolve

I'm trying to use the Frenet–Serret formulas to find the curve that matches the torsion and curvature I specify numerically with an InterpolatingFunction. The ...
11
votes
4answers
4k views

DSolve not finding solution I expected

Try to solve the following ODE via DSolve $$ \left\{\begin{aligned} y'(x)+2 y(x) e^x-y(x)^2 &= e^{2 x}+e^x \\ y'(0) &=1 \end{aligned}\right. $$ The ...
22
votes
2answers
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I failed to solve a set of one-dimension fluid mechanics PDEs with NDSolve

The fluid here has been assumed as single component perfect gas i.e. it obeys the equation $p=ρ R T$, the thermal conductivity is assumed as a constant, so the equation set is: ...
13
votes
2answers
456 views

Boundary Condition for Schrödinger Equation in Infinite Range

I am trying to simulate the movement of a coherent state in a quantum harmonic oscilator, but for some reason the answer diverges and there is a warning about not enought boundary conditions. Also, ...
11
votes
3answers
656 views

Vector form using NDSolve

Michael E2 wrote a wonderful solution for my question. Now I am considering the system: $$ \begin{align*} x'&=x^2 y,\ x(0)=1\\ y'&=-x y^2,\ y(0)=1 \end{align*} $$ I am wondering how I can ...
4
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1answer
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NDSolve with vector function

(Possible duplicate yet I still can't understand.) Basic 2D revolving around origin: ...
9
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3answers
1k views

How to solve the differential equation with Duhamel's integral?

How do I solve a differential equation with Duhamel's integral? I tried to solve it with NDSolve, but failed: ...
16
votes
3answers
9k views

Solving a time-dependent Schrödinger equation

I want to solve the time-dependent Schrödinger equation: $$ i\partial_t \psi(t) = H(t)\psi(t)$$ for matrix, time-dependent $H(t)$ and vector $\psi$. What is an efficient way of doing this so that ...
20
votes
2answers
5k views

Elegant way of obtaining the envelope of oscillating function [duplicate]

I am solving a differential equation numerically and the output is an oscillating function with the amplitude of the oscillation decaying in time. I would like to extract the power law governing this ...
22
votes
1answer
3k views

2D Heat equation: inconsistent boundary and initial conditions

I'm attempting to use NDSolve on a 2D boundary value problem with initial conditions. Upon running my code, I get the following message: "NDSolve::ibcinc: Warning: Boundary and initial conditions are ...
18
votes
3answers
1k views

Only final result from NDSolve

Finally, I started to play with differential equations in Mathematica. And I have faced the problem, which seems to me so basic that I'm afraid this question is going to be closed soon. However, I'...
17
votes
4answers
7k views

Change variables in differential expressions

I have a fairly complicated differential expression in terms of a variable r and two unknown functions of r, B[r] and n[r]. I want to do a Taylor expansion of this around r=infinity. I want to do this ...
12
votes
2answers
701 views

How to draw the image of a circle under the action of a transformation of the phase flow?

How to draw the image of a circle $x^2+(y-1)^2<1/4$ under the action of a transformation of the phase flow for the equation $\dot{x}=y,\ \dot{y}=-\sin x$? Here $\dot{x}$ means $dx/dt$. Any help or ...
7
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2answers
1k views

DSolve misses a solution of a differential equation

[Note that in the cited duplicate, DSolve not finding solution I expected, the general solution returned by DSolve is missing a solution for quite different reasons ...
8
votes
3answers
588 views

Position of discontinuous coefficient influences the solution of PDE

This issue is raised in the discussion under this post about heat flux continuity and I think it's better to start a new question to state it in a clearer way. Just consider the following example: <...
2
votes
2answers
1k views

How to visualize slope fields of differential equations without vectors?

I'm looking to visualize slope fields of differential equations for my differential equations course. Every example I see draws them as vectors, adding unnecessary "arrows" that, to me, are visually ...
6
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1answer
369 views

Implement finite Fourier transforms

Recently I came across finite Fourier transforms, which can be used for solving certain type of boundary value problem (BVP) of linear partial differential equation (PDE) with constant coefficient. ...
39
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2answers
2k views

Variable naming changes everything

Bug fixed in 10.0.0 I am having a rather unusual problem I do not understand with Mathematica where renaming one of the variables of my function causes the function to stop "working". Here is the ...
17
votes
3answers
13k views

NDSolve with Euler method

I want to solve this equation with NDSolve[] using the Euler method: x'[t] == 0.5*x[t]-0.04*(x[t])^2 with initial condition ...