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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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### 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 ...
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### 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 ...
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### Numerically solving Helmholtz equation in 3D for arbitrary shapes

Context While studying manifold Learning I got interested in finding the eigenvectors of the Laplacian. (also in connection to this problem of solving the heat equation) Following this and that ...
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### A geometric multigrid solver for Mathematica?

Cross posted to community.wolfram.com Mathematica ships a variety of linear solvers through the interface LinearSolve[A, b, Method -> method] the most ...
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### Mathematica vs. Comsol for finite element analysis?

Being relatively new to finite element analysis, I was wondering how expert users assess Mathematica's capabilities in solving PDEs via the finite element method compared to other commercial tools (e....
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### 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. ...
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### Symbolic solution(s) to generalized Heat equation

Symbolic solution(s) to Heat equation? or more generally,(eventually) Green functions to known PDEs I am interested in variations of the heat equation: or more generally or even more generally (<...
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### In what way can M11.3's system modelling features be used by those who don't have SystemModeler?

Mathematica 11.3 includes some functionality from SystemModeler, but not the full SystemModeler environment. I assume that most people here are familiar with Mathematica, but not SystemModeler, or ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### Efficient Langevin Equation Solver

This question is not about good algorithms for solving stochastic differential equations. It is about how to implement simple codes in Mathematica efficiently exploiting Mathematica's programming ...
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### Calculating a potential function using the finite element method

This is my first attempt to use the Finite Element method available in version 10. There are questions and I am very open to suggestions. My example is flow around a cylinder which is a well known ...
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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 ...
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### 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 ...
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### How to use NDSolve to track equilibrium?

I am looking for an extension of NDSolve where integration runs until certain variables are settled at an equilibrium. Now I have a working solution in my sleeves ...
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### Finite element simulation of Airy waves

I am attempting to solve for waves on a water surface starting with a two dimensional solution. The equations are that the water must satisfy Laplace's equation everywhere with a time dependent ...
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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, ...
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### 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 <...
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### Phase portrait on a cylinder

It is very nice and very easy to make a sketch of a phase portrait with StreamPlot. For example, for the classical pendulum, defined by \begin{eqnarray*} \dot x&...
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### Is there any possibility to implement a structure like a ProgressIndicator into NDSolve?

It is already formulated in the title. NDSolve takes sometimes a considerable piece of time. It would be very practical to have some information on how long it is still to wait. So, any ideas? To ...
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### Ball Bouncing on Hilly Terrain

There is a maple code for the bouncing ball on the given curve. I tried to make this animation by using Mathematica Why does the following code not work? ...
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### Optimizing a Numerical Laplace Equation Solver

Laplace's Equation is an equation on a scalar in which, given the value of the scalar on the boundaries (the boundary conditions), one can determine the value of the scalar at any point in the region ...
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### Stress calculations using finite elements

A standard engineering problem is to calculate stresses in a structure due to applied forces. With the inclusion of the finite element method in version 10 this question attempts to investigate how ...
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### Mismatch between Mathematica and COMSOL in 3D FEM problem

I would like to solve an advection-diffusion problem on a torus domain. There are three Dirichlet conditions: One at the inlet (concentration $c=0$), one at the outlet ($c=0.5$) and one at the wall (\$...