Questions tagged [differential-equations]

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

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3
votes
2answers
922 views

Plotting several numerical solutions plus the analytic solution of ODE in one plot

I want to be able to plot several numerical solutions of an ODE, plus its analytical solution in one plot, in order to see how the numerical solutions converge towards the analytical one with respect ...
14
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3answers
5k views

NDSolve with vectors

I'm stumped. I'm trying to write this using vectors, but the 2nd derivative isn't being expanded like I expected it to be. This is a system of equations for a projectile with quadratic drag and ...
5
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1answer
1k views

Solve system of ordinary differential equations that doesn't have an initial condition (t=0), but has an inifinity condition (t=infinity)?

I have a question for solving t -> Infinity on Mathematica. First, I have a system of ODEs: ...
3
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1answer
328 views

How to solve a system of ordinary differential equations contain a interpolating function?

I'm wondering how I can solve a system of ODE that has a interpolating function? For example, z and y are ...
13
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1answer
4k views

Numerically solving an inhomogeneous partial differential equation

I'm trying to solve a cylindrical partial differential equation with boundary conditions. But I got an error message saying ...
39
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3answers
62k 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 ...
7
votes
2answers
23k views

How do I solve coupled ordinary differential equations?

I have four coupled ODE's. I am not sure how to plot and solve them using Mathematica. I won't give the exact problem, but the following is something analogous: The equations ...
6
votes
1answer
2k views

How do I find the best parameter to fit my data if the model is a interpolating function?

Hi I have a question regarding to find the best parameters for my model to fit my data. I have 3 ordinary equation, and I now just picked some parameters (...
2
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2answers
7k views

How to decouple a first order ODE system by eliminating coupled functions

Let's see an example of a first order ODE system : $$\begin{align*} y_1'&=a_{11}\cdot y_1+a_{12}\cdot y_2 \qquad(1)\\ y_2'&=a_{21}\cdot y_1+a_{22}\cdot y_2 \qquad(2) \end{align*...
1
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4answers
6k views

How do I plot x[t] vs. x'[t] (where x[t] and x'[t] are solutions to NDSolve)?

I have a differential equation which I solved using NDSolve. I can easily plot x[t] vs. t, x'[t] vs. t, but.... how do I plot x[t] vs. x'[t]? I tried using the Evaluate function to simplify things, ...
1
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0answers
734 views

NDSolve Convergence test failure and significant effect of DifferenceOrder on eventual results

I am solving a non linear partial differential equation with what I call free boundary conditions (solid mechanicists would know this as simply supported). I realized that this boundary condition ...
1
vote
1answer
1k views

How to plot population growth model?

How would I draw the graph of a function $\frac{dy}{dt}=(Ry^2/T)-Ry$ in Mathematica? I have tried a few times but the constants are confusing me.
15
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2answers
1k views

The only usage for the option InterpolationOrder in NDSolve is to be set to All?

We know that changing the option InterpolationOrder in ListLinePlotListPlot3D、...
3
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2answers
1k views

Problem with NDsolve for a system of equations

I want to solve a system of differential equations which is not very complicated, but I cannot handle the problem with mathematica!! Please have a look at the problem and result and help me with your ...
3
votes
0answers
635 views

Numerically solving PDE with high precision

I want to numerically solve the PDE $\partial_t u(t,x)=c\partial_x u(t,x)+(mx-l)u(t,x)$ with some initial and boundary conditions and given parameters $c$, $m$ and $l$. Consider the code ...
7
votes
3answers
1k views

Numerical solution of a differential equation with NIntegrate coefficients

I am trying to solve a linear ODE with a variable coefficient which is given in terms of an integral I can only do numerically. That is, I have an equation of the form $$ u'(t)=u(t)\int_0^t\sqrt{1+e^{-...
3
votes
1answer
923 views

Solving the Sine Gordon PDE in mathematica

how can i solve this equation in mathematica? this is sine-gordon eq. but the boundary condition can not recognized by mathematica . thank you for you attention. ...
4
votes
1answer
225 views

Construct DifferentialMatrices and Kernel for LevinRule for this integral and ODE set

I've made a lot of progress on my problem the last few days thanks to all the help I've received on here. I think I'm upto the final step of greatly improving the performance of NIntegrate[..] on my ...
2
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2answers
4k views

No result from DSolve

I don't get any answer when I evaluate the following expression: ...
2
votes
1answer
597 views

Vector Analysis: Why are the Grad, Laplacian and Div being evaluated to zero

For the non linear partial differential equation below, why are the Gradient, Laplacian and Divergence being evaluated to zero despite using the VectorAnalysis ...
4
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1answer
5k views

Finding the Minimum value of an interpolating function

I can't seem to use FindMinValue to find the min. value of a curve represented by an interpolating function. For instance the below code generates an interpolating ...
15
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3answers
1k views

How to perturb a Dynamic System?

I'm trying to model a basic feedback system with delayed feedback. I've done the initial setup and now want to add a few more advanced features to my system. Currently, it's just a simple delayed-...
3
votes
1answer
364 views

Why is NDSolve solving in term of two 1st order ODE slower than 2nd order?

As mentioned in the documentation for NDSolve it's often convenient to reduce a 2nd order ODE to a system of first order equations. When I do this however I seem to see a significant speed reduction ...
6
votes
1answer
1k views

NDSolve runs out of memory

I need to solve a second order ODE numerically. The ODE depends on two parameters (a,b). Things work fine when 'a' is small, but for large 'a' the solutions are oscillating rapidly and Mathematica ...
20
votes
3answers
17k 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 ...
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 ...
29
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3answers
3k views

1D Euler equations (fluid dynamics) with NDSolve

Is it possible to accurately solve the 1D Euler equations in Mathematica using NDSolve? For example, let us consider the Sod shock tube problem. Introduction to ...
6
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1answer
962 views

Unexpected results from NDSolve

I am trying to solve a stiff reaction diffusion system with NDSolve. However, it does not produce the expected results. My problem is a spherical cell with 5 ...
7
votes
1answer
2k views

Improving NDSolve speed for heavily stiff problems

Having looked around the intergoogles and Mathematica.SE, I thought I'd pose a question with a minimum working example. Here is the situation I am trying to improve: I am solving a 4th order non ...
20
votes
2answers
16k views

Integral equation numerical solution with NDSolve

I'm trying to solve something like: f[x] == Integrate[f[x]*g[x]] where g[x] is known and f[x]...
3
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2answers
750 views

Second Order Non Linear Differential Equation

I'm trying to solve the following differential equation numerically: ...
2
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1answer
305 views

tricky memoization

Let's say I have the NDSolve example for documentation involving splitting 2nd order into set of 1st order ODEs: ...
4
votes
1answer
523 views

Efficient way to perform elementary integration step with NDSolve internal method

I'm trying to tweak the NDSolve function to perform one elementary integration step (using some explicitly selected stepping algorithm via ...
4
votes
3answers
2k views

Solving a PDE containing DiracDelta

I want to get the answer from a PDE: $$\begin{align*} \frac{\partial \rho(r,t)}{\partial t}&=Dr^{-2}\frac{\partial}{\partial r}r^2h(r)e^{-U(r)}\frac{\partial}{\partial r}e^{U(r)}\rho(r,t)-\left(\...
0
votes
1answer
884 views

Solving Differential Equation depending on variables solved by NDSolve

How to solve a differential equation which consists of variables depending upon another differential equation?
2
votes
1answer
175 views

Using MaxStepFraction as ticks on plot

Is there any way I could use the MaxStepFraction (or grid size) as used in NDSolve in the example below as ticks on the 3d Plot? ...
1
vote
2answers
783 views

Replacing variable in an equation with an Interpolating function polynomial and plotting residual

I was trying to plot the residual for the solution of my PDE. However, I was unsure about a couple of things. I imported the data and created an Interpolation polynomial with ...
2
votes
0answers
277 views

EventLocator with LSODA?

Is the EventLocator option not compatible with LSODA on NDSolve. Below is what I tried to do ...
5
votes
3answers
1k views

How could I get the value of y[t] at each specific interpolation point?

sol = NDSolve[{Derivative[2][y][t] + Sin[y[t]] == 0, Derivative[1][y][0] == 0, y[0] == 1}, y, {t, 0, 2}] the above-mentioned differential equations can be solved ...
2
votes
4answers
308 views

How can I get the value of a at "t=2.4985352432136567" in the following expression?

By running the following code: ...
25
votes
5answers
3k 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
votes
3answers
6k views

How to discretize a nonlinear PDE fast?

I wish to numerically solve the following PDE. Although there are some complete discussions for solving PDEs in tutorial/NDSolvePDE, there is no hint for the nonlinear case by discretization. Thus, I ...
4
votes
2answers
3k views

export data points from differential equation system?

I am solving a dynamical three equation system. Besides plotting the individual effects for each of the state variables in an array and in a tridimensional graph, I would like to export the data ...
1
vote
1answer
2k views

Problem while solving system of two second order non linear coupled differential equations using NDSolve function

I am a completely new to Mathematica, and I am sorry if this question is dumb. I have to solve a system of two second order non linear coupled differential equations (that I got from the Lagrangian ...
6
votes
3answers
8k views

DSolve gives complex function although the solution is a real one

I have a problem with the DSolve[] command in mathematica 8. Solving the the following 4th order differential equation spits out a complex solution although it should be a real one. The equation is: ...
3
votes
2answers
1k views

How to avoid this kind of numerical error caused by extreme parameters when using NDSolve?

Here I use a one-dimensional heat conduction equation as the example. I found that when the thermal diffusion coefficient is small enough, Mathematica will give a result against the second law of ...
1
vote
1answer
344 views

Particular solutions of a Differential Equation not evaluated in a given case

Below first case which gives particular solutions of an OED correctly: ...
6
votes
2answers
8k views

How to handle NDSolve::ndsz problem (singularity problem)

I have 2 second order differential equations (non-linear). The physics behind them is correct. I verified the equations many times. It is a solid pendulum with a mass-spring at the end of it. Now, ...
4
votes
2answers
2k views

Manipulate a Differential Equation result

I want to Manipulate the result of Differential Equation like : ...
9
votes
2answers
465 views

Differentiating an unknown solution to a PDE

Sorry if this question is too basic -- I'm not very familiar with Mathematica. I am interested in a way to systematically address the following sort of problem: Suppose that $u=u(x,y)$ is a function ...