Matlab PDEs: Differential Equations vs. PDEs

[end3r7]

I know how to do differential equations and a plot a phase plane with pplane7. But I have no clue how to do the same for pde's.
Is it similar or not at all?

 

[Chris Hillman]

I know how to do differential equations and a plot a phase plane with pplane7.

ANY ode? Aren't you forgetting something?

But I have no clue how to do the same for pde's.
Is it similar or not at all?

If you answered my question you probably now know the answer to yours.

 

[end3r7]

Ha, good one. =)

Hopefully it won't be too dissimilar. Any hints or should I just read the docs at mathworks.com?

 

[Chris Hillman]

For what kinds of ordinary differential equations can you plot a phase portrait? (Hint: not just for any old ODE!)

BTW, a friendly moderator should probably move this entire thread to the differential equations subforum.

 

[end3r7]

Basically I use pplane7, so any system of two first order differential equations I believe.

 

[Chris Hillman]

What does pplane7 do? You tell me!

Basically I use pplane7, so any system of two first order differential equations I believe.

Let's back up. When you say "plot a phase plane", I think you mean "sketch a phase portrait". I have used MATLAB in the past, but I haven't used in recently and I am not familiar with "pplane7".

I have been assuming that given a second order ODE for y in terms of x, pplane7 obtains the corresponding autonomous first order system of ODEs $u=y, v=y\prime$ and numerically plots the phase portrait in the u,v plane. For example, given the van der Pol equation governing a nonlinear spring
$$y\prime \prime + y = \mu \, (y - y^2) y\prime$$
the first order autonomous system is
$$\dot{u} = v, \; \; \dot{v} = -u + \mu \, (1-u^2) \, v$$
and the corresponding flow on $R^2 = \left{ (u, \, v): u, \, v \in R \right}$ is generated by the vector field
$$v \, \partial_u + \left( -u + \mu \, (1-u^2) \, v \right) \, \partial_v$$
The integral curves of this vector field are the phase curves, and plotting a judicious selection of phase curves (in this case, there is a unique closed phase curve, and the other phase curves approach it as time increases, so it is a limit cycle) gives the phase portrait. This phase portrait gives a vivid picture of the behavior of solutions to the original ODE.

Does this look familiar? (See Arnold, Ordinary Differential Equations for many more examples.)

From your responses I am guessing you are not sure what pplane7 does either, and that this is part of the problem. I was trying to get you to realize that figuring out exactly what pplane7 is the first step in answering your own question. Do you have some on-line help which explains what is acceptable input for pplane7?

Once you understand why whatever restrictions on the acceptable input are mathematically necessary (don't forget the possibility that you don't want to consider higher dimensional phase portraits!), you will be in a better position to start thinking about whether phase portraits make sense if you start with a PDE instead of an ODE.

 

[end3r7]

http://www.math.hmc.edu/~depillis/PCMI2005WEBSITE/DAY4/phaseplanes.pdf

Here is pplane7

You input in a system, first order, with two equations.

 

[end3r7]

Oh, and sorry for the terminology... english is not my first language, so I probably will say something and maybe mean another at times =P

 

[Chris Hillman]

My attempt to mimic Socrates has evidently run afoul of a technical limitation: I don't have at hand a recent installation of matlab. FWIW my expection is that the answer to your (refined and restated question) will be that you shouldn't expect to make phase portraits except under the circumstances where this is standard practice.

I suggest that you ask a friendly moderator to move this thread to the Computers forum at PF.