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Poison

Guest

Poisson equation

I have to solve a poisson equation

d^2u=-1 in a unit square with boundary condition u=0. (here d denotes a triangular which is upside-down )

It's pretty easy in a unit circle but how can I do it in a square?

Poison

Guest

Re: Poisson equation

Anyone?

polylog

Member

Registered: 2006-09-28

Posts: 162

Re: Poisson equation

I guess you are working in Cartesian coordinates?

Then

But I'm not exactly sure what you mean by 'how can I do it in a square?'.

In the meantime, until someone here with expertise on this topic posts, you could also post this on another forum where advanced things are routinely answered:

http://www.chatarea.com/Mathematics

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Poisson equation

Use Matlab for a good approximation.

Please give your answer if you find one. Actually this is a hard and challenging topic indeed.

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X'(y-Xβ)=0

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: Poisson equation

It's been a long time since I took a differential equations class, so I don't remember all the tricks for solving general second order problems like this. I have dealt with special applications using the poisson function, but they always allowed us to make simplifications - either spherical symmetry, or uncorrelation between x and y, etc.

In your case you're dealing with a 2d version of the Poisson equation - just in x and y. If you are allowed to say u is uncorrelated between x and y (often you are in really life, though probably not if you're assigned this problem in school), you can write u(x,y) = A(x) + B(y), and require A and B to satisfy your boundary conditions separately in their own dimensions. Then, do a fourier decomposition of A and B. I'll just solve the case of A; B is done in exactly the same way:

[align=center]

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You're requiring the function to be zero at the box boundaries - say x=0,L, so these give you the conditions:

[align=center]

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[align=center]

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and

[align=center]

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This gives you two conditions on the two infinite series of allowed a and b values - there's not enough information to solve for the other constants. You can get similar restrictions on the y breakdowns of the B function: say B = sum(c_n sin(nwy) + d_n cos(nwy) ), you get sum(d) = 0 and sum(c_n sin(nwL) + d_n cos(nwL) = 0 ).

Applying the poisson requirement, you can get one final condition:

[align=center]

[/align]

And even after applying all your conditions that's still all you know about your function, even in this simplified case. Perhaps you can also come up with a more direct formal solution to this than a fourier breakdown - I'd be very interested to see the answer if you come up with it. In real life though, I think this is often the easier approach. And usually when dealing with fluids or electromagnetic waves you're allowed to require not only u=0 at the boundaries, but also that all derivatives of u are zero at the boundaries, which lets you solve for almost all of your unknowns.

Maybe this helps you think about it anyway?

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: Poisson equation

Forget it-i'm not curious about the answer any more. Guess the answer of this complex question is a waste for me.

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X'(y-Xβ)=0

fgarb

Member

Registered: 2006-03-03

Posts: 89

Re: Poisson equation

I wouldn't be surprised if there's a trick that makes this problem much easier in this special case, I just don't know what it is.