Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'm struggling here, trying to understand how to do this, and after 4 hours of reading, i still can't get around the concept and how to use it.

Basically, i have this problem:

A={(x,y) / x≥0, 0≤y≤pi

So U(x,0) = B; U(x,pi) = C; U'x(0,y) = 0;

I know that inside A, the laplace operator of U is 0. So i have to find U, and U must meet those requirements.

I don't have to use any form of differential equation. I'm supposed to find some sort of conformal transformation in order to make the domain a little more.. easy to understand. And then i should just get a result.

The problem is, i think i don't know how to do that. If any of you could help me understand how to solve this one, i might get the main idea and i could try to reproduce the resolution in similar cases.

Thank you very much, and i'm sorry for my english.

share|improve this question
    
Perhaps you could try to map your domain conformally to the unit disk and then use Poisson's formula? –  Alex R. Sep 8 '12 at 19:22
    
I haven't seen Poisson's Formula in class, so i don't think it's necessary. Or, ar least, it can be solved without it. I thought about converting that into a disk, using the sin(z) transformation (as y is constant, it could give me an elipse). But i will have a condition in the point (0,0) and another in the border. And i don't think i know how to know the function i'm looking for with those two conditions (without using poisson, i think). Thank you for your answer! –  Fede Sep 8 '12 at 20:06
add comment

1 Answer

up vote 0 down vote accepted

The domain is simple enough already. Observe that there is a function of the form $U=\alpha y+\beta$ which satisfies the given conditions.

share|improve this answer
    
Oh, but.. what about the restriction of x≥0? That's my main problem. I solved one before, but without that restiction (it was a infinite region, limited by y=0 and y=pi . But.. i don't see how it could be applied here. Could you give me some more.. clues? Thank you! –  Fede Sep 10 '12 at 16:40
    
@Fede I don't see what's the problem with having a smaller domain. If $U$ satisfies the equation in a larger domain, it also satisfies it in the smaller domain. The only extra thing to check is the boundary condition $U_x=0$ when $x=0$. But this is obviously true because $U_x$ is identically zero. –  user31373 Sep 10 '12 at 16:54
    
i was writing a whole response to that, saying that I didn't get it. Then it hit me. U'x. Not U'y. Nobody cares if U varies in 'y'. This whole time I was thinking that as U'x was 0, U had to take a constant D.. but.. oh, I feel very stupid now. Thank you very, very much. You are awesome. (And sorry for my english). –  Fede Sep 10 '12 at 17:39
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.