Laplace's Equation PDE FUN TIMES

[TheOmegaCode]

Introduction to PDE's were really, really short and I can't for the life of me figure this out. No matter how much of the text or notes I read.

It's a simple Laplace Equation with initial boundary conditions provided. It should be solvable with separation of variables.

I've put a few feelers out there for some guidance. Just need a push in the right direction to get the problem rolling.

THE PROBLEM!!

Any help or negative comments would be greatly appreciated.

 

[theflyingpig]

That's an odd numbered problem. You should post what book you got it from and see if someone has the solution guide for it. That's the best solution here. Why waste time trying to figure something out when you can just copy it? Everyone knows this.

 

[TheOmegaCode]

If it is, I don't know what book they pulled it out of. It's a study guide for a final review. Only question I can't figure out. I looked through our book but it's not one of the problems.

 

[theflyingpig]

Well, that sucks then. I wish you luck, my friend. Close your eyes, let the lord guide your hand, and you will solve your problem.

 

[Fenixgoon]

looks about right. i think your answers may come out to be sine/cosine, but don't take my word for it

 

[dotcom173]

I'm currently pulling my hair out over that same class i think...linear analysis II, fuck this. Good luck with Laplace's trannies

 

[RESmonkey]

Looks like it's from my textbook. The way it's written and the variables used is too familiar. let me go grab my book. Is that a number 10 or 19?

 

[Matthiasa]

You could probably check with that umm stupid wolfram alpha search site. Math is like the one thing it can do.

 

[RESmonkey]

Wait, can you rewrite that problem make every variable legible?

 

[MovingTarget]

Sounds like something straight out of Lawrence Evans' book. Have you tried using the method of charictaristics? It seems doable, but starting from the general solution(s) is really a pain...

 

[RESmonkey]

Originally posted by: MovingTarget
Sounds like something straight out of Lawrence Evans' book. Have you tried using the method of charictaristics? It seems doable, but starting from the general solution(s) is really a pain...

I can't tell whether he wrote a 4, a t, or a y. HANDWRITING!!! /wrists

Oh well, I'll check back in a few minutes. If no OP response, I'm sleeping.

 

[Born2bwire]

This is a very common type of problem I think. Why don't you see if you can't find an example for a resonant cavity? The boundary conditions and PDE will not be the same but the process should be similar. You already seem to have an idea of the fact that you should get modal solutions.

 

[frostedflakes]

Hmm, I can't seem to figure it out either, where did you get stuck OP? When I'm solving for the initial conditions I end up with constants canceling out.

Maybe the solution I have for the PDE is incorrect (I'm pretty sure it's right, though).

u(x,y)=(C1*cos(vx)+C2*sin(vx))*(C3*cos(vy)+C4*sin(vy))

v=sqrt(-?)

edit: Nevermind, think I goofed in the solution. Working on fixing it and recalculating the constants.

 

[TheOmegaCode]

The solution, I think, should end up being some series solution with sinh and sin in it, the cos and cosh should cancel out. I'm reworking it from a few examples I saw.

Problem in text.

I appreciate the input.

 

[RESmonkey]

Originally posted by: TheOmegaCode
The solution, I think, should end up being some series solution with sinh and sin in it, the cos and cosh should cancel out. I'm reworking it from a few examples I saw.

Problem in text.

I appreciate the input.

When do you need it solved by?

 

[frostedflakes]

Well then I'm stumped, because it's not simplifying like that for me.

Sorry man, wish I could have been more help.

 

[TheOmegaCode]

This is more or less what I worked out. I don't really understand it entirely, but it's sinking in.

Solution Part 1
Solution Part 2

Except that integral should be with respect to y... I think...