What are the three similar cases for examining Laplace's Equation boundaries?

[LHS]

Can anyone help me think of the three similar cases I need to examine, I was thinking 0<x<pi/2 0<y<pi/2, 0<x<pi 0<y<pi/2, 0<x<pi/2 0<y<pi, with the same boundaries as those parts of the original square, but it doesn't really work for me, any help would be greatly appreciated!

[PLAIN]http://img145.imageshack.us/img145/5541/77950426.png

 

[tiny-tim]

Welcome to PF!

Hi LHS! Welcome to PF!

(have a pi: π )

Can anyone help me think of the three similar cases I need to examine …

Well, the obvious three cases are for the same square but with a different side having T = 1

 

[LHS]

Ah thank you! that certainly makes sense, I seem to get very complicated expressions after I work those 3 new cases out.. I assume set x=y=pi/2 and equate? Doesn't seem to be happening unfortunately!

 

[tiny-tim]

Hi LHS!

What are your other three expressions for T(x,y)?

 

[LHS]

For one of the new cases I get the same expression as in the question when x=y=pi/2
For the other two cases however I get

T(pi/2,pi/2)= (summation from 0 --> infinity) (4/pi)*sin((2*n+1)*(PI/2))*(sinh((2*n+1)*(PI/2))-tanh((2*n+1)*(PI))*cosh((2*n+1)*(PI/2)))/((2*n+1)*tanh((2*n+1)*PI))

http://www.wolframalpha.com/input/?...h((2*n+1)*(PI/2)))/((2*n+1)*tanh((2*n+1)*PI))

Sorry.. don't know how to use latex yet

 

[tiny-tim]

oh hold on … because it was 3 days ago, i'd forgotten what this question was all about …

start again …

if you add all four solutions, what is that the solution of? ​

 

[LHS]

If you add all four solutions do you get the solution to T=1 on all sides, e.g.
So it becomes a 1x1x1 cube?

4T(pi/2,pi/2)=1
=> T(pi/2,pi/2)=1/4?

 

[tiny-tim]

If you add all four solutions do you get the solution to T=1 on all sides,

yes!

e.g.
So it becomes a 1x1x1 cube?

cube?

 

[LHS]

ergh.. sorry, that was me being an idoit. I was saying the distribution of T, would the surface be flat? at T(x,y)=1 for 0<x,y<pi, so you can say 4*T(pi/2,pi/2)=1

 

[tiny-tim]

Yup!

If T satisfies Laplace's equation, and is constant on the boundary, then it's constant.

 

[LHS]

Brilliant! thank you very much for helping me with this.