HELP : Laplace 2D equation on a square

[sarahisme]

HELP!: Laplace 2D equation on a square

Hi everyone,

I quite get the answer out for this question and i just feel like i have been beating my head into a wall for 4 hours! aghh...

http://img167.imageshack.us/img167/3579/picture12vv7.png

I get stuck when i try to solve the sub-problem which has boundary conditions:

http://img172.imageshack.us/img172/8765/picture13gn5.png

I do the usual thing of seperating varibales on http://img167.imageshack.us/img167/2107/picture14eh5.png which leads to the equations:

http://img167.imageshack.us/img167/6631/picture15oe2.png

so i solved these equations and applyed the boundary conditions:

X'(0) = 0, X'(1) = 0, Y(1) = 0

this gives the equation http://img147.imageshack.us/img147/634/picture16pd3.png

but then if i sub in the non-homogenous BC, http://img147.imageshack.us/img147/8363/picture17zg4.png

i get 0 = x

lol, what have i done this time! :P

any help would be so very much appreciated! :)

Cheers
Sarah

 

[StatusX]

I don't think you solved for Y correctly. You want Y(1)=0, and you have Y(0)=0 instead.

 

[sarahisme]

I don't think you solved for Y correctly. You want Y(1)=0, and you have Y(0)=0 instead.

ok, well what i did was try to solve

http://img154.imageshack.us/img154/5458/picture19sw2.png

which has general solution

http://img154.imageshack.us/img154/8532/picture20ix0.png

and applying the BC Y(1) = 0 gives

http://img138.imageshack.us/img138/101/picture21ap4.png

which means that k_1 = 0 and so solution for Y is:

http://img172.imageshack.us/img172/6020/picture22ba4.png


i think this is correct, but yeah, i guess it isn't hey? ;) could you please help me to find where i am going wrong?

thanks

sarah

 

[StatusX]

Why does that mean k_1=0? cosh(n pi) and sinh(n pi) are both positive numbers.

 

[sarahisme]

Why does that mean k_1=0? cosh(n pi) and sinh(n pi) are both positive numbers.

isnt it because cosh(n pi) is never zero and sinh(n pi) is equal to 0 for n = 0 and so since we need the whole expression to be equal to 0 therefore k_1 needs to be 0?

 

[StatusX]

But you need to pick a k_1, k_2 for each n such that that expression is zero. Note that you can rewrite A sinh(x)+B cosh(x) as C sinh(x+d) for some C and d, just like for sin and cos.

 

[sarahisme]

hmm ok i think i see now...

ok i get this answer for the solution to the sub-problem now

http://img85.imageshack.us/img85/9914/picture23hb4.png

does that look better? :)

 

[StatusX]

I don't think so, but it's hard to tell. Take one step at a time. What is Y(y), and is it zero at y=1? EDIT: I'll be gone for the night, hopefully someone else can help you if you still need it right now.

 

[sarahisme]

i do i do, lol, any parting hints? ;)

 

[sarahisme]

I don't think so, but it's hard to tell. Take one step at a time. What is Y(y), and is it zero at y=1? EDIT: I'll be gone for the night, hopefully someone else can help you if you still need it right now.

ok, i get now get
http://img136.imageshack.us/img136/764/picture24tf0.png