# HELP : Laplace 2D equation on a square

• sarahisme
In summary, Sarah was stuck on a problem involving boundary conditions and solving for Y. She was able to solve the equation for X and Y, but when she tried to solve for Y(1), she got an error. After some research, she found that she needed to solve for Y(1) using a general solution and then apply the boundary conditions. Once she did that, she was able to solve the problem.
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

Last edited by a moderator:
I don't think you solved for Y correctly. You want Y(1)=0, and you have Y(0)=0 instead.

StatusX said:
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

Last edited by a moderator:
Why does that mean k_1=0? cosh(n pi) and sinh(n pi) are both positive numbers.

StatusX said:
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?

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.

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? :)

Last edited by a moderator:
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.

Last edited:
i do i do, lol, any parting hints? ;)

StatusX said:
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

Last edited by a moderator:

## 1. What is the Laplace 2D equation on a square?

The Laplace 2D equation on a square is a partial differential equation that describes the steady-state distribution of temperature or potential within a square-shaped region. It is often used in physics and engineering to model diffusion, electrostatics, and other physical phenomena.

## 2. How is the Laplace 2D equation on a square solved?

The Laplace 2D equation on a square can be solved using various methods, such as separation of variables, finite difference methods, and the method of eigenfunction expansion. These methods involve breaking down the equation into simpler equations and solving them iteratively or analytically.

## 3. What are the boundary conditions for the Laplace 2D equation on a square?

The boundary conditions for the Laplace 2D equation on a square specify the values of the function at the edges of the square region. These values can be either fixed or dependent on the values of the function at neighboring points. Common boundary conditions include Dirichlet, Neumann, and Robin conditions.

## 4. Can the Laplace 2D equation on a square be applied to non-square shapes?

Yes, the Laplace 2D equation can be extended to non-square shapes, such as rectangles, circles, and irregular polygons. However, the boundary conditions and solution methods may vary depending on the specific shape being considered.

## 5. What are some applications of the Laplace 2D equation on a square?

The Laplace 2D equation on a square has various applications in fields such as physics, engineering, and mathematics. It is commonly used to model heat transfer, fluid flow, electrostatics, and diffusion processes. It is also used in image and signal processing for edge detection and denoising.

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