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-   -   Laplace's equation on a rectangle with mixed boundary conditions (http://www.physicsforums.com/showthread.php?t=256652)

nathan12343 Sep15-08 11:47 PM

Laplace's equation on a rectangle with mixed boundary conditions
 
1. The problem statement, all variables and given/known data
Solve Laplace's equation inside the rectangle [itex]0 \le x \le L[/itex], [itex]0 \le y \le H[/itex] with the following boundary conditions

[tex] u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u_y(x,0) = 0\text{, and } u(x,H) = 0[/tex]

2. Relevant equations



3. The attempt at a solution

I know that with Dirichlet boundary conditions one can simply superpose 4 solutions to 4 other problems corresponding to one side held fixed and the others held at 0. Can the same technique be generalzed for mixed boundary conditions, like I have above? I don't think so, because when I do that the solution I get for
[tex] u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u(x,0) = 0\text{, and } u(x,H) = 0 [/tex]
does not satisfy [itex]u_y(x,0) = 0[/tex].

Does anyone have a hint for how I might find solutions which simultaneously satisfy the boundary condition at [itex]u(0,y)\text{ and for }u_y(x,0)[/itex]?

gabbagabbahey Sep16-08 02:11 AM

Re: Laplace's equation on a rectangle with mixed boundary conditions
 
Quote:

Quote by nathan12343 (Post 1875270)
1. The problem statement, all variables and given/known data
Solve Laplace's equation inside the rectangle [itex]0 \le x \le L[/itex], [itex]0 \le y \le H[/itex] with the following boundary conditions

[tex] u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u_y(x,0) = 0\text{, and } u(x,H) = 0[/tex]

2. Relevant equations



3. The attempt at a solution

I know that with Dirichlet boundary conditions one can simply superpose 4 solutions to 4 other problems corresponding to one side held fixed and the others held at 0. Can the same technique be generalzed for mixed boundary conditions, like I have above? I don't think so, because when I do that the solution I get for
[tex] u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u(x,0) = 0\text{, and } u(x,H) = 0 [/tex]
does not satisfy [itex]u_y(x,0) = 0[/tex].

Does anyone have a hint for how I might find solutions which simultaneously satisfy the boundary condition at [itex]u(0,y)\text{ and for }u_y(x,0)[/itex]?

Why not find the general 2D solution to Laplace's equation, using separation of variables (i.e. [tex]u(x,y) \equiv X(x)Y(y)[/tex])and then substitute your boundary conditions to find the particular solution?

HallsofIvy Sep16-08 08:35 AM

Re: Laplace's equation on a rectangle with mixed boundary conditions
 
Let v(x,y)= u(x,y)- xg(y)/L

Then [itex]\nabla^2 v= \nabla^2 u- xg"(y)/L= -xg"(y)/L[/itex] since [itex]\nabla^2 u= 0[/itex].

The boundary conditions on v are v(0,y)= 0, v(L, y)= g(y)- g(y)= 0, vy(x, 0)= -xg'(0)/L, v(x,H)= -xg(H)/L.

Because the boundary conditions on x are both 0, you can write v as a Fourier sine series:
[tex]v(x,y)= \sum_{n=1}^\infty A_n(y)sin(n\pi x/L)[/tex]

You will need to write -xg"(y)/L as a Fourier sine series in x so you can treat g"(y) as a constant.


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