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gabbagabbahey
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#2
Sep16-08, 02:11 AM
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Quote Quote by nathan12343 View Post
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?