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

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, v_{y}(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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