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P: 5,004
 Quote by nathan12343 1. The problem statement, all variables and given/known data Solve Laplace's equation inside the rectangle $0 \le x \le L$, $0 \le y \le H$ with the following boundary conditions $$u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u_y(x,0) = 0\text{, and } u(x,H) = 0$$ 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 $$u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u(x,0) = 0\text{, and } u(x,H) = 0$$ does not satisfy $u_y(x,0) = 0[/tex]. Does anyone have a hint for how I might find solutions which simultaneously satisfy the boundary condition at [itex]u(0,y)\text{ and for }u_y(x,0)$?
Why not find the general 2D solution to Laplace's equation, using separation of variables (i.e. $$u(x,y) \equiv X(x)Y(y)$$)and then substitute your boundary conditions to find the particular solution?