# Solve Laplace equation on rectangle domain

## Homework Statement

I'm having issues with a Laplace problem. actually, I have two different boundary problems which I dont know how to solve analytically.
I couldn't find anything on this situations and if anybody could point me in the right direction it would be fantastic.
It's just Laplace's equation on the square [0,a]x[0,b] with a mixed boundary.

## Homework Equations

1. the first one is:

Uxx+Uyy=0
Ux(0,y)= 0
Ux(a,y)= f(y) (some function)
U(x,0)= 0
Uy(x,b)= v (some constant)

1. the second is:

Uxx+Uyy=0
Ux(0,y)= 0
U(a,y)= g(y) (some function)
U(x,0)= 0
Uy(x,b)= v (some constant)

## The Attempt at a Solution

I tried to do ordinary separable solution but I don't really know how to do this in such problems.

I know the variable separation method.
But I'm not sure how to do it when in one side of the rectangle there is a Dirichlet boundary U(x,b)=const and in the other one I have Neumann boundary Uy(x,0)=0

pasmith
Homework Helper
Laplace's equation is linear. You can always write the solution as a sum of solutions satisfying simpler boundary conditions. For example in the second problem you can set $U = U_1 + U_2$ where $U_1$ satisfies $$U_x(0,y) = 0 \\ U(a,y) = f(y) \\ U(x,0) = 0 \\ U_y(x,b) = 0$$ and $U_2$ satisfies $$U_x(0,y) = 0 \\ U(a,y) = 0 \\ U(x,0) = 0 \\ U_y(x,b) = v$$

I'm not sure how to solve it when I have one side with Dirichlet boundary and the other side with Neumann boundary.
U ( x , 0 ) = 0
U y ( x , b ) = 0
or
U ( x , 0 ) = 0
U y ( x , b ) = v

pasmith
Homework Helper
We're looking for $U = \sum_n X_n(x)Y_n(y)$

For $U(x,0) = U_y(x,b) = 0$ you need $Y_n(0) = 0$ and $Y_n'(b) = 0$. Therefore set $Y_n(y) = \sin k_ny$ so $Y_n(0) = 0$ is satisfied and $k_n$ is determined by $Y_n'(b) = k_n\cos k_nb = 0$.

For $U(x,0) = 0$ and $U_y(x,b) = v$, you should regard $v$ as a function of $x$ to be expanded as a fourier series; accordingly $Y_n$ will be an exponential function with $Y_n(0) = 0$ and (for convenience) $Y_n'(b) = 1$. The combination of exponentials which satisfies these is $Y_n(y) = \sinh(k_ny)/\sinh(k_nb)$ where $k_n$ is determined by the boundary conditions on $X_n$.