Solve Laplace equation on rectangle domain

[Dor]

Homework Statement

I'm having issues with a Laplace problem. actually, I have two different boundary problems which I don't 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.

 

[drvrm]

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

you can look up the following to see how variable separation can be done...
ref;

Rn 3.3 Solving Laplace's Equation on Bounded Domains
https://math.dartmouth.edu/~m126w13/notes1.pdf

 

[Dor]

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]

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 <br /> U_x(0,y) = 0 \\ U(a,y) = f(y) \\ U(x,0) = 0 \\ U_y(x,b) = 0 and U_2 satisfies <br /> U_x(0,y) = 0 \\ U(a,y) = 0 \\ U(x,0) = 0 \\ U_y(x,b) = v

 

[Dor]

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]

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&#039;(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&#039;(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&#039;(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.