- #1
hermish
- 10
- 0
Homework Statement
Use separation of variables to find the solution to Laplaces equation satisfying the boundary conditions
u(x,0)=0 (0<x<2)
u(x,1)=0 (0<x<2)
u(0,y)=0 (0<y<1)
u(2,y)= asin2πy(0<y<1)
The Attempt at a Solution
I am able to perform the separation of variables technique on the wave equation. The heat equation is a little harder, I struggle a bit, but eventually I get there. Laplace's equation is pretty much impossible. From my understanding, the method is very similar in all three cases, but I think there are some differences which I don't see, which is why I can't do this question.
So I managed to separate the variables, deriving two ODE's, one in terms of x and one in terms of y, with the separation constant λ.
F''(x) - λF(x) = 0
G''(x) - λG(x) = 0
For the case where λ=0, there are no solutions because nothing can satisfy the last boundary condition listed.
For the case where λ<0, I think there are no solutions, I could sort of tell by having a look at the final answer given. I don't understand how to show this?
For the case where λ>0
I get F(x) = A*cosh(sigma*x) + B*sinh(sigma*y)
G(y) = (Ccos(sigma*y) + Dsin(sigma*y))
where sigma is the roots of the ODE's.
so now u(x,t) = F(x)*G(y)
I have this function with 5 unknowns, A,B,C,D, and sigma
When I apply all the boundary conditions, I don't really get anywhere. No helpful information appears.
What am I doing wrong? Or, what am I not doing?