A long cylinder of material with a with a rectangular cross-section occupies the region 0 < x < a, 0 < y < b in space. The faces at x = 0 and x = a are held at temperature u=0 , and the face at y = b is thermally insulated. The face at y = 0 is held at temperature (const.) until a steady temperature u(x,y) is reached in the cylinder. Show that
for 0 < x < a, 0 < y < b.
I am lost with what to do for the boundary conditions and . All the 2- D Laplace equations I have done have had boundary conditions like , ect.
I have never seen or done one with .
Anyway, here is all I have done so far....
(trivial)
unsure what to do with the other to BC's....
For the PDE
dividing by
(separation constant)
So
thats all ive been able to do w/out knowing what to do with the other BC's. Can anyone please show me?
That's a good start. Next thing is to look at what you know about the function F(x). You have and F(a)=F(0)=0. That's a standard simple harmonic motion equation, and you should be able to deduce that is a solution for every natural number n. Therefore .
Now look at what you know about the function G(x). You have . That has a solution for any constant A. You also have a boundary condition (from ). But , and this will be 0 if . So we have the solution .
Putting that information together, we know that
. . . . . .
is a solution, for all choices of the coefficients C_n, and we want to choose these coefficients so that the remaining boundary condition holds. So we want
. . . . . . . . . . .(*)
whenever . That is where the Fourier sine series will come in. Since sin is an odd function, if a Fourier sine series is u_0 when 0<x<a then it must take the value -u_0 when -a<x<0.
So there is still some work to do. You need to calculate the Fourier series for the function and compare the answer with (*), to find what the coefficients C_n should be. (From the given solution, it looks as though only the odd coefficients will be nonzero.)