# Thread: Heat Equation with Neumann Boundary conditions

1. ## Heat Equation with Neumann Boundary conditions

So essentially, I am given these conditions:

Satisfy laplace's equation inside a rectangle for $u(x,y)$

$(0
$\frac{du}{dx}(0,y)= \frac{du}{dx}(L,y)= 0$
$\frac{du}{dy}(x,0)= 0$
$\frac{du}{dy}(x,H)= f(x)$

the problem comes in three parts:

1) without solving this problem, briefly explain the physical condition underwhich there is a solution to this problem.

I think I understand this one. $\int_{0}^{L}f(x)dx=0$ so you can have a steady state, which is necessary when solving a heat equation with the Laplace equation.

2) Solve this problem by the method of separation of variables. Show that the method works only under the condition of part (1).

I can solve the problem easily (as far as PDE's go, it's not too bad). But I don't have the theory knowledge to answer the second part. I don't know how to mathematically prove that it is necessary for the integral of F(x) to be zero over (0,L)
The solution I get is:

$$U(x,y)= \sum_{n=1}^{\infty} A_n\cos(\frac{nx\pi}{L})\sinh(\frac{ny\pi}{L})$$

$$A_n= \frac{2}{L\sinh(nH\pi/L)}\int_{0}^{L}f(x)\cos(\frac{nx\pi}{L})dx$$

3)The solution (part 2) has an arbitrary constant. Determine it by consideration of the time dependent heat equation subject to the initial condition:
U(x,y,0)=g(x,y) (the 0 is t=0)

I have no clue how to start this at all. help!