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Homework Statement
What is the stationary (steady state) solution to the following reaction diffusion equation:
[tex]
\frac{\partial C}{\partial t}= \nabla^2C - kC
[/tex]
Subject to the boundary conditions C(x, y=0) = 1, C(x = 0, y) = C(x = L, y) (IE, periodic boundary conditions along the x-axis, the value at x=0 is the same as at x=L). Also, at y = 0 and y = L, [tex]\frac{\partial C}{\partial x} = \frac{\partial C}{\partial y} = 0[/tex].
Homework Equations
With
[tex]\frac{\partial C}{\partial t} = 0[/tex],
rearrange to:
[tex]
\nabla^2C = kC
[/tex]
...
The Attempt at a Solution
I believe I can solve this PDE without the boundary conditions, at least the one equation is satisified by a sum of hyberbolic sine or cosine functions. I have absolutely no idea how to incorporate the boundary conditions though. That they are periodic across x tells me that the solution should be symmetric about x = L / 2, but I have no mathematical reasons for this. I have never taking a PDE class before so I am a bit out of my element... any help would be very useful. I know that there IS an analytic solution with these constraints, but I haven't a clue what it is.