(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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].

2. Relevant equations

With

[tex]\frac{\partial C}{\partial t} = 0[/tex],

rearrange to:

[tex]

\nabla^2C = kC

[/tex]

...

3. 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.

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# Homework Help: Stationary solution to reaction-diffusion equation with certain boundary conditions

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