Advancing from 1D diffusion model to 2D and 3D

Questions
, ,
1

I have a sample of 1D diffusion model, and I wish to upgrade it to a 2D and than 3D.

Diffusionsample.bmp (2 MB)

I simply tried to add another coordinate in the code of my sample but from some reason it doesn’t work.

2DDiffusion.bmp (2.01 MB)

Can someone tell what am I doing wrong?
I uploaded both files, so I would appreciate if someone could give some advice on this one.
Thanks

2

Make the following changes in the diffusion differential equation:
For 1D → 2D, make substitutions x → r and d^2/dx^2 → d^2/dr^2 + (1/r)d/dr
For 1D → 3D, x → r and d^2/dx^2 → d^2/dr^2 + (2/r)d/dr

3

Thanks for reply, but I am not sure I understand why is this so. How would it look in code? r is radius? What is with diffusion constant?
And also, I still don’t understand why my solution is not working?

4

Thanks for reply, but I am not sure I understand why is this so.

Mr. Blinder presumes that you use in 2D a point source for diffusion, so the problem has (for a homogenous material) radial symmetry and he mentions the radial form of the Laplacian, the angle does not matter because of the radial symmetry. You can use Mathematica to find out the form of the Laplacian in diverse co-ordinate systems. Symmetry is always an important point in reducing computational complexity of problems, you know.

5

Thanks, once again you step in to help. I have managed though to do this other way in the meantime, since I didn’t understand the advice of Mr. Blinder completely, I did this diffusion with the use of Distribution of random walks. It worked just fine I think.

Anyway I very much appreciate your effort to help me.

Thanks once again to you and Mr. Blinder.