1. ## Solving Diffusion PDE

Dear all,

I'm trying to solve the diffusion PDE for my system, shown below:

$\frac{\partial C}{\partial t} = D (\frac{\partial^2 C}{\partial r^2} + \frac{1}{r} \frac{\partial C}{\partial r})$

where C is the concentration, changing with time t and radius r. D is the diffusion coefficient.

I'm solving this using seperation of variables, giving me two ODE.

One of the ODE is:

$\frac{d^2R}{dr^2} + \frac{1}{r} \frac{dR}{dr} + lamda^2 R = 0$

where lamba is a constant.

Does any one have any ideas on how to solve this ODE tro obtain a general solution? I tried using a change in variables.

Any help would be appreciated. Thanks in advance!

Regards,

Billy

2. First multiply through by $r^2$ to get $r^2\frac{d^2R}{dr^2}+ r\frac{dR}{dr}+ \lambda^2r^2 R= 0$. That's Bessel's equation of order 0. Look up "Bessel's function".

3. Thanks for the quicky reply HallsofIvy.

I'm not too sure how i am to solve Bessel's function. Could you give me an example?

Cheers.

4. Also, am i solving the Bessel function for integer v, or non integer v? As of now, i do not have a value for my lamda.

5. Well, I can't give an entire course on Bessel's equation. Doesn't your text book deal with it? Here is a link to the Wikipedia article:
Bessel function - Wikipedia, the free encyclopedia

Typically, you determine what values the eigenvalue can take by looking at the boundary conditions.

6. Thanks again for your direction.

After going through the derivation, i know what Jo(x) is, which is for Bessel's equation of order 0. So is it true to say that R(r) = Jo(x)? Or is it more complicated than this?

7. Originally Posted by wenzhe2092

After going through the derivation, i know what Jo(x) is, which is for Bessel's equation of order 0. So is it true to say that R(r) = Jo(x)? Or is it more complicated than this?
There's actually a second solution $Y_0$ but it has a singularity at $r = 0$ so yes, you are right.

8. Originally Posted by wenzhe2092
Thanks for the quicky reply HallsofIvy.

I'm not too sure how i am to solve Bessel's function. Could you give me an example?

Cheers.
Originally Posted by Danny
There's actually a second solution $Y_0$ but it has a singularity at $r = 0$ so yes, you are right.
Thanks for the reply. Just to clarify, why is it that we can just get rid of $Y_0$? What do you mean by singularity?

9. Update:

So far i have established my principal solution for my ODE, as shown below:

$
u(r,t)=\sum_{n=1}^{\infty}A_n \cdot J_0(\lambda_n^(0.5) r) exp (-\lambda_n Dt)$

10. Update:

So far i have established my principal solution for my ODE, as shown below:

$
c(r,t)=\sum_{n=1}^{\infty}A_n \cdot J_0(\lambda_n^{0.5} r) exp (-\lambda_n Dt)$

Now to solve it. The initial condition is given as c(r,0) = 0.0398 mol/m^3 for a<r<b where a and b are the inner and outer radius respectively.

Any tips on how to apply this condition?

Next, the boundary condition is such that:

c= o at r = b
c= 0.433 at r = a

Any help appreciated! Thanks.

11. bump