Solving Diffusion PDE in a Hollow Cylinder

[wenzhe2092]

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 separation of variables, giving me two ODE.

$$T = Aexp (-\lambda^2 D t)$$

where $$-\lambda^2$$ is the separation constant and A is the integration constant

$$R(r) = BJ_{0}(\lambda r) + CY_{0} (\lambda r)$$

Principle solution given by:

$$C(r,t) = Aexp (-\lambda^2 D t) [BJ_{0}(\lambda r) + CY_{0} (\lambda r)]$$

My question is, given the boundary conditions and initial conditions of:

C(0.0135,t) = 0.433
C(0.0185,t) = 0
C(r,0) = 0.0398

How would i be able to solve this? I'm really stuck and any help would be appreciated. Note my system is a hollow cylinder.

 

[matematikawan]

Dear all,

Principle solution given by:

$$C(r,t) = Aexp (-\lambda^2 D t) [BJ_{0}(\lambda r) + CY_{0} (\lambda r)]$$

My question is, given the boundary conditions and initial conditions of:

C(0.0135,t) = 0.433
C(0.0185,t) = 0
C(r,0) = 0.0398

How would i be able to solve this? I'm really stuck and any help would be appreciated. Note my system is a hollow cylinder.

Look like the four unknowns A, B, C and $\lambda$ can just be reduced to three unknown. You only have three boundary and initial conditions.