What Type of PDE is This Modified Diffusion Equation?

[Clouis523]

Hi I'd appreciate any help on identifying the type of PDE the following equation is...

*This is NOT homework, it is part of research and thus the lack my explanation of what this represents and boundary conditions. I have a numerical simulation of the solution but I'm looking to have a math win on my thesis.*

dC/dt = D(del^2(C)+$y_{1}$del(σ))

I've used separation of variables into a space function and a temporal function (which I've already solved since it's the exact same as the standard Diffusion equation). Long story short this is very similar to a Sturm-Louisville Problem but instead of getting a characteristic equation with lambda squared times the space function I end up with after subbing in the conditions for sigma.

d^2U/dr^2+(1/r)dU/dr+(ψ*δ(r-$r_{0}$)+λ^2)*U=0

I have two issues with getting a solution here first is the dirac delta before the eigenvalues (lambda) and second is that I have no idea what type of PDE this falls under other than it is very similar to S-L problems. I'm positive that the solution will include bessel functions (as you can probably tell this is in cylindrical.

If someone could point me to the type of PDE or even better a text that I could reference for this type's solution method you would make my day.

Cheers.

 

[SteamKing]

I don't know what 'have a math win on my thesis' means.

It's 'Sturm-Liouville problem'. I don't think Sturm ever was in Louisville in his life.

 

[Clouis523]

Having a math win will mean I have a general solution to the set of boundary conditions in my experiment. I will be able to represent the solution as a plug and play formula.

As for Sturm-Liouville I actually don't know anything about the guy, I just studied the method of solution. Good to know for the future though. I'm really just looking for what class of PDE this is so I can read up on solution method. I assume it will be series solution similar to a Fourier series.

 

[the_wolfman]

You're close:

Lets consider a similar equation in rectangular coordinates:

$\frac{d C}{d t} = \nabla ^2 C + S$

Here S is a source term

To solve this equation expand C onto a series of orthogonal functions.
For example
$C = \sum_n \sum_m C_{nm} \sin {nx} \sin {my}$

Next we also expand the source onto the same basis functions:
$S = \sum_n \sum_m S_{nm} \sin nx \sin my$

Finally we plug this expansion back into our regular equation.
$\frac{d \sum_n \sum_m C_{nm} \sin {nx} \sin {my} }{d t} = \nabla ^2 \sum_n \sum_m C_{nm} \sin nx \sin my + \sum_n \sum_m S_{nm} \sin nx \sin my$.

However because the basis functions are orthogonal we end up with a series of easily solvable ODEs

$\frac{d C_{nm} }{d t} = (n^2+m^2) C_{nm} + S_{nm}$.

Thus solving your PDE amounts to solving this equation for each $C_{nm}$.

In polar or cylindrical coordinators the procedure is exactly the same but you have to use a different set of orthogonal basis functions.

You want to pick your basis functions such that
$\nabla ^2 \phi = \lambda \phi$
and that your boundary conditions are satisfied.

Its the end of the day and my brain stopped working but I think the right basis function will be something similar to
$\phi = \sum_i \sum_m J_m(x_{mi}r) \cos m\theta$
where $x_{mi}$ is the i-th zero of J_m(r).