Modified diffusion equation PDE

[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).

 

[blue_raver22]

Based on the form of the equation, it appears to be a modified diffusion equation with a source term. This type of PDE is often used in mathematical models to describe the diffusion of a substance in a medium with a varying concentration or a source of the substance. The source term, y1del(σ), represents the rate at which the substance is being added or removed from the system.

As for the type of PDE, it is difficult to determine without knowing the specific boundary conditions and the meaning of the variables. However, based on the presence of the Dirac delta function, it is possible that this could be a singularly perturbed PDE, where the solution changes rapidly near a certain point or region.

In terms of finding a solution, it seems like you are on the right track with using separation of variables. As you mentioned, this is similar to a Sturm-Louisville problem, which is a type of eigenvalue problem. The eigenvalues, λ^2, will depend on the specific boundary conditions and may need to be determined numerically.

I would suggest consulting with a textbook on partial differential equations, such as "Partial Differential Equations: An Introduction" by Walter A. Strauss, which covers the solution methods for various types of PDEs including eigenvalue problems and singularly perturbed problems. Additionally, seeking guidance from a colleague or professor who is familiar with this type of PDE may also be helpful. Good luck with your research!