# Solution to Second Order Coupled PDE in x,y,z, and time

1. Aug 13, 2010

### landon244

I'm trying to solve equation in the attached pdf, which describes anistropic diffusion in 3D with an additional term to account for hydrogen bonding and unbonding of the diffusing substance to the medium. I've considered Laplace transforms, then solving in the Laplace domain, then inverting numerically. Ideally I would get a fully analytical solution, but I'm not sure how to approach it. Solving it numerically in three dimensions (using Crank-Nicolson or something similar) would undoubtedly be very computationally expensive, correct?

If anyone has any suggestions on how to arrive at an analytical solution to this, or an efficient way to implement it numerically, I would greatly appreciate your input. I’ve been struggling with this for some time now, and just haven’t gotten very far. Thanks in advance.

#### Attached Files:

• ###### 3D Coupled PDE.pdf
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2. Aug 25, 2010

### Eynstone

The equation simplifies considerably if we separate the variables, i.e., n=p(x)q(y)r(z)s(t)
and N= M(x,y,z)S(t). On substitution & elimination of n/N , we end up with a system in
s,s' and S,S'.
I can't presume to advise you on modelling the diffusion. Yet, the resulting equations are amenable to analytic methods if you either assume S' <<beta S(to linearise ) or assume something about n/N.