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What is the mathematical form called for PDEs used with the MATLAB function pdepe?

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The MATLAB function pdepe (documentation here) requires that the form of the PDE being solved for is
Where is the flux term, and is the source term. Would this be considered an elliptical/parabolic PDE in conservative form? The terms and are what are throwing me off. The page references a source but I when looking through it (granted with little scrutny and rather quickly) I did not see any reference to this form. I think the reference is just for the integrator used in the algorithm employed. If anyone can confirm that this is a conservative form OR can tell me the name of this form, I'd be greatly appreciative.

Accepted Answer

Torsten on 24 Apr 2024
Edited: Torsten on 24 Apr 2024
Would this be considered an elliptical/parabolic PDE in conservative form?
No. "Conservative" usually refers to the first spatial derivative, not the second spatial derivative.
A PDE in conservative form usually reads
du/dt + d(f(u))/dx = 0
The terms x^-m and x^m are what are throwing me off.
The terms x^-m and x^m refer to the coordinate system in which you want to solve your equations.
The second-order derivative in cartesian coordinates is
d/dx (D*du/dx)
in cylindrical coordinates
1/r * d/dr (r*D*du/dr)
and in spherical coordinates
1/r^2 * d/dr (r^2*D*du/dr)
If you write this coordinate independent as
div(D grad u)
, no such m appears - only after translating in a special coordinate system:
I think the required form of the PDE for "pdepe" has no special name. Only the restriction that the second-order derivatives have to be specified in flux-form (which is directly related to the form on how the boundary conditions are to be given) could be noted.
And you are correct: the code is suited for parabolic-elliptic PDEs in one space dimension and not for hyperbolic PDEs . That means that the second-order flux term f must be different from 0.
The type of equations solved with "pdepe" are usually called "reaction-convection-diffusion equations".

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