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
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1678221/image.png)
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.
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1678226/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1678231/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1678236/image.png)
![](https://www.mathworks.com/matlabcentral/answers/uploaded_files/1678241/image.png)
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Torsten
2024 年 4 月 24 日
編集済み: Torsten
2024 年 4 月 24 日
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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