This is similar; in this case it is a concatenation of 1st order operators

[$]\frac{\partial }{\partial z} - i\frac{\partial }{\partial \overline{z}}[$] and [$]\frac{\partial u}{\partial z} + i\frac{\partial u}{\partial \overline{z}}[$]. This is is the elliptic case.

The hyperbolic case is similar but in real space.

[$]\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2}[$] = [$](\frac{\partial }{\partial x} - \frac{\partial }{\partial {y}})[$] [$](\frac{\partial u}{\partial x} + \frac{\partial u}{\partial {y}})[$].

Then [$]x = (\xi + \eta), y = (\xi - \eta)[$] leads to [$]\frac{\partial^2 u}{\partial \xi \partial \eta } [$] .

Putting in a convection/drift term seems to cause meltdown fubar.

//

used \overline for complex conjugation, better than \bar?

Statistics: Posted by Cuchulainn — October 1st, 2017, 4:36 pm

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Statistics: Posted by ppauper — October 1st, 2017, 1:29 pm

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[$]\frac{\partial^2 u}{\partial x^2} - \frac{\partial^2 u}{\partial y^2}[$]

to

[$]\frac{\partial^2 u}{\partial \xi \partial \eta } [$]

by the change of coordinates

[$]x = (\xi + \eta), y = (\xi - \eta)[$]

Now the open question is if this works in 3 independent variables. Not sure if an uneven number of independent variables carries over.

[$]\frac{\partial^2 u}{\partial x^2} [$] [$] - \frac{\partial^2 u}{\partial y^2}[$] [$] - \frac{\partial^2 u}{\partial z^2}[$]

This PDE looks more benign for some reason

[$]\frac{\partial^2 u}{\partial x^2} [$] [$] - \frac{\partial^2 u}{\partial y^2} - \frac{\partial^2 u}{\partial z^2} - \frac{\partial^2 u}{\partial p^2}[$]

I think applications can be found in anisotropic waves.

// BTW the second derivatives are a bit shifted. It that the way it works here or am I missing a bracket etc?

Statistics: Posted by Cuchulainn — September 30th, 2017, 7:49 am

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[$]\nabla^{2}u=f[$] becomes [$]4\frac{\partial^2 u}{\partial z\partial\bar{z}} = f[$] with [$]z=x+iy[$] and [$]\bar{z}=x-iy[$]

Statistics: Posted by ppauper — September 29th, 2017, 11:25 am

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Anyone know if this PDE has a name?

Statistics: Posted by Cuchulainn — September 26th, 2017, 10:05 pm

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Bonsoir tout le monde. It would also be nice to have the possibility to add a picture from the clipboard.

Statistics: Posted by outrun — September 18th, 2017, 7:52 pm

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Statistics: Posted by tagoma — September 12th, 2017, 9:50 pm

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https://twitter.com/realDonaldTrump/sta ... 3161939968

Statistics: Posted by outrun — September 12th, 2017, 7:24 pm

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viewtopic.php?f=44&t=100861

Statistics: Posted by Cuchulainn — September 3rd, 2017, 6:45 pm

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Statistics: Posted by Cuchulainn — September 2nd, 2017, 10:35 am

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