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Kamil90
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Joined: February 15th, 2012, 2:02 pm

Fichera for transport equation

August 29th, 2016, 5:39 pm

Hi,
I think I understood how to apply Fichera criteria to the Black Scholes type of equation and I see it is mentioned it was originally developed for elliptic type of equations. We can consider BS eq as elliptic such that it degenerates for all t and therefore we need terminal condition for t=T.
However, what about transport equation say u_t+u_x=0? This is not elliptic but can it be considered one by arguing that it degenerates for all t and for all x and therefore we need to calculate the Fichera function?
I thought "degenerate" meant to define some points(regions) that lower the order of the equation from n to n-1 but that transport equation I have written is of first order to begin with and it doesn't degenerate at any x. Or, shall I say the Fichera theory only applies to those degenerate equations such that happen to become of order 1 from 2 at certain points in the region? So that transport equation could be of order 2?
 
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Cuchulainn
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Re: Fichera for transport equation

August 29th, 2016, 6:56 pm

I remember Fichera giving such an example (i.e. the whole boundary is degenerate in x).  A domain transformation makes it even more interesting.
https://forum.wilmott.com/viewtopic.php ... rt#p435024
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