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Stuck with a PDE systemby L0r3n20
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#1
Jul512, 09:50 AM

P: 19

Hi all! I'm stuck with a system of PDE. I'm not sure I want to write it here in full, so l'll write just one of them. I've found a solution to this equation but I'm not sure it's the most general one since when I plug this solution in to the other eqs, I get a trivility condition for the coefficients
[tex] 2\bar{k}^1\left(\bar{s},\bar{t},\bar{u}\right)2 k^1\left(s,t,u\right)+\left(s\bar{s}\right)\left(\partial_s k^1\left(s,t,u\right) + \bar{\partial}_{\bar{s}}\bar{k}^1\left(\bar{s}, \bar{t}, \bar{u}\right)\right) =0 [/tex] Can someone help? 


#2
Jul512, 08:53 PM

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P: 9,656

Not sure I've understood the equation. Is this simplification valid:
2u(x)  2v(y) + (yx)(∂v/∂y + ∂u/∂x) = 0 ? If so: ∂v/∂y + ∂u/∂x = 2(uv)/(xy) Consider (u, v+v') is also a solution. So ∂v'/∂y = 2v'/(yx) v' = (yx)^{2}f(x) where f is an arbitrary function of x. Does that help? 


#3
Jul612, 05:32 AM

P: 19

Thank you for the answer. Ok that was useful, at least a bit. In fact, although it is correct, I need a u and a v depending ONLY from one of the two variables (x and y). In fact I'm dealing with (anti)holomorphic functions and I need them to respect the holomorphicity condition.



#4
Jul712, 11:17 PM

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P: 9,656

Stuck with a PDE system



#5
Jul812, 02:08 AM

P: 19

Even thought I set f(x) constant I get the xdependence from yx, right?



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