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Stuck with a PDE system

by L0r3n20
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L0r3n20
#1
Jul5-12, 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?
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haruspex
#2
Jul5-12, 08:53 PM
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Not sure I've understood the equation. Is this simplification valid:
2u(x) - 2v(y) + (y-x)(∂v/∂y + ∂u/∂x) = 0
?
If so:
∂v/∂y + ∂u/∂x = 2(u-v)/(x-y)
Consider (u, v+v') is also a solution. So
∂v'/∂y = 2v'/(y-x)
v' = (y-x)2f(x)
where f is an arbitrary function of x.
Does that help?
L0r3n20
#3
Jul6-12, 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.

haruspex
#4
Jul7-12, 11:17 PM
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Stuck with a PDE system

Quote Quote by L0r3n20 View Post
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.
OK, so make f(x) constant. You can do likewise for (u+u', v).
L0r3n20
#5
Jul8-12, 02:08 AM
P: 19
Even thought I set f(x) constant I get the x-dependence from y-x, right?


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