Hi !
I need some help in finding a simple method to transform any linear PDEs in IR^2 into linear complex differential equation or pseudo-linear complex differential equation.
Mohamed> ...any PDEs...
What makes you think this is a reasonable request for help?
Pr Olaussen ,
I'm sorry, not " any PDEs "; I meant any linear PDEs , I've re-edited it.
and I'm asking if there's a method, or not ?
Sorry again.
That may not be your real intent ( I presume). Otherwise complexify that is given
linear PDEs in two variables: $Lu=f,$ then $Lv=g$ write $w= u+i v$ as a solution of
$Lw = f+ig.$ So it works on a real domain, but $w$ is complex. ( It is like the converse part of Schrodinger equation, split into two PDEs taking real and imaginary part.)
Moreover, given any harmonic function in 2D, one can get its complex congugate and together one obtains an analytic function. Converse also holds.
Do you mean get a PDE on the complex plane ?
Many thanks Pr Pani
First, yes i want to get a PDE on the complex plane if that's possible for linear operator L,
second, I want to study the growth (using Nevanlinna Theory) of solutions to some PDE of Degree 2, and for that, I thought to transform the PDE to simple linear differential equation.
So, I ask again, if that is possible ?
Thanks again
You may want to keep in mind the classical example by Lewy:
https://en.wikipedia.org/wiki/Lewy's_example
The best known (and useful) case is the Laplace equation in two dimension, where f.i. the real or imaginary part of any analytic function is a solution. In that case you don't seem to need any complex differential equation at all! But, of course, the fact that the function is analytic implies that it satisfies the Cauchy-Riemann equation.
So my question is: Would you allow the complex conjugation operator to occur in your complex differential equation? If you do, then it is just a matter of a coordinate transform from coordinates x, y to coordinates z=x + i y, z* = x- i y, which is always possible (but hardly very useful). If your new equation does not depend on z*, then you have made significant progress.
Now, that is not the end of the story: Perhaps you may first make a coordinate transform (x, y) -> (X, Y) such that with the new coordinates Z = X + i Y, Z* = X - i Y, your equation does not depend on Z*. How can one know if that is possible or not? I have no idea, but it would mean that your problem allows a complex structure^*. Anyway, I think one needs more knowledge of the original 2D-equation to be able to make progress.
^* One should also consider the possibility of post-transformations (transformations of the solution):
H = F + i G , with (F, G) -> (f,g).
Dear Pr Olaussen
So many thanks for your valuable remarks
Exactly; the original problem is deduced from Laplace equation and in this case we can just pass through the complex plane to get the solution which is Poisson Formula (I guess), similarly, when I want to study some properties of solutions of PDE, it's hard to pass through complex plane, as same as in Laplace equation, because, the solution is not always harmonic to be real part or imaginary part of an analytic function. For that, if I win to separate the real coordinates (x,y) in PDE, to get a new suitable coordinate X and i*Y , to achieve a Complex Differential Equation without complex conjugation operator, in this case, it's okay.
By this discussion, I've made a good idea in my mind, but I'm not sure yet about the possibility of progress.
Just Hope !
Sincerly yours
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