ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sat, 16 May 2020 09:36:47 +0200Can I simplify a PDE with Sagemath?https://ask.sagemath.org/question/51422/can-i-simplify-a-pde-with-sagemath/I'm a beginner in Sagemath and I search for help with a calculation problem with some derivatives and basic operations.
I need to simplify a PDE that depends of a smooth function $f:U\subset\mathbb{C}\to\mathbb{C}$, something like,
$$Af^2+2fB+C=0\tag{1}$$
where $A$, $B$ and $C$ depends of $f$ and $\bar{f}$ and their complex derivatives, of order until 3.
Of course I can compute this by hand, but in my problem it will be a very long computation and I thought about to try Sagemath for this computation.
I would like to know if I can to define a symbolic complex function on Sagemath and to do basic operations as to compute derivatives in $z$ and $\bar{z}$. For example, if
$A=(f_{z\bar{z}}+|f_{z}|^2+f_zf_{\bar{z}})_z$
$B=((\bar{f}_{z\bar{z}})^2+|\bar{f}_{z}|^4+(\bar{f}_z\bar{f}_{\bar{z}})^2)_z$
$C=((f_{z\bar{z}})^3+|f_{z}|^6+(f_zf_{\bar{z}})^3)_z,$
Can I simplify equation $(1)$ using Sagemath?
Thank for your attention.
I appreciate any help.
----------
Notation:
The derivatives on $z$ and $\bar{z}$ are the [Wirtinger derivatives][1]. Then, given a smooth function $f:U\subset\mathbb{C}\to\mathbb{C}$, we have
$f_z = \frac{\partial f}{\partial z}, \bar{f}_z = \frac{\partial \bar{f}}{\partial z}, f_{\bar{z}} = \frac{\partial f}{\partial {\bar{z}}},\bar{f}_{\bar{z}} = \frac{\partial \bar{f}}{\partial {\bar{z}}}, |f_z|^2=f_z\bar{f}_{\bar{z}}, |\bar{f}_z|^2=\bar{f}_z f_{\bar{z}}, f_{z\bar{z}}=(f_z)_{\bar{z}}$ and $\bar{f}_{z\bar{z}}=(\bar{f}_z)_{\bar{z}}$.
[1]: https://en.wikipedia.org/wiki/Wirtinger_derivativesSat, 16 May 2020 09:36:47 +0200https://ask.sagemath.org/question/51422/can-i-simplify-a-pde-with-sagemath/