Can I simplify a PDE with Sagemath?

asked 2020-05-16 02:36:47 -0500

Irddo gravatar image

updated 2020-05-16 13:36:27 -0500

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. 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 $\bar{f}_{z\bar{z}}=(\bar{f}_z)_{\bar{z}}$.

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