# Thread: System of non-linear partial differential eqs from electrostatics

1. ## System of non-linear partial differential eqs from electrostatics

I have an electrostatics problem wich leads to the following system of differential equations:

$\frac{\partial E_z}{\partial z}=\frac{\rho}{\epsilon_0}$ (1)

$Z_i E_r \frac{\partial \rho}{\partial r}+(u_g+ Z_i E_z) \frac{\partial \rho}{\partial z} + \rho Z_i \frac{\partial E_z}{\partial z}=0$ (2)

Substituting eq. (1) into eq. (2):
$Z_i E_r \frac{\partial \rho}{\partial r}+(u_g+ Z_i E_z) \frac{\partial \rho}{\partial z} + \frac{\rho^2 Z_i}{\epsilon_0}=0$ (3)

Therefore I have a system of 2 equations (1 & 3) with 2 unknowns, the axial field $E_z$ and the charge density $\rho(z,r)$. The rest of the variables are known so they can be supposed as constants.

The Neumann boundary conditions in the axial axis, when r=0 are:
$\frac{\partial E_z}{\partial r}=0; \frac{\partial E_z}{\partial z} \neq 0$
$\frac{\partial \rho}{\partial r}=0; \frac{\partial \rho}{\partial z} \neq 0$

I'm not sure on how to solve it, I'm considering two options:

- derivate eq. (3) with respect to $z$ to substitute in eq. (1), but I don´t get rid of $E_z$ and the eq. (3) becomes more complicated.

- Solve by semi-implicit method, considering that $z=du_z/dt$, but since is an equation in partial derivatives I'm not sure on how to manage the term in $r$

I'm totally stuck on this, I'm asking for a direction of solving it, not for a solution, so any help would be grateful.