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Math Help - System of non-linear partial differential eqs from electrostatics

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    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 donīt 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.

    Thanks in advance.

    P.S.: I forgot to say that the problem is solved in cylindrical coordinates (it occurs due to a discharge tip inside a cylindrical tube) and it has axial symmetry.
    Last edited by Madoro; December 17th 2012 at 05:18 AM.
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