I have a system of coupled non-linear ODEs.
$$J = \mu*e*n(x)*E(x) + \mu*K*T*\frac{dn(x)}{dx}$$ $$\frac{dE(x)}{dx} = \frac{4*\pi*e}{\epsilon}[N_D(x) -n(x)]$$
The first equation is drift-diffusion, the second is Gauss' Law. I am interested in solving this system self-consistently for the carrier profile, $n(x)$, and the electric field, $E(x)$, using finite differences. The current, $J$, the mobility, $\mu$, elementary charge, $e$, are known. The doping profile $N_D(x)$ is known, and I have boundary conditions for the carrier concentration $n(0) = N_D(0)$ and $n(L) = N_D(L)$. I am unsure how to proceed with solving these equations.
I am following the analysis of Baranger and Wilkins in the attached review. Appendix B.
Thanks in advance!
You can decouple your system by injecting your equation (2) into your equation (1). Doing so, you will get a nonlinear second order differential equation in terms of the electric field $E$. The second equation reads as,
$$n = N - \frac{\epsilon}{4\pi e} \dot{E}$$
from which you derive,
$$\dot{n} = \dot{N} - \frac{\epsilon}{4\pi e} \ddot{E}$$
Letting,
\begin{eqnarray*} a(x) &=& \frac{4\pi e}{\epsilon}\dot N(x) -\frac{4\pi}{\epsilon \mu kT}J\\ b(x) &=& \frac{4\pi e^2}{\epsilon kT}N(x)\\ c &=& \frac{e}{kT} \end{eqnarray*}
The second-order DE can be rewritten, $$\ddot{E}=a(x) + b(x) E + c E\dot{E}$$
This equation can be casted into the form of Abel's equation of the second kind (of the first derivative order) considering as new dependent variable $E$ and new dependent field $u(E)=\dot E$ so that $\ddot{E}=u\dot u$. It can also be casted into the form of Abel's equation of the first kind considering the further change of dependent field $v(E)=\frac{1}{u(E)}$.
Unfortunately I was not able to integrate any of them in closed form.