# How to solve numerically a system of 3 interdependent non-linear ordinary differential equations?

As per title, I need to solve this:

$$\begin{cases} \frac{d^2V}{dx^2} = -\frac{q}{\epsilon}\left[p - n + \frac{N_0}{1+c_pp+c_nn}\right] \\\\ \frac{d}{dx}\left[\mu_nn\frac{dV}{dx} + D_n\frac{dn}{dx}\right] - \left[\frac{n-n_0}{\tau_n} - G_{op}\right] = 0 \\\\ \frac{d}{dx}\left[\mu_pp\frac{dV}{dx} - D_p\frac{dp}{dx}\right] + \left[\frac{p-p_0}{\tau_p} - G_{op}\right] = 0 \end{cases}$$

where V (electrostatic potential), n (total free electrons density) and p (total free holes density) are the unknown functions.

Being in 1D and in static conditions, all the functions depend on just the spatial variable "x" (the domain goes from "0" to "L").

Finally, the boundary conditions are:

$$\begin{cases} V(x=0) = V_{left} (known) \\\\ V(x=L) = V_{right} (known) \\\\ \frac{dp}{dx}(x=0) = 0 \\\\ \frac{dn}{dx}(x=0) = 0 \\\\ \frac{dp}{dx}(x=L) = 0 \\\\ \frac{dn}{dx}(x=L) = 0 \end{cases}$$

It is basically a Poisson equation coupled with drift-diffusion + continuity equations for electrons and holes in a crystalline semiconductor.

• Device Modeling? There are several monographs published by the Wien branch of Springer Verlag that describe comprehensively the theory and the finite difference schemes used to numerically solve the drift-diffusion equations coupled. with the nonlinear Poisson equation, for example "Semiconductor equations" authored by Markowich, Ringhofer and Schmeiser (1990) – Daniele Tampieri Feb 25 '20 at 18:47