I want to solve steady state diffusion with constant production term (in the source $[-L_s, 0]$) and a non-linear degradation term, where degradation takes place over the whole domain $[-L_s, L]$, but production only takes place if $x<0$. This is ensured by the Heaviside function $H(x)$. So if $x<1$ then $H(x) = 1$.
Parameters: $D$ Diffusion coefficient, $\alpha$ degradation rate, $p$ production rate and $n>1$. $D, \alpha, p$ are all constant.
The equation looks as follows:
\begin{equation} 0 = D\Delta C(x) - \alpha C(x)^n + p H(x < 0). \end{equation}
I impose zero flux at the respecitve ends of the domain. Let's say $L_s$ is the beginning of the source domain (left side of the domain) and $L$ is the right hand side of the domain then,
\begin{equation} \frac{\partial C(L_s)}{\partial x} = 0 = \frac{\partial C(L)}{\partial x}. \end{equation}
I was wondering if one could maybe use different methods than for example Laplace-transforms to solve the equation as it is not linear.
Thanks for your help.
Welcome to math stack. 1. You will need to specify whether $C$ is to be continuous at $0$ (probably yes) and same for $C'$ (maybe also yes). Your equation then insures that $C''$ will not be continuous at $0$. 2. Steady state diffusion in one dimension brings to mind simple things like constants, and in fact there is a constant value, easy to find, that satisfies the requirement on the left hand side. That might or might not be the correct partial answer, and you won't know until you work out more about the right hand side. 3. On the right hand side, your equation just says $DC'' = \alpha C^n$. A first integral can be found by first multiplying by $C'$ and then integrating. 4. After that comes a nonlinear but first order ODE, that would probably have to be done numerically.