Four governing equations concerning the reaction occurred in the porous electrode are \begin{equation} \nabla \cdot i_1 + \nabla \cdot i_2=0 \end{equation}
\begin{equation} i_2 = -\kappa \nabla \phi_2 \end{equation}
\begin{equation} i_1 = -\sigma \nabla \phi_1 \end{equation}
\begin{equation} \nabla \cdot i_2 = a i_{o}\biggl\{\exp\biggl(\frac{\alpha_{a} F}{RT}(\phi_1-\phi_2)\biggr) - \exp\biggl(-\frac{\alpha_{c} F}{RT}(\phi_1-\phi_2)\biggr) \biggr\} \end{equation}
For large positive overpotential ($\phi_1-\phi_2$), the forth equation can be approximated as
\begin{equation} \nabla \cdot i_2 = a i_{o} \exp\biggl(\frac{\alpha_{a} F}{RT}(\phi_1-\phi_2)\biggr) \end{equation} The boundary conditions in one-dimensional case are \begin{align} i_2=-I_{app} \quad &\textrm{at} \quad x=0\\ i_1=0 \quad &\textrm{at} \quad x=0\\ i_2=0 \quad &\textrm{at} \quad x=L\\ i_1=-I_{app} \quad &\textrm{at} \quad x=L\\ \end{align} Arbirary reference potential can be made as \begin{align} \phi_2=0 \quad &\textrm{at} \quad x=L\\ \end{align} Also everywhere in the system we have \begin{equation} i_1 + i_2 = -I_{app} \end{equation}
Now I try to solved the equations numerically by first rewrite the equation in terms of $\phi_1$ and $\phi_2$ \begin{equation} \frac{d^2\phi_1}{dx^2} = \frac{ai_{o} \exp\biggl(\frac{\alpha_{a} F}{RT}(\phi_1-\phi_2)\biggr)}{\sigma}\\ \frac{d^2\phi_2}{dx^2} = -\frac{ai_{o} \exp\biggl(\frac{\alpha_{a} F}{RT}(\phi_1-\phi_2)\biggr)}{\kappa} \end{equation} The boundary conditions above form only pure Neumann conditions for $\phi_1$ (through $i_1$), so I suppose there will be infinite solutions (if some conditions is achieved).
Now if I differentiate equation \begin{equation} \frac {di_2}{dx} = a i_{o} \exp\biggl(\frac{\alpha_{a} F}{RT}(\phi_1-\phi_2)\biggr) \end{equation} \begin{equation} \frac {d^2i_2}{dx^2} =\frac{\alpha_{a} F}{RT} \frac{d}{dx}(\phi_1-\phi_2) \cdot ai_{o} \exp\biggl(\frac{\alpha_{a} F}{RT}(\phi_1-\phi_2)\biggr) \end{equation} \begin{equation} \frac {d^2i_2}{dx^2} =\frac {di_2}{dx} \frac{\alpha_{a} F}{RT} (\frac{I-i_2}{\sigma}-\frac{i_2}{\kappa}) \end{equation} Suddenly it transforms to a well-posed second order equation in terms of $i_2$! How come? My problem is if I don't approximate the over potential term then I wouldn't be able to do this trick. I wonder whether this is a well-posed or an ill-posed problems.