| 197 | Now since the second equation has no spatial dependence, we can solve it for |

| 198 | [[latex(e^{n+1}_i = \frac{1}{ 1 +\psi \phi^n_i}\left \{ \left ( \psi \epsilon^n_i \right )E^{n+1}_i + \left ( 1 - \bar{\psi}\phi^n_i \right ) e^n_i + \left ( \bar{\psi} \epsilon^n_i \right ) E^n_i-\theta^i_n \right \} )]] |

| 199 | and plug the result into the first equation to get a matrix equation involving only one variable. |

| 200 | |

| 201 | [[latex(\left [ 1 + \psi \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i-\frac{\phi^n_i \psi \epsilon^n_i}{ 1 +\psi \phi^n_i}\right ) \right ] E^{n+1}_i - \left ( \psi \alpha^n_{i+1/2} \right ) E^{n+1}_{i+1} - \left ( \psi \alpha^n_{i-1/2} \right ) E^{n+1}_{i-1} |

| 202 | = |

| 203 | |

| 204 | \left [ 1 - \bar{\psi} \left( \alpha^n_{i+1/2} + \alpha^n_{i-1/2} + \epsilon^n_i \right ) \right ] E^n_i +\frac{\psi \phi^n_i}{ 1 +\psi \phi^n_i}\left ( \bar{\psi} \epsilon^n_i \right ) E^n_i |

| 205 | |

| 206 | + \frac{ \phi^n_i}{ 1 +\psi \phi^n_i} e^n_i |

| 207 | |

| 208 | + \frac{1}{ 1 +\psi \phi^n_i}\theta^i_n |

| 209 | |

| 210 | )]] |

| 211 | |

| 212 | |

| 213 | |