## coupled linear PDE system

Hi all, I have a system of 3 coupled linear PDEs which can be expressed in matrix form as:

$$\left( \begin{array}{ccc} \alpha_1 \partial_{\theta} & \alpha_2 & \alpha_3 \\ \beta_1 \partial_r & \beta_2 & \beta_3 \\ 0 & \gamma_2 \partial_{\theta} & 1 + \gamma_3 \partial_r \\ \end{array} \right) \left( \begin{array}{c} \psi \\ E_r \\ E_{\theta} \end{array} \right) = \left( \begin{array}{c} -\alpha_4 \\ -\beta_4 \\ 0 \\ \end{array} \right)$$

where the coefficients $\alpha,\beta,\gamma$ are functions of position $r,\theta$.

I believe its possible to decouple this system and end up with a single, 2nd order equation for $\psi$, but I don't know how to proceed to do this. Somehow it must be possible to diagonalize the matrix operator.

If its not possible, then does anyone know of any references for solving such a system with finite difference methods? I'm familiar with FD methods for a single equation but haven't done coupled equations before.
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 Recognitions: Homework Help This is a first order system, I would recommend method of characteristics.