Solving 2nd Order PDE System with Crank-Nicholson

[hunt_mat]

I have the following system of PDEs:
$$\hat{\rho}\hat{c}_{th}\frac{\partial\hat{T}}{\partial\hat{x}}-\alpha_{1}\frac{\partial}{\partial\hat{x}}\left(\hat{k}(\hat{x})\frac{\partial\hat{T}}{\partial\hat{x}}\right)=\alpha_{1}\hat{\sigma}(\hat{x})\hat{E}$$
$$\frac{\partial}{\partial\hat{x}}(\hat{\varepsilon}(\hat{x})\hat{E})=-\beta\hat{c}$$
$$\frac{\partial\hat{c}}{\partial\hat{t}}-\gamma_{1}\frac{\partial}{\partial\hat{x}}\left(\hat{D}(\hat{x})\frac{\partial\hat{c}}{\partial\hat{x}}\right)= \gamma_{2}\left(\frac{\partial\hat{E}}{\partial\hat{x}}+\frac{\partial\hat{c}}{\partial\hat{x}}-\frac{\partial\hat{T}}{\partial\hat{x}}\right)$$

I would like to solve this system using the Crank-Nicholson method. Now for a linear equation, the CN scheme is well defined, MATLAB has some very nice algorithms for this.

However the first equation has a nonlinear term in E, and I have no equation which time steps E. I suppose that I could use a Newton-Raphson scheme to get the solution. Would that be the correct way forward?

 

[Orodruin]

What is your question?

 

[hunt_mat]

What would be the best way forward? As stated in my post.

 

[Orodruin]

What would be the best way forward? As stated in my post.

Sorry about that.
For some reason no text after your first equation is visible in Safari on iOS.

 

[Dr Transport]

a perturbation expansion for $E$ (kill all the $\hat{}$, it makes the equations hard to read and is confusing, unless they are all vector quantities,l then you have a mess and an intractable system).