# A second order PDE

• MATLAB
Homework Helper
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?

## Answers and Replies

Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
What is your question?

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

Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
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
Science Advisor
Gold Member
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).