A second order PDE

  • #1
hunt_mat
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I have the following system of PDEs:
[tex]
\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}
[/tex]
[tex]
\frac{\partial}{\partial\hat{x}}(\hat{\varepsilon}(\hat{x})\hat{E})=-\beta\hat{c}
[/tex]
[tex]
\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)
[/tex]

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

  • #2
Orodruin
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What is your question?
 
  • #3
hunt_mat
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What would be the best way forward? As stated in my post.
 
  • #4
Orodruin
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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.
 
  • #5
Dr Transport
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a perturbation expansion for [itex] E [/itex] (kill all the [itex] \hat{}[/itex], 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).
 

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