Numerical solution to coupled diff. eq.

[mrandersdk]

I have three equations

$$(\frac{\partial}{\partial t} + c \frac{\partial}{\partial z}) E_p(z,t) = i N_r(z) \sigma_{ba}(z,t)$$

$$\frac{1}{i E_c^*(z,t)}(\frac{\partial}{\partial t} + \Gamma_{bc}) \sigma_{bc}(z,t) = \sigma_{ba}(z,t)$$

$$-\frac{E_p(z,t)}{E_c(z,t)} + \frac{1}{i E_c(z,t)}(\frac{\partial}{\partial t} + \Gamma_{ba}) \sigma_{ba}(z,t) = \sigma_{bc}(z,t)$$

where the functions $$E_p(z,t)$$ and $$N_r(z)$$ is know, and all other things that do not depend on time or position are known constants.

How can I solve this numericaly with some give initial conditions. I have MATLAB 7.0 or mathematica to my disposal but can't make anything work.

 

[Chris Hillman]

Why Numerical?

Did you try finding an exact solution? And are these complex functions or what? Is * Hermitian transpose?

 

[mrandersdk]

yes the function can be complex, but are of real variable, so not need for complex function theory.

the 'i' in the first equation is the complex i.

Do you think it is possible to find and exact solution, that would of cause be great.

 

[Rainbow Child]

It is not really too hard to find an exact solution if $$E_p(z,t)$$ and $$N_r(z)$$ are known.

From the first equation you have $$\sigma_{ba}(z,t)$$ in terms of the known funtions $$E_p(z,t)$$ and $$N_r(z)$$.

From the 3nd equation you have $$\sigma_{bc}(z,t)$$ in terms of the known funtions $$E_p(z,t),\,N_r(z)$$ and the unknown function $$E_c(z,t)$$.

Plugging these informations into the 2nd equation you have a PDE which involves $$E_c(z,t) \, and \, E^*_c(z,t)$$, say it (A).

• If you are looking for real solutions, i.e. $$E_c(z,t) =E^*_c(z,t)$$ then equation (A) is just an ODE with respect to $$E_c(z,t)$$ since it involves only the derivative $$\partial_t E_c(z,t)$$. It looks like
  
  
  $$\partial_t E_c(z,t) \sim \alpha_1\,E_c(z,t) +\alpha_2\,E^3_c(z,t)$$​
  
  where $$\alpha_i$$ are known funtions of $$(z,t)$$. It can be full integrated either by hand or with the help of Mathematica.
  
• If $$E_c(z,t)$$ is a complex funtion you have to split every term in (A) at it's Real and Imaginary part, in order to end up with two messy[/] DE. Fortunately, in this case too you can integrate the resulting system, by imposing the integrabity conditions on $$Re[E_c(z,t)]\, and\, Im[E_c(z,t)]$$
  

I hope that was useful for you!

 

[momentum_waves]

Wave phenomena are predicted. Thankfully, at this point, they look to be linear (if gamma & c are constant).

If so, then you may be reasonably safe in using simple numeric strategies.