I want to solve the following differential equation from a paper with the boundary condition:
The paper used the Crank–Nicolson method for solving it. I think I understand the method after googling it, but most websites discussing it use the heat equation as an example. Here we can replace the usual t
variable with Xi
, and the usual x
as Rho
. Now unlike the heat equation, this equation has an additional first derivative of Xi
term (the first term gives both first and second derivative of Xi
). Does it matter what finite difference scheme I use to approximate this term?
Another thing is that the equation has nonlinear terms involving |A|^2
and |A|^4
. How can I incorporate this to the matrix equation I am going to solve? (PS: A
is complex)
I am pretty new to numerical methods so forgive me if I have neglected anything significant.