I have to write down the ODE system for this PDE with the method of lines:
The pde is: d(u)/dt=D(d^2(u)/dx^2)+u(1-u) with an IC u(0,x)=1-x and 2 BCs: u(t,0)=1 and u(t,1)=0.
D is a constant.
What is an ODE system? I do not understand that term.
I know how to do this d(u)/dt=D(d^2(u)/dx^2); I can change d^2(u)/dx^2 into [(u_i+1)-(2u_i)+(u_i-1)]/delta(x)^2
so, the whole thing will become
d(u)/dt=D[(u_i+1)-(2u_i)+(u_i-1)]/delta(x)^2
or
[(u_i with a superscript of n+1)-(u_i with a superscript of n)]/delta(t)=D[(u_i+1)-(2u_i)+(u_i-1)]/delta(x)^2
then i can solve for u_i with a superscript of n+1 explicitly.
However, in my question there is an extra term u(1-u), how should i deal with it.