How can I use Fourier Transform to solve Fisher-Kolmogorov Equation in 1D?
\begin{equation} u_t(x,t) = u_{xx}(t) + u(1-u) \end{equation}
\begin{equation} u(0,x) = \phi(x) \end{equation}
with Dirichlet \begin{equation} u(0,t)=0 \\ u(1,t)=0 \end{equation}
and Neumann boundary conditions \begin{equation} u_x(0,t)=0\\ u_x(1,t)=0 \end{equation}
The problem was partially solved following this tutorial.
Solving the equation \begin{equation} u_t = u_{xx} \\ \end{equation}
from $t$ to $\Delta t$ gives
\begin{equation} \hat{u}_t = (ik)^2 \hat{u} \\ \tilde{u}(x,t+\Delta t) = F^{-1} \left\{e^{-k^2 \Delta t} \hat{u} \right\} \end{equation}
Now solving the equation
\begin{equation} u_t = u \, (1-u) \\ \end{equation}
Using the splitting operator method we can write
\begin{equation} u(x,t+\Delta t) = e^{-u(1-u)\Delta t} \tilde{u}(x,t+\Delta t) \end{equation}
which gives the solution
I need to know:
How can I impose Dirichlet and Neumann boundary conditions?