Solution for the One-dimensional Diffusion Equation
The boundary conditions are
\begin{equation*}
u(0,t)= 0 \hspace{0.5cm} t \ge 0, \hspace{1cm} u(L,t)= 0 \hspace{0.5cm} t \ge 0,
\end{equation*}
We assume that we have solutions of the form (separation of variable)
\begin{equation*}
u(x,t)=F(x)G(t).
\end{equation*}
which inserted in the partial differential equation results in
\begin{equation*}
\frac{F''}{F}=\frac{G'}{G},
\end{equation*}
where the derivative is with respect to
x on the left hand side and with respect to
t on right hand side.
This equation should hold for all
x and
t . We must require the rhs and lhs to be equal to a constant.