Explicit Scheme
In one dimension we have the following equation
$$
\begin{equation*}
\nabla^2 u(x,t) =\frac{\partial u(x,t)}{\partial t},
\end{equation*}
$$
or
$$
\begin{equation*}
u_{xx} = u_t,
\end{equation*}
$$
with initial conditions, i.e., the conditions at \( t=0 \),
$$
\begin{equation*}
u(x,0)= g(x) \hspace{0.5cm} 0 < x < L
\end{equation*}
$$
with \( L=1 \) the length of the \( x \)-region of interest.