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.