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Explicit Scheme, solving the equations

Since all the discretized initial values \begin{equation*} u_{i,0} = g(x_i), \end{equation*} are known, then after one time-step the only unknown quantity is u_{i,1} which is given by \begin{equation*} u_{i,1}= \alpha u_{i-1,0}+(1-2\alpha)u_{i,0}+\alpha u_{i+1,0}= \alpha g(x_{i-1})+(1-2\alpha)g(x_{i})+\alpha g(x_{i+1}). \end{equation*} We can then obtain u_{i,2} using the previously calculated values u_{i,1} and the boundary conditions a(t) and b(t) . This algorithm results in a so-called explicit scheme, since the next functions u_{i,j+1} are explicitely given by Eq. (7).