Now it is time for the finite difference approximations of derivatives: $$ \begin{equation} u'(t_n) \approx \frac{u^{n+1}-u^{n}}{t_{n+1}-t_n}\tp \label{decay:FEdiff} \end{equation} $$ Step 3: Replacing derivatives by finite differences . ) ) }}\,(x-a)_{k}=\sum _{k=0}^{\infty }{\binom {x-a}{k}}\,\Delta ^{k}[f](a),}, which holds for any polynomial function f and for many (but not all) analytic functions (It does not hold when f is exponential type a {\displaystyle \displaystyle d

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