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Inserting the finite difference approximation in $$ u'(t_n) = -au(t_n)$$ gives $$ \begin{equation} \frac{u^{n+1}-u^{n}}{t_{n+1}-t_n} = -au^{n},\quad n=0,1,\ldots,N_t-1 \tag{5} \end{equation} $$
(Known as discrete equation, or discrete problem, or finite difference method/scheme)
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