First derivatives, implicit method

Let us solve the equation  

$${ dq \over dt } \eq 2\ r\ q$$ (24)

by numerical methods. The most obvious (but not the only) approach is the basic definition of elementary calculus. For the time derivative, this is  

$${ dq \over dt } \ \ \ \approx \ \ \ { q(t+ \Delta t )\ -\ q ( t ) \over \Delta t }$$ (25)

Using this in equation (24) yields the the inflation-of-money equations (22) and (23), where $2\,r = .1$.Thus in the inflation-of-money equation the expression of dq/dt is centered at $t+ \Delta t / 2$,whereas the expression of q by itself is at time t. There is no reason the q on the right side of equation (24) cannot be averaged at time t with time $t+\Delta t$,thus centering the whole equation at $t+ \Delta t / 2$.When writing difference equations, it is customary to write $q(t+\Delta t)$ more simply as q_t+1. (Formally one should say $t=n\Delta t$ and write q_n+1 instead of q_t+1, but helpful mnemonic information is carried by using t as the subscript instead of some integer like n.) Thus, a centered approximation of (24) is  

$$q_{{t+1}}\ -\ q_t \eq 2\,r \,\Delta t \ \ { q_{{t+1}}\ +\ q_t \over 2 }$$ (26)

Letting $\alpha = r \Delta t$, this becomes  

$$( 1- \alpha )\ q_{{t+1}}\ \ - \ ( 1+ \alpha )\ q_t \eq 0$$ (27)

which is representable as the difference star

$$\begin{tabular} {cc} &\begin{tabular} {\vert c\vert} \hline... ...r} {c} t \\ $\downarrow$ \\ \end{tabular} & \\ \end{tabular}$$

For a fixed $\Delta t$ this star gives a more accurate solution to the differential equation (24) than does the star for the inflation of money. The reasons for the names ``explicit method'' and ``implicit method'' above will become clear only after we study a more complicated equation such as the heat-flow equation.