First derivatives, explicit method

The inflation of money q at a 10% rate can be described by the difference equation

$$\begin{eqnarray} q_{ t+1 }\ -\ q_t\ \ \ &=&\ \ \ .10 \ q_t \\ ( 1.0 )\ q_{ t+1 }\ \ +\ \ ( -1.1 )\, q_t\ \ \ &=&\ \ \ 0\end{eqnarray}$$ (17) (18)

This one-dimensional calculation can be reexpressed as a differencing star and a data table. As such it provides a prototype for the organization of calculations with two-dimensional partial-differential equations. Consider $$\begin{tabular} {ccc} \hspace{.2in}{\rm Differencing Star} \... ...\\ $\downarrow$ \\ \end{tabular}\end{tabular}%\EQNLABEL{two2}$$

Since the data in the data table satisfy the difference equations (17) and (18), the differencing star may be laid anywhere on top of the data table, the numbers in the star may be multiplied by those in the underlying table, and the resulting cross products will sum to zero. On the other hand, if all but one number (the initial condition) in the data table were missing then the rest of the numbers could be filled in, one at a time, by sliding the star along, taking the difference equations to be true, and solving for the unknown data value at each stage.

Less trivial examples utilizing the same differencing star arise when the numerical constant .10 is replaced by a complex number. Such examples exhibit oscillation as well as growth and decay.