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Let us solve the equation

| |
(19) |

by numerical methods.
The most obvious (but not the only) approach is the basic definition of
elementary calculus.
For the time derivative, this is
| |
(20) |

Using this in equation (19) yields the
the inflation-of-money equations (17) and (18),
where .Thus in the inflation-of-money equation
the expression of *dq*/*dt* is centered at ,whereas the expression of *q* by itself is at time *t*.
There is no reason the *q* on the right side of equation (19)
cannot be averaged at time *t*
with time ,thus centering the
whole equation at .When writing difference equations,
it is customary to write more simply as *q*_{t+1}.
(Formally one should say 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 (19) is
| |
(21) |

Letting , this becomes
| |
(22) |

which is representable as the difference star
For a fixed this star gives
a more accurate solution to the differential
equation (19) 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**.

** Next:** Explicit heat-flow equation
** Up:** FINITE DIFFERENCING IN (omega,x)-SPACE
** Previous:** First derivatives, explicit method
Stanford Exploration Project

12/26/2000