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Let us solve the equation
| ![\begin{displaymath}
{ dq \over dt } \eq 2\ r\ q\end{displaymath}](img69.gif) |
(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
| ![\begin{displaymath}
{ dq \over dt } \ \ \ \approx \ \ \
{ q(t+ \Delta t )\ -\ q ( t )
\over \Delta t }\end{displaymath}](img70.gif) |
(25) |
Using this in equation (24) yields the
the inflation-of-money equations (22) and (23),
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 (24)
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 qt+1.
(Formally one should say
and write qn+1 instead of
qt+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
| ![\begin{displaymath}
q_{{t+1}}\ -\ q_t \eq 2\,r \,\Delta t \ \ { q_{{t+1}}\ +\ q_t \over 2 }\end{displaymath}](img76.gif) |
(26) |
Letting
, this becomes
| ![\begin{displaymath}
( 1- \alpha )\ q_{{t+1}}\ \ - \ ( 1+ \alpha )\ q_t \eq 0\end{displaymath}](img78.gif) |
(27) |
which is representable as the difference star
![\begin{displaymath}
\begin{tabular}
{cc}
&\begin{tabular}
{\vert c\vert} \hline...
...r}
{c}
t \\ $\downarrow$ \\ \end{tabular} & \\ \end{tabular}\end{displaymath}](img79.gif)
For a fixed
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.
Next: The explicit heat-flow equation
Up: FINITE DIFFERENCING
Previous: First derivatives, explicit method
Stanford Exploration Project
10/31/1997