Computer Algebra Recipes

2.3. NUMERICAL SOLUTION OF ODES 89
2.3.1 Finite Di®erence Approximations
A tool knows exactly how it is meant to be handled, while the user of
the tool can only have an approximate idea.
Milan Kundera, Czech author, critic (1929{)
Consider a general function y(x) as schematically depicted by the solid curved
line in Figure 2.21. The independent variable has been taken to be x, butit
could just as well be the time t. It is desired to ¯nd, say, the ¯rst and second
Q
x–h
h
y(x–h)
P
y(x)
h
R
y(x+h)
x x+h
Figure 2.21: Obtaining ¯nite di®erence approximations to derivatives.
derivatives of y at an arbitrary point P on the curve located at the horizontal
position x. Assuming that h is small, two neighboring points R and Q located
at x + h and x ¡ h, respectively, are considered. At point R, the vertical height
is y(x + h) andatQ it is y(x ¡ h).
Since h is assumed to be small, y(x + h) can be Taylor expanded in powers
of h about h = 0, neglecting terms of, say, order h 4 (i.e., O(h 4 )) and higher.
> restart:
> eq1:=y(x+h)=taylor(y(x+h),h=0,4);
eq1 := y(x + h) =
y(x)+D(y)(x) h + 1
2 (D(2) )(y)(x) h2 + 1
6 (D(3) )(y)(x) h3 +O(h4 )
The O(h4 )termineq1 can be removed with the convert(polynom) command.
> eq1b:=convert(eq1,polynom);
eq1b := y(x + h) =y(x)+D(y)(x) h + 1
2 (D(2) )(y)(x) h2 + 1
6 (D(3) )(y)(x) h3 Similarly y(x ¡ h) is Taylor expanded and the fourth-order term removed.