Graphics Programs Reference
In-Depth Information
If y i arespecifiedat different values of x , the problemiscalleda boundary value
problem
.
For example,
y =−
y (0)
y
y (0)
=
1
=
0
is an initial value problem since both auxiliary conditions imposed on the solution
are givenat x
=
0. On the other hand,
y =−
y
y (0)
=
1
y (
π
)
=
0
is aboundary value problembecause the twoconditions arespecifiedat different
values of x .
In thischapterweconsider only initial value problems. The more difficult bound-
ary value problems are discussedinthe nextchapter. We also make extensive use of
vectornotation, which allows us manipulate sets of first-order equations in a concise
form. For example, Eqs. (7.4) are writtenas
y =
F ( x
,
y )
y ( a )
= α
(7.5a)
where
y 2
y 3
.
y n
f ( x
F ( x
,
y )
=
(7.5b)
,
y )
A numerical solution of differentialequations is essentiallyatable of x - and y -values
listedat discrete intervals of x .
7.2
Taylor Series Method
The Taylor series methodisconceptually simple and capable of high accuracy. Its
basis is the truncated Taylor series for y about x :
1
2! y ( x ) h 2
1
3! y ( x ) h 3
1
n ! y ( m ) ( x ) h m
y ( x ) h
y ( x
+
h )
y ( x )
+
+
+
+···+
(7.6)
Because Eq. (7.6) predicts y at x
h from the informationavailable at x , it is also a
formula for numerical integration. The last term kept in the series determines the
order of integration .For the series in Eq. (7.6) the integration orderis m .
The truncation error, due to the termsomitted from the series, is
+
1
1)! y ( m + 1) (
) h m + 1 ,
E
=
ξ
x
<ξ<
x
+
h
( m
+
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