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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