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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