Comparison of numerical approximation methods for the solution of first order differential equations
by Leon J D Rouge
A THESIS Submitted to the Graduate Faculty In partial fulfillment of the requirements for the degree
of Master of Science In Applied Mathematics
Montana State University
© Copyright by Leon J D Rouge (1952)
Abstract:
Since the solution of an nth order differential equation can be reduced to the solution of a system of
first order differential equations, we shall concern ourselves only with the solution of the latter. In our
discussion we shall consider the basic conditions needed to assure a solution by Picard’s Method, a
short description of the method, determination of the error inherent in the method, extensions of the
method within the region of convergence, procedures to minimize errors, and an illustration of its
application.
Parallelling Picard's Method, we shall analyze the method of Taylor's series. In similar manner the
difference methods are presented, pointing out in particular that, although these methods are more
accurate than the analytic methods such as Picard's and Taylor's, they are step-by-step numerical
approximations and, unless the results are fitted into an expression such as Newton's formula, they
cannot produce a solution in analytic form.
An analytic discussion follows, outlining recommended procedures to be followed in the solution of
first order differential equations, emphasizing the need for careful preliminary analysis of error terms,
spacings, and the inherent characteristics of each method in order to lead most efficiently to desired
results.
As an illustration of the problems encountered and the accuracy obtained by each method, an example
is presented and results compared.
COMPARISON OF NUMERICAL APPROXIMATION
METHODS FOR THE SOLUTION OF
FIRST ORDER DIFFERENTIAL EQUATIONS
by
LEON J . D. ROUGE
A THESIS
SulMnltted t o t h e G rad u ate F a c u lty
In
p a r t i a l f u l f i l l m e n t o f th e re q u ire m e n ts
f o r th e d e g re e o f
M aster o f S c ie n ce in A p p lied M athem atics
at
Montana S ta te C o lleg e
Approved:
id, MajMl D epartm ent
Examii
iean, G ra d u a te D iv is io n
Bozeman, Montana
J u n e , 1952
*.
2
£ (Ljp., 2 -
TABLE OF CONTENTS
Page
A b s tr a c t
3
I
I n tr o d u c tio n
A
I I P i c a r d 's Method
6
1.
2.
3.
Ae
5.
6.
E x iste n c e o f s o lu tio n
E r r o r d e te rm in a tio n
Example
E x te n sio n s o f t h e method w ith in R*
O p tio n a l p ro c e d u re s t o m inim ize e r r o r s i n
e x te n s io n s o f d a ta
Illu s tra tio n
I I I Method o f T a y lo r 's s e r i e s
1.
2.
3.
15
In tro d u c tic m
E r ro r E stim a te s
M odified method o f T a y lo r 's s e r i e s
19
IV D iffe re n c e M ethods
1.
2.
3.
In tr o d u c tio n
Adams Method
Rxmge-Kutta Method
V A n a ly sis o f Methods
28
T I Comparlscm o f m ethods by i l l u s t r a t i v e problem
30
V II C onclusion
39
B ib lio g ra p h y
AO
103050
3
ABSTRACT
S in ce t h e s o lu tio n o f an n th o rd e r d i f f e r e n t i a l e o u a tio n can be
reduced t o th e s o lu tio n o f a system o f f i r s t o rd e r d i f f e r e n t i a l e q u a tio n s ,
we s h a l l concern o u rs e lv e s o n ly w ith t h e s o lu tio n o f t h e l a t t e r . In o u r
d is c u s s io n we s h a l l c o n s id e r th e b a s ic c o n d itio n s needed t o a s s u re a
s o lu tio n by P ic a r d ’ s M ethod, a short, d e s c r ip ti o n o f th e m ethod, d e te rm in a tio n
o f th e e r r o r in h e r e n t i n th e method, e x te n s io n s o f th e method w ith in th e
re g io n o f convergence, p ro c e d u re s t o m inim ize e r r o r s , and an i l l u s t r a t i o n
o f i t s a p p lic a tio n .
P a r a l l e l l i n g P i c a r d 's M ethod, we s h a l l a n aly z e th e method o f T a y lo r 's
s e r i e s . In s im ila r manner th e d if f e r e n c e methods a r e p re s e n te d , p o in tin g
o u t i n p a r t i c u l a r t h a t , a lth o u g h th e s e m ethods a r e more a c c u r a te th a n th e
a n a l y t i c m ethods such a s P i c a r d 's and T a y l o r 's , th e y a r e s te p - b y - s te p
n u m e ric a l ap p ro x im a tio n s a n d , u n le s s t h e r e s u l t s a r e f i t t e d i n t o an ex­
p r e s s io n such a s N ew ton's fo rm u la, th e y cannot produce a s o lu tio n in an a­
l y t i c form .
An a n a l y t i c d is c u s s io n fo llo w s , o u tlin in g recommended p ro c e d u re s to
be fo llo w ed i n th e s o lu tio n o f f i r s t o rd e r d i f f e r e n t i a l e q u a tio n s , em­
p h a s iz in g th e need f o r c a r e f u l p re lim in a ry a n a ly s is o f e r r o r te rm s , s p a c in g s ,
and th e in h e r e n t c h a r a c t e r i s t i c s o f each method i n o rd e r t o le a d most
e f f i c i e n t l y t o d e s ir e d r e s u l t s .
As an i l l u s t r a t i o n o f th e problem s en co u n tered and t h e accu racy
o b ta in e d by each m ethod, an example i s p re s e n te d and r e s u l t s compared.
4
I.
INTRODUCTION
Given an n th o rd e r d i f f e r e n t i a l e o u a tio n o f th e f o r a
y ( n ) (x ) = f ( x ,y ,y » , , , . y ( * ~ l ) )
s u b je c t t o i n i t i a l c o n d itio n s y (x Q) - y 0 , y '( x 0 ) a
y o (n - l ) .
. . . . y ^ 11” 1 ^ )
-
Qy in tr o d u c tio n o f p a ra m ete rs y i , T g , . . . . y ^ , t h e eq u atio n
can be reduced t o th e system
y' - yl» yl' = y2*......yA-I = f^x»y»yi,....yn„2.)
each o f which i s a d i f f e r e n t i a l e q u a tio n o f f i r s t o r d e r .
F or exam ple, th e second o rd e r d i f f e r e n t i a l e q u atio n
y" s f ( x , y , y ' )
w ith i n i t i a l c o n d itio n s y(xQ) _ y Q, y '(x 0) - y«, by th e s u b s t i t u t i o n
y ' (x )
z
p ( x ) , re d u c e s t o th e system
P1Cx) - f(x ,y ,p )
y '(x ) s p(x)
Thus, o u r b a s ic problem i s th e s o lu tio n o f f i r s t o r d e r d i f f e r e n t i a l
e q u a tio n s o f t h e f o r a
y ’(x) - f(x ,y )
w ith i n i t i a l c o n d itio n y (x 0 ) e y0 .
Di o u r d is c u s s io n we s h a l l c o n sid e r
th e b a s ic c o n d itio n s needed t o a s s u re a s o lu tio n by each method p re s e n te d ,
a s h o r t d e s c r ip tio n o f each m ethod, an a n a ly s is o f th e a c c u ra cy o b ta in e d ,
5
and an a n a l y t i c com parison o f th e v a rio u s m ethods.
F in a l l y , a s an
i l l u s t r a t i o n o f th e a p p lic a tio n o f each m ethod, a problem i s p re s e n te d .
6
II.
PICARD'S METHOD
1# E x is te n c e o f s o l u t i o n .
The e x is te n c e and u n iq u e n e ss o f a s o lu tio n
y (x ) o f th e d i f f e r e n t i a l e q u a tio n
a )
y
=
w ith i n i t i a l c o n d itio n F(X0 ) = J 0 a re g u a ra n te e d i f t h e fo llo w in g th r e e
b a s ic assu m p tio n s a r e s a t i s f i e d :
(a )
f ( x ,y ) i s s in g le v a lu e d in a re g io n R o f th e x y -p la n e which i n ­
c lu d e s th e p o in t
(XotJ0).
(b ) f ( x ,y ) i s c o n tin u o u s in R, hence
(2)
Jfcx.y) I i M
f o r a l l ( x ,y ) in R.
( c ) f ( x ,y ) s a t i s f i e s a L ip s c h itz c o n d itio n f o r any two p o in ts (x>y^)
and ( x ,y 2 ) i n R:
(3)
I
K
w here K i s a c o n s ta n t dependent on f ( x ,y ) b u t in d ep en d en t o f (x ,y ^ ) and
( x , y 2) .
We d e f in e th e sequence o f fu n c tio n s :
( a)
y-
We w i l l show t h a t f o r x w ith in a c e r t a i n i n t e r v a l t h i s sequence has a
7
l i m i t a s n -» od , which s a t i s f i e s ( I ) .
From (4 ) i t fo llo w s :
(5 )
I y,00 - > j =
\£f(*,y.)dK I
\
6 / V Zx-X6/
Hence i f we choose a c o n s t a n t s sm all enough, th en
(6)
Ix-X9UcXj ly.W-yj <M«
d e f in e s a new re g io n R* ab o u t (X0 Jy0 ) w hich w i l l be c o n ta in e d in R.
By in d u c tio n we th e n o b ta in from (4 )
( 7)
I
y > <xJ - : t i * 1 * 1/ ( W - u V M x ^ M / * - * „ K / S
so t h a t yn (x ) w i l l ro n a in i n R* f o r
jx - X0 J^1o(.
To prove lira yn (x ) r y (x ) f o r a l l x i n
(3 )
I
|x - X0 I
, o b serv e t h a t
-RwId x i f a / y , W -W dx
y»<*>->(y I
I^KMh-Xotdx
=
S im ila r ly
iy ^ )- ^ l ^
(? )
S in ce:
do)
I
- y ./ i |
y
.
+
- y ^/ +
+ ly ,w -y.[
we e v id e n tly have
(U )
I y j o - I J i I M K ^ 1T r 1*
k*-i
8
C o n seq u en tly , t h e sequence
( 12 )
y„ (x) » y 04 ^
(y kM
J
converges a s n -* oo , and I t s convergence i s uniform f o r a l l x in
|x - X0I ^ s in c e i t i s m a jo riz e d by th e e x p o n e n tia l s e r i e s .
Jj™,
( 13 )
=
y Cr)
and y (x ) c l e a r l y i s a s o lu tio n o f ( I ) o v e r th e i n t e r v a l
y (x o ) - y 0 .
2.
Thus we have
k - x jz < w ith
The u n iq u e n e ss o f y (x ) i s proved by s ta n d a rd m ethods.
E r r o r d e te r m in a tio n .
L et u s assume th e e x a c t s o lu tio n o f y ’ -
f ( x ,y ) i s y (x ) and l e t y k (x ) = T (x ) be an approxim ate s o lu tio n o b ta in e d
by
K
a p p lic a tio n s ( s te p s ) o f P i c a r d 's M ethod.
Both y (x ) and Y (x) s a t i s f y
th e i n i t i a l c o n d itio n s y (x Q) - yQ> I (X 0 ) - y 0 » however,
(14)
Ym ) 4-Afx.Ycv;)
where A (x,Y (x)) i s th e e r r o r te rm , in tro d u c e d by th e f a c t t h a t Y(x) i s
an approxim ate s o lu tio n .
L et u s a p p ly P ic a r d ’ s method o f s u c c e s s iv e a p p ro x im a tio n s t o th e
d if f e r e n c e y (x ) - Y (x ).
( 15)
y,
Then
[ f ( x , y e) - f a y . ) -A fa y o)Jd* -
S in ce A(x,Y) i s bounded in B,
-J ^ A fa y 0) dy
!a ( x ,Y)| 4 N , we o b ta in
(16) Iy,^ - Y(*)l ~£/A(y,y.)ldx < /s/ |x-v»l
9
S im ila r ly
(17)
yjf)
I
- Y
( O
) I
Y)fa IA(Yl Y)IJdy
^ Jl*[kly,(*>-Y I + I M o.YnJfa
and
(is)
I W -Yf*) ( < Kh^zV
-+•••• -f a/ (x-xej
T aking l i m i t s we fin d
| 6 1 - Iy
(1 9 )
w here
€ Is
(0 )- Y C o) I -
J i m Iy„f»)
- Y< K )
I<
N e * 1* '* " 1
th e e r r o r i n th e k - t h ap p ro x im atio n yk (x ) - Y (x ).
T h e re fo re ,
i f we w ish t o d e te rm in e th e "maximum e r r o r " * due to k a p p lic a tio n s o f
P i c a r d 's Method o v e r an i n t e r v a l [x - X0JXoc , we must f i r s t f in d an u p p er
bound N f o r A (x ,y ) i n R*, and th e n s o lv e (1 9 ) f o r € f o r th e i n t e r v a l i n
q u e s tio n .
The v a lu e o f N, o f c o u rs e , depends upon k , th e number o f s te p s
u sed in d e te rm in in g yk (x ) from ( 4) .
I f we w ish t o o b ta in an approxim ate s o lu tio n o f a c e r t a i n a cc u ra cy ,
we can d e c re a s e N by in c r e a s in g k , o r , more e a s i l y , red u ce t h e i n t e r v a l
J x - X 0J t o t h e e x te n t n e c e s s a ry .
3 . Example.
The fo llo w in g w i l l i l l u s t r a t e P i c a r d 's Method o f
*The term "maximum e r r o r " i s used h e re and t h e r e a f t e r i n th e sen se o f an
u p p e r bound f o r th e e r r o r .
10
s u c c e s s iv e a p p ro x im a tio n s .
C o n sid e r t h e e q u a tio n
y '-y + x
( 20)
which we propose t o so lv e u n d e r th e i n i t i a l c o n d itio n y (0 ) - 0 ,
C le a r ly ,
f ( x »7) ~ y + x s a t i s f i e s c o n d itio n s ( a ) , ( b ) , and (c ) in th e e n t i r e x y p la n e .
L et u s ta k e a re g io n R Q z,-2 , 1 , —I) j th e n K — 3 i s th e maximum v alu e
o f f ( x ,y ) i n R.
From (6 ) i t fo llo w s t h a t x must be r e s t r i c t e d t o th e
i n t e r v a l |x / - l / 3 in o rd e r t h a t y rem ain w ith in R*
Thus t h e new re g io n
R* i s d e fin e d a s R* [ l / 3 , - l / 3 , l , - l j .
S o lv in g th e e q u a tio n s (4 ) s u c c e s s iv e ly , we o b ta in
w hence, by ta k in g l i m i t s we fin d
( 22)
y lt)
-
J j ” y.(*J
=
Zf- ' -X
T h is i s th e e x a c t s o lu tio n o f (20) s a t i s f y i n g y (0 ) = 0 .
F or th e e r r o r ap p ro x im atio n a f t e r two s te p s (k = 2) we c a lc u la te
(2 3 )
and
(2 4 )
,x ) [ = ( - £ [ <
M
11
Then, s in c e K = |f y ( x , y ) | -
\e[6
(2 5 )
I in R (and R *), we f in d
^
-zo.oo/13
Thus, th e maximum e r r o r o b ta in e d in t h i s c ase i s I 0.00163 w h ile
t h e a c tu a l e r r o r i n ( 23) i s o n ly - 0 . 00005.
4.
E x te n sio n s o f th e method w ith in R*.
I f we w ish to e v a lu a te
y ( x ) a t s e v e r a l p o in ts w ith in th e i n t e r v a l o f convergence, two methods a r e
a v a il a b le :
(a)
U sing yk(x ), th e k - th ap p ro x im atio n t o y ( x ) , d e te rm in e i t s
n u m e ric a l v a lu e f o r x ^ , Xg
t o in c r e a s e in
(b )
....
w ith consequent in c r e a s e i n e r r o r due
fx - x0 | .
D eterm ine a new yk ( x ) , u s in g th e ap p ro x im ate v a lu e 7k ( ^ ) a s
in i« ia l re n d itio n .
Here we in c o r p o r a te an e r r o r due t o ap p ro x im ate i n i t i a l
v a lu e , b u t red u ce th e e r r o r due t o th e m ethod, a s / x - XjJ i s l e s s th a n
Ix - x ol •
The e f f e c t o f an e r r o r * i n th e i n i t i a l c o n d itio n can b e determ ined
a s fo llo w s :
l e t y (x ) be t h e e x a c t s o lu tio n o f ( I ) w ith y ( x Q) - y Q and l e t
Y (x) be th e e x a c t s o lu tio n o f ( l ) which s a t i s f i e s th e i n i t i a l c o n d itio n
Y( xo)
Z Y0+* •
d if f e r e n c e
(26)
07 a p p ly in g P ic a r d ’ s s u c c e s s iv e ap p ro x im a tio n s t o th e
y (x ) - l ( x ) we o b ta in , a s i n (1 8 ) ,
Iy„<t) -Y.w I
Iel Kk
k.-o
whence, by ta k in g l i m i t s .
K /V-Vo I
( 27)
Iy r x ) ..
y
Yc*A- In** I
H->e6
^
T h e re fo re , t h e maximum e r r o r due to u se o f approxim ate
I n i t i a l c o n d itio n
i s d i r e c t l y p r o p o r tio n a l t o 6 , th e e r r o r i n th e i n i t i a l c o n d itio n .
Sum m arizing, we see t h a t by e x ten d in g P i c a r d 's Method o v e r two
i n t e r v a l s , we have accum ulated t h r e e e r r o r s :
( a ) An e r r o r 6 ^ due t o t h e f i r s t a p p lic a tio n o f P ic a r d ’ s Method w ith
k s te p s and g iv en ( c o r r e c t ) i n i t i a l c o n d itio n :
] 6 , ( S M 1C k l^ * 1
(2 8 )
(b ) An e r r o r ^ due t o second a p p lic a tio n o f P i c a r d 's Method w ith
k s te p s and approxim ate i n i t i a l c o n d itio n :
(c ) An e r r o r 6^ due t o approxim ate i n i t i a l c o n d itio n i n s te p (b)
above:
(30)
K
U
l € , | e K ' <- <' l < K * W
T h is e r r o r in c o r p o r a te s t h e e r r o r d e s c rib e d i n ( a ) .
Hence, combining th e s e e r r o r s , we o b ta in f o r an i n t e r v a l o f le n g th 2h:
Ca)
K U K U K K ZV,elk 4Me**
5.
O p tio n a l p ro c e d u re s t o m inim ize e r r o r s in e x te n s io n s o f d a ta .
13
I n view o f th e above e r r o r ap p ro x im a tio n s we have two p r i n c i p a l methods
o f p ro c e d u re f o r th e c a lc u la tio n o f y (x ) a t s e v e ra l p o in ts w ith in R*:
(a )
The c o n tin u e d u se o f th e i n i t i a l ap p ro x im atio n y%(x) o v er R*.
The consequent need f o r g r e a t e r acc u ra cy i n yk (x ) e v e n tu a lly w i l l
n e c e s s i t a t e more s te p s in th e c a lc u la tio n o f f u r t h e r v a lu e s in o rd e r to
red u ce th e e f f e c t o f a l a r g e r exponent i n (1 9 ) o r ( 2 8 ) .
We can reduce th e
v a lu e o f N, b u t a f t e r m s te p s , th e r e d u c tio n i n N i s n o t a p p re c ia b le ; on
t h e c o n tr a r y , th e e f f e c t o f an in c r e a s e in Jx - X0 I g r e a tly outw eighs any
f u r t h e r re fin e m e n ts o f N.
(b )
The e x te n s io n by new a p p ro x im a tio n s y%(x) w ith i n i t i a l c o n d itio n s
o b ta in e d from y% (x).
In t h i s c a s e , th e exponent in (1 9 ) o r (2 8 ) can be
made t o rem ain c o n s ta n t by u se o f eq u al sp a c in g and Njc can be made very
sm all by in c r e a s in g th e number o f s te p s i n each ap p ro x im a tio n , b u t th e s e
re fin e m e n ts i n o rd e r t o be e f f e c t i v e must outw eigh th e a d d itio n a l e r r o r s
(3 0 ) due to approxim ate i n i t i a l c o n d itio n s .
A n a tu ra l lim ita tio n of t h is
e x te n s io n i s th e e r r o r we d e c id e t o a c c e p t.
However, an ad v an tag e t o t h i s
p ro c e d u re a c c ru e s in t h a t we a re no lo n g e r lim ite d in o u r e x te n s io n s to
th e bounds o f R*, b u t may p ro ceed t o th e bounds o f R.
In a g iv en problem , we must th e r e f o r e weigh th e work in v o lv e d by
e i t h e r method o f p ro c e d u re w ith consequent a c c u ra c y i n o rd e r t o d eterm in e
th e b e s t a p p ro a ch .
I t can be shown t h a t a "most a c c u ra te " o r "best p o s s ib le "
method e x i s t s f o r each problem , where com binations o f ip a c in g s and r e ­
a p p lic a tio n s o f t h e method could be d ev elo p ed i n o rd e r t o ex ten d th e
d a ta o v e r t h e g r e a te s t p o s s ib le i n t e r v a l w ith in a p re v io u s ly d e sig n a te d
14
maximum e r r o r .
6.
Illu s tra tio n .
In o rd e r t o i l l u s t r a t e th e two m ethods o f
p ro c e d u re d is c u s s e d above, we pro p o se t o c a lc u la te y ( 0, 2 ) i f
( 20)
and y ( 0 ) - 0 .
U sing th e f i r s t m ethod, we d i r e c t l y o b ta in
(3 2 )
whence y 2( 0 .2 ) = 0.0 2 1 3 3 .
U sing th e second m ethod, we have to c a lc u la te f i r s t y g fO .l) - 0.00517
whence, a f t e r s u b s t i t u t i o n i n t o ( 20) and subsequent i n t e g r a t i o n , we fin d
( 33 )
( 0 . 1 ) ~ 0 . 0 1 /3 ?
The maximum e r r o r i n (3 2 ) i s found t o be
e r r o r i n (3 3 ) i s
- 0.0 0 1 9 9 .
- 0.00 1 6 3 , th e maximum
Hence, th e maximum e r r o r due t o two a p p lic a tio n s
i s s l i g h t l y l a r g e r th a n t h a t due t o one a p p lic a tio n ; how ever, th e e x ac t
v a lu e o f y ( 0 . 2) i s 0 . 02138, d e m o n stra tin g t h a t by b o th m ethods f a r g r e a te r
t o le r a n c e s have been allo w ed th a n n eed ed .
15
III.
I.
I n tr o d u c tio n .
METHOD OF TAYLOR’S SERIES
The method o f a p p ro x im atio n by T a y lo r’ s S e rie s
i s based on T a y lo r 's s e r i e s expansion o f a s o lu tio n y (x ) o f ( I ) about
x z x0 , and r e s u l t s in approxim ate s o lu tio n s a ti s f y i n g th e i n i t i a l con­
d i t i o n y (x 0 ) s y Q.
S u f f ic ie n t c o n d itio n s which p e rm it u se o f th e method
a r e ( a ) s in g le v a lu e d n e s s o f f ( x ,y ) in R and (b ) e x is te n c e o f a l l d e r i­
v a tiv e s o f f ( x ,y ) in R.
From T a y lo r 's theorem we o b ta in
(34) j / f x ;
=> y ( Vo) + y 'f t y C x - x * ;
where th e d e r iv a tiv e s o f
^
%
a r e c a lc u la te d a c c o rd in g t o ( I ) and
i s th e rem ain d er te rm .
0 < <9 < /
2.
E r r o r e s tim a te s .
The maximum e r r o r in c u rre d by u s in g only th e
f i r s t n f l te rm s i n (34) can be d eterm in ed from th e maximum v a lu e o f Rfi
o v e r th e i n t e r v a l c o n s id e re d .
The e r r o r due t o approxim ate i n i t i a l c o n d itio n s i s c a lc u la te d in th e
same manner a s fo r P ic a r d ’ s M ethod.
As an i l l u s t r a t i o n l e t u s a g a in c o n s id e r th e e q u a tio n
( 20 )
+X
16
w ith i n i t i a l c o n d itio n y ( 0 ) - 0 .
06)
We have
J /V ,;=
y
=
y //
/
^
y <ti0- - y w --y
J ck40Co) =x /
Thus we o b ta in from (34)
y (x)
(3 7 )
57
J 7 ..................
and y ( O .l) - 0 .0 0 5 1 7 . S im ila r ly ,
(3 8 )
J 1(Oi)
-
o-/ o n 7> JC 0-O*
'o f'? , ....... y * (° -0 '
d'
ooS''7 + o. 10^n(X-OJ) +■~ y
Z /o r/^
44 -
whence
(3 9 )
y (*)
(x-o-0 -J- Jotr'I^x -o.J-j.
and y ( 0 . 2 ) - 0 . 02139.
T o c a lc u la te th e e r r o r i n (37) we o b se rv e t h a t y ( ^ v )( x )< 1 .5
o v e r th e i n t e r v a l O ^ x ^ .0 .1 .
Hence th e maximum e r r o r i n (3 7 ) i s l e s s
th a n ±0 . 00001.
The maximum e r r o r i n (3 9 ) i s found by adding t h e e r r o r due t o th e
approxim ate v a lu e o f y ( O .l) t o th e e r r o r due t o th e u se o f f o u r term s o f
T a y lo r’ s s e r i e s i n (3 9 ) .
T h is maximum e r r o r tu r n s o u t t o be ±0.000035,
and i s th u s s l i g h t l y l a r g e r th a n th e e r r o r t h a t would have been o b ta in e d
by c a lc u la tin g y ( 0 . 2) d i r e c t l y from ( 37) .
3.
M odified method o f T a y lo r’ s s e r i e s .
I n th e s ta n d a rd method o f
a p p ro x im atio n by T a y lo r’ s s e r i e s , we o b ta in a s e r i e s f o r y ( x ) , which i s
17
e v a lu a te d a t x - x ^ .
T h is v a lu e y (x ^ ) -
i s used a s i n i t i a l v a lu e in
d e te rm in in g y (x g ) by r e a p p lic a tio n o f th e method:
o r , i n g e n e r a l:
(u )
JZivi ^ y ( X )
J
Afao
Thus, we can approxim ate t h e v a lu e s y ^ , y 2 , . . . .y m w ith in th e i n t e r v a l
o f convergence i n R.
T h is p ro c e s s , how ever, r e q u ir e s e v a lu a tio n o f n
d e r i v a t i v e s f o r each a p p lic a tio n o f th e m ethod, an o p e ra tio n which i s
som etim es q u ite c o m p lic a te d .
As an a p p ro x im atio n t o t h e s ta n d a rd p ro c e d u re , a p o s s ib le a l t e r n a t i v e
i s to d iffe re n tia te
>
(4 2 )
J (X )= J ^
Ar*o
W
( x - x ,/
(X ,)
frf
s u c c e s s iv e ly and c a lc u la te th e d e r iv a tiv e s y ^ x ^ ) , y " ( x g ) , . ..y ^ n ^(x2)
from th e r e s u l t i n g e x p re s s io n s .
( 43)
y o; ^
J
k-o
Then a n o th e r ap p ro x im atio n can be s e t up
t ( X ) cj^ t j
•
w hich may be t r e a t e d s i m i l a r l y t o o b ta in y (x ^ ) and y '( x ^ ) , y #(x ^ ) ,
. . . y (n )(x3) .
P ro ceed in g t h i s way, we can approxim ate y (x ^ ),...y (x ^ ).
To i l l u s t r a t e t h e d if f e r e n c e betw een th e two methods o f a t t a c k , l e t
u s a g a in c o n s id e r th e e q u a tio n
(20)
18
Above we found y ( 0 .2 ) ^0.02139 I f y (0 ) - 0 .
Here we u se a s ap p ro x im atio n
(3 7 ) ,
which we d i f f e r e n t i a t e t o o b ta in
y'o-0^ o. /osr/y
y% (J
r
t./osrt?
J tu(O J ) V I . (O S ’
O J ) - ''
( (
yY tu(iV'ij
y * (O.t)
—/
Then from (4 3 ) we f in d
(4 6 )
y (* ) -O-OosrO +o./osrr?( x-~o*() +
............
whence y ( 0 . 2 ) = 0 . 02139, which i s e x a c tly t h e v a lu e c a lc u la te d b e fo re .
C le a r ly , t h e second method o f p ro c e d u re i s to be p r e f e r r e d i f th e
c a lc u la tio n o f t h e s u c c e s s iv e d e r iv a tiv e s becomes ex tre m ely d i f f i c u l t .
19
IV . DIFFERENCE METHODS
1 . I n tr o d u c ti o n .
In t h e d if f e r e n c e m ethods we a r e p r im a r ily concern­
ed w ith s te p - b y - s te p ap p ro x im a tio n s o f t h e n u m e ric al v a lu e s o f y (x ) o f
( I ) w ith o u t d e r iv in g an a n a l y t i c e x p re ss io n f o r y ( x ) .
A lthough th e p re ­
v io u s ly d e sc rib e d m ethods a r e s u ita b le f o r c a lc u la tio n o f approxim ate
v a lu e s o f y (x ) w ith in th e i n t e r v a l o f co n v erg en ce, g r e a t e r a c c u ra cy i s
u s u a lly o b ta in e d by th e d if f e r e n c e m ethods.
We s h a l l c o n s id e r tw o, th e
Adams Method and th e R unge-K utta Method, to g e th e r w ith t h e i r a l l i e d d i f f ­
e re n c e form ulae (S im p so n 's R u le, M iln e 's R u le , e t c . ) .
F or th e d e r iv a tio n o f th e d if f e r e n c e methods a s u f f i c i e n t c o n d itio n
i s t h a t f ( x ,y ) be o f c la s s Cn i n R.
For a c t u a l c a lc u la tio n s i t i s only
n e c e s s a ry t o have ta b u la te d d a ta .
Th". d if f e r e n c e methods a r e e x tre m ely u s e f u l when g r e a t d i f f i c u l t y i s
in v o lv e d i n c a r ry in g o u t th e re q u ire d o p e ra tio n s in P i c a r d 's Method o r in
th e method o f T a y lo r 's s e r i e s .
The p r i n c i p a l d isa d v a n ta g e t o th e Adams
Method i s t h a t s e v e r a l v a lu e s o f ( x ,y ) must be known i n o rd e r t o secu re
a c c u r a te e x tr a p o la tio n s o r i n t e r p o l a t i o n s .
To overcome t h i s d is a d v a n ta g e ,
i t i s n e c e s s a ry t o employ o th e r methods a s " s t a r t e r " m ethods and th e n t o
c o r r e c t th e s e ap p ro x im a tio n s by d if f e r e n c e fo rm u lae .
Once s u f f i c i e n t d a ta
a r e a v a il a b le , th e d if f e r e n c e methods g e n e r a lly f u r n is h more a c c u ra te
a p p ro x im a tio n s o v er a l a r g e r i n t e r v a l .
On th e o th e r hand, R unge-K utta
Method r e q u ir e s o n ly one i n i t i a l c o n d itio n , b u t th e n e c e s s a ry c a lc u la tio n s
a r e somewhat more in v o lv e d .
2.
Adams M ethod.
In th e d e r iv a tio n o f th e Adams Method we approxim ate
20
y (x ) in ( l ) a t th e p o in t (Xn^tv Zn4h) by two o r more te rm s o f a T a y lo r 's
s e r i e s ab o u t ( Xyv Z^)
( 47)
( X ) t - A / ( X j / . ........
D enoting y ( ^ +h) by Tfi4 l, T(Xfi) by yfi, y ' (Xfi) by f n , r t c . ,
we o b ta in f o r t h e f i r s t two te rm s
(W )
JU
-x
/„ =
^
s
A /;
or
(4 9 )
which r e p r e s e n ts t h e a v era, e slo p e from yn t o Zn^i*
S im ila r ly , we o b ta in f o r t h e f i r s t t h r e e term s o f th e T a y lo r 's s e r ie s
(so )
Here
)l
-y ,
s
A /. f 4 7 ;
- y"(% ^) which can be f u r t h e r e x p re sse d a s
th e a v erag e r a t e o f change o f slo p e f r a n yn«x to Yn .
The n u m erato r i s
d e fin e d a s t h e " f i r s t backward d if f e r e n c e " o r Vfn ,
U sing t h i s nom em clature, we o b ta in t h e form ula o f f i r s t d if f e r e n c e s :
( 52) ) U
-y „
= A /„ + J-A
J
S im ila r ly we o b ta in th e form ula o f f o u r th d if f e r e n c e s :
(53) X , , - X s A ^ ^ i A y / ,
vX
A 9%
21
T h is method r e q u ir e s knowledge o f th e v a lu e s o f n+1 p o i n t s , where
n i s t h e o rd e r o f d if f e r e n c e s d e s ir e d .
In s im ila r manner we can d e riv e th e fo u r b a s ic form ulae le a d in g t o
t h e s ta n d a rd d if f e r e n c e fo rm u la e , a s in d ic a te d below:
( 54)
y .+ ,
( 55)
>
( 56)
y*-i
(57)
y>w
=
=/>,-3
ft,
- I v^
f t*
+^(f* ~ £ VA
~
^
Ti? ^
-
^ ^
"f
~^Ttov6
+ /w o
-/^vo
^^
E r r o r s a t t r i b u t e d t o th e u se o f th e d if f e r e n c e form ulae can be approx­
im ated by th e f i r s t n e g le c te d te rm .
The e r r o r ap p ro x im atio n w i l l be p a r t i ­
c u l a r l y a c c u ra te i f th e d if f e r e n c e term im m ed iately p re c ed in g t h e f i r s t
n e g le c te d term h as zero c o e f f i c i e n t .
Even w ith o u t knowledge o f t h e d i f f ­
e re n c e term used f o r t h e e r r o r a p p ro x im a tio n , we can c lo s e ly e s tim a te th e
e r r o r by e v a lu a tio n o f th e co rre sp o n d in g d e r i v a t i v e o f y a t Xn .
C o rre c tn e ss
o f t h i s e s tim a te i s based on t h e r e l a t i o n s h i p between d if f e r e n c e s and
d e r i v a t i v e s when th e sp ac in g h i s s m a ll.
I t can b e shown t h a t th e n th
d e r i v a t i v e e v a lu a te d a t a p o in t i s th e l i m i t o f th e q u o tie n t o f t h e n th
d if f e r e n c e o f th e fu n c tio n by th e n th power o f h when h te n d s t o z e r o .*
Exam ination o f (5 4 ) th ro u g h (5 7 ) r e v e a ls t h a t
(a )
( 54) a lo n e i s th e "backward d if f e r e n c e " form ula a s d e riv e d from
*C ours d ' A nalyse I n f i n i t e s i m a l e . 6h. J . de l a V a lle P o u is s in
22
Adams M ethod.
U sing f o u r th d if f e r e n c e s o n ly , th e e r r o r ap p ro x im atio n
i f (4 7 5 /l4 4 0 )h V^fn o r (475/l440) h ^ y ^ ( x ^ ) .
E q u atio n (54) may a ls o be
w ritte n
(M )
> ,
= >
+l
V1 < ^A
V 7 lZx-J
where second d if f e r e n c e s a r e u s e d , th e e r r o r b ein g th e t h i r d d if f e r e n c e
c o n v e rte d t o d e r iv a tiv e form a s d e sc rib e d above.
T h is form ula i s used
in o n e -s te p e x tr a p o la tio n from yn t o yn+]_.
(b ) adding (5 4 ) to (5 5 ) prod u ces:
(5 9 ) 7»'«» = X-#
3
Absence o f th e f i r s t d if f e r e n c e le a d s to th e re a so n a b ly a c c u r a te "p re ­
d i c t o r " form ula:
( 60)
-2 V ^
+
3
" 'C y ' )
A nother u s e f u l e x tr a p o la tio n form ula can be d e riv e d from (5 9 ) by u sin g
o n ly t h i r d o rd e r d if f e r e n c e s :
+ £ -(7-f* -Zfh-,
( 61)
( c)
(6 2 )
^ tC - J
!> 1
Adding ( 5 4 ) , ( 5 5 ) , (5 6 ) , and ( 5 7 ) , we o b ta in
^ 4
- Y #
P%
Y
from which we d e r iv e M iln e 's R ule:
(°3) j t f / =■ jt-3 Tt ^ (zf» -X v Y-2^ C j
y ^rJ
23
(d )
Sim pson’ s R ule r e s u l t s from th e a d d itio n o f (5 5 ) and (56)
+X
- A
p ro d u cin g :
(65 )
—X'1-
6C ^
y
Absence o f t h i r d d if f e r e n c e s makes Sim pson’ s R ule an a c c u r a te c o r r e c to r
o f p re v io u s ly o b ta in e d a p p ro x im a tio n s.
I t can a ls o be u sed a s an i n t e r ­
p o la tio n form ula f o r f n _^.
(e )
(6 6 )
Adding (5 6 ) and (5 7 ) , we o b ta in
= % ., v I
/- / » f 3
^ / / < t ■/ J y y r j - / , - J
which can be u sed a s a c o r r e c t o r .
(f)
Adding th e c o rre sp o n d in g d if f e r e n c e form ula f o r Fn+2 to (5 4 ),
we o b ta in
(6 7 )
=
('if, -i° L *
w hich can be used a s a tw o -s te p p r e d i c t o r .
/ V
" N
I t i s p a r tic u la r ly u sefu l
i f th e v a lu e o f th e f o u r th d e r iv a tiv e i s s m a ll.
We may d eterm in e t h e maximum e r r o r due t o r e a p p lic a tio n o f d if f e r e n c e
fo rm u lae by u s e o f methods p re v io u s ly d e s c r ib e d .
S in c e , i n th e s e fo rm u lae,
we combine s e v e r a l p re v io u s ly determ in ed v a lu e s o f y in o rd e r t o e x tr a p o la te
f u r t h e r v a lu e s o f y , t h i s h a s a " b rid g in g " e f f e c t and th e re b y red u ces th e
a c t u a l e r r o r c o n s id e ra b ly .
To i l l u s t r a t e Adams M ethod, we a g a in c o n s id e r th e e q u a tio n
)
24
J t-z. y
(2 0 )
w ith i n i t i a l c o n d itio n s y (0 ) * 0 and y ^ ( O .l) - 0.00517.
U sing (5 2 ) ,
we o b ta in y ( 0 .2 ) = 0 .0 2 0 9 5 , w ith a n e r r o r o f - 0.00 0 4 6 .
U sing two s te p s o f P ic a r d ’ s ap p ro x im atio n we o b ta in e d y ( 0 .2 ) %
0 .0 2 1 3 3 , which i s c o n s id e ra b ly more a c c u r a te .
S ta r t in g w ith c o r r e c t v a lu e s f o r t h r e e p o in ts , we may ap p ro x im ate
th e f o u r th by ( 6 1 ) .
- 0 .0 0 0 0 4 .
We o b ta in y ( 0 .3 ) s 0 .0 4 9 8 1 , w ith an e r r o r o f
The a c c u ra cy i s c o n s id e ra b ly g r e a t e r th a n t h a t o b ta in e d by
u s e o f f i r s t d if f e r e n c e form ula (5 2 ) .
E x tr a p o la tin g by M iln e ’ s R ule ( 63) t o y ( 0 .4 ) by u se o f t h e above
d a ta , we o b ta in y ( 0 .4 ) s 0 .0 9 1 8 1 , w ith an e r r o r o f - 0.000004, p lu s
e r r o r due t o e r r o r i n one i n i t i a l v a lu e .
I f we u se c o r r e c t i n i t i a l
v a lu e s , we o b ta in y ( 0 .4 ) = 0 .091 8 2 , n o t an a p p re c ia b le im provem ent.
The c o r r e c t v a lu e i s 0 .09182.
I f we u se v a lu e s o f y ( 0 . l ) , y ( 0 .2 ) , and y ( 0 .3 ) o b ta in e d by two
s te p s o f P ic a r d ’ s M ethod, th e approxim ate v alu e o f y ( 0 .4 ) by M iln e 's Rule
i s 0 .09173, w hereas by P i c a r d 's Method i t i s o n ly 0 .0 9 0 6 ? .
T h is serv es
t o em phasize t h e g r e a t e r a c c u ra cy o f t h e d if f e r e n c e fo rm u la e .
25
4*
M lK e -K u tta M ethod,
The R unge-K utta Method, based on T a y lo r 's
s e r i e s , b u t r e q u ir in g no e v a lu a tio n o f d e r i v a t i v e s , has t h e advantage
o i r e q u ir in g o n ly one s e t o f i n i t i a l v a lu e s f o r ( x ,y ) i n a d d itio n to ( I ) .
F u rth erm o re , i n th e absence o f an a n a l y t i c e x p re ssio n f o r y '( x ) , s o lu tio n s
may be o b ta in e d by u se o f ta b u la te d d a ta .
Formulae employed in th e method a r e :
(a )
(6 8 )
Second o rd e r a c c u ra c y :
JU
-= X
-f -4- (+(*>»!/»)+
+
+
J
(b ) T h ird o rd e r a c c u ra c y :
^ y*-+ i
( 69)
^
^
y>«+i ai)
a5r A A ,
(c )
F o u rth o rd e r a c c u ra q r:
(7 0 ) 7 h+< '
7« + £ ( A1 + 2 Ai -f
A, = AA(xM,y h)
L = A f C x ^ iA 1y . + ^ A J
A,=• Affxw+y , y„+-iAv)
Ay - A f f xh +
, y.
+ Ai J
E r r o r s due t o th e method a r e n o rm ally computed by u s e o f a com parison
te s t:
l e t th e e r r o r a s s o c ia te d w ith u s in g a p ro ced u re o f n th o rd e r a ccu racy
k tim e s w ith sp ac in g h be ex p re ssed in th e form K khn41.
i t may be assumed
t h a t t h e c o r r e c t r e s u l t i s o b ta in e d by ad d in g t h i s e r r o r t o th e c a lc u la te d
r e s u lt,.
F u r th e r , assum ing t h a t K i s n o t s tro n g ly dependent on th e o th e r
26
v a r i a b l e s , we can compare t h i s e r r o r
w ith t h a t o b ta in e d b y d o u b lin g
th e spr c in e h o v e r th e i n t e r v a l in q u e s tio n ,
Si
Kkhnt'
*
_ _/_
< 71)
I f we s u b tr a c t th e o r d in a te (jr
(2 )
y) , o b ta in e d by d o u b le sp a c in g ,
from t h a t o b ta in e d by s in g le sp ac in g (y
(72)
)
f - f - ' Kkhntl(^ -I)
w hich i s (Sn- I ) tim e s th e e r r o r in
m ate th e e r r o r i n y W
(7 3 )
T h e re fo re , we can a p p ro x i­
by
/■
y 0 ,- y ^
6i = ^ T /
and add t h l a e r r o r t o
t o o b ta in a c o r r e c te d v a lu e .
E r r o r s d u e t o r e a p p lic a tio n o f th e method a r e e v a lu a te d I n th e
same manner a s f o r o th e r m ethods.
To i l l u s t r a t e t h e H unge-K utta M ethod, l e t u s a g a in c o n s id e r
(2 0 )
y ' ^ y + *
w ith i n i t i a l c o n d itio n y (0 ) - 0 .
U sing t h i r d o rd e r a c c u ra c y , we o b ta in
(74)
y ( o . f ) - ~ ( o . o i + o . o / / ) ■= 0 . o o t r t y
The c o r r e c t v a lu e i s y ( 0 .1 ) * 0.00517*
S im ila r ly ,
27
(75)
y (o.i) ^ O.O-2-/V0
The e r r o r i s found t o be
€ ( 0 .2 )
< 0 .0 0 0 0 0 5 .
(o.ov9?sr)
(76)
y (0.9)= 0.09191
w here
(O.OT-/YO)
<£(0.4) < 0.000005.
(0.091*?-)
L ik ew ise,
28
V.
ANALYSIS OF METHODS
Given a d i f f e r e n t i a l e q u a tio n o f f i r s t o rd e r w ith i n i t i a l c o n d itio n
^r(X0 ) - y0 » th e f i r s t problem i s t o d e te rm in e th e re g io n R* i n which we
can e x p e c t t o f in d a s o lu tio n .
E xam ination o f f ( x ,y ) t o d e term in e i t s
l n t e g r a b i l i t y and d i f f e r e n t i a b i l i t y w i l l d is c l o s e th e f e a s i b i l i t y o f
a p p ly in g e i t h e r P i c a r d 's Method o r th e Method o f T a y lo r 's s e r i e s .
Should
f ( x ,y ) be o f a ty p e d i f f i c u l t t o i n t e g r a t e o r d i f f e r e n t i a t e , we may tu rn
im m ed iately t o th e R unge-K utta Method e i t h e r co m o letely o r lo n g enough
f o r e v a lu a tio n o f s u f f i c i e n t ap p ro x im a tio n s o f yCx^) in o rd e r to make
p o s s ib le a p p lic a tio n o f d if f e r e n c e fo rm u lae.
I f we d e c id e t o a p p ly P i c a r d 's Method o r to c o n s tr u c t a T a y lo r 's s e r i e s ,
th e a c c u ra c y d e s ir e d in th e l a s t e x tr a p o la tio n w i l l d eterm in e th e number
o f a p p lic a tio n s o f th e m ethod, each in v o lv in g a c e r ta in number o f s te p s .
I f more th an one a p p lic a tio n i s r e q u ir e d , th e spacing m ust be d e term in e d .
H ere, we must re a c h a compromise between la r g e sp acin g w ith f a i r l y r a p id ,
b u t more in a c c u r a te r e s u l t s , and v e ry sm all sp acin g w ith slo w e r, more
a c c u r a te r e s u l t s i f th e e r r o r s due t o approxim ated i n i t i a l c o n d itio n s do
n o t c o m p letely o f f s e t th e a d v an ta g e s o f sm all sp a c in g s.
In th e ev en t we a r e u n a b le to s e c u re th e a ccu racy d e s ir e d , we must
abandon th e s e methods e i t h e r co m p letely o r a f t e r enough ap p ro x im atio n s
(w ith in a c c u ra c y l i m i t s ) to ap p ly d if f e r e n c e fo rm u lae.
F or each r e a p p lic a tio n o f P i c a r d 's o r T a y lo r 's m ethods, we may s e le c t
a new K in th e L ip s c h itz c o n d itio n to be s a t i s f i e d by f ( x , y ) , th e re b y
29
d e c re a s in g th e maximum e r r o r e s tim a te f o r a l l I n t e r v a l s e x ce p t p o s s ib ly
th o s e In which K i s n e a r i t s u p p er bound o v e r th e e n t i r e r e g io n .
Once we have determ ined two o r more p o in ts ( x ,y ) beyond ( x0>y 0 ) ,
we can b e g in t o apply t h e d if f e r e n c e fo rm u lae w ith p ro p e r o rd e r a c c u ra c y ,
such a s (5 8 ) w ith i t s r e l a t i v e l y h ig h a c c u ra c y , (59) and (6 7 ) which p o s se ss
f a i r a c c u ra c y w ith a d d itio n a l advantage o f s e rv in g a s two s te p p r e d ic to r s .
W ith a t o t a l o f fo u r p o in ts ( x ,y ) in c lu d in g (x 0 ,y 0 ) , we may a p p ly
M iln e 's R ule w ith i t s h ig h degree o f a c c u ra c y a s a o n e -s te p p r e d ic to r ,
fo llo w e d by S im pson's R ule a s a c o r r e c t o r .
T h is p ro ced u re i s c a r r ie d o u t t o
th e f i n a l p o in t ( x ,y ) i f t h e d e s ir e d a c c u ra c y i s a t t a i n a b l e .
Should th e d e s ir e d a c c u ra c y a t (%t»y%) n o t be r e a liz e d by th e d i f f ­
e re n c e fo rm u lae , i t w i l l become n e c e s s a ry t o ap p ly th e R im ge-K utta Method.
Here we m ust d e te rm in e th e l a r g e s t sp a c in g c o n s is te n t w ith th e d e s ir e d
a c c u ra c y .
A fte r each s te p o f th e m ethod, we w i l l d e term in e th e e r r o r term
a s p re v io u s ly d e s c r ib e d , adding t h i s t o t h e approxim ated v a lu e s , and con­
t in u in g t o th e n e x t p o in t.
I f an a n a l y t i c e x p re ss io n i s d e s ir e d t o approxim ate t h e s o lu tio n
betw een (x 0 ,y o ) and (x^ yn )> t h i s may b e o b ta in e d by u se o f N ew ton's
i n t e r p o l a t i o n form ula:
( 77)
y(*) c a 0+ a, Cx-XmJ 4 av(
x
J - *.........
In g e n e r a l, c a r e f u l p re lim in a ry a n a ly s is o f e r r o r te rm s , sp a c in g s,
and t h e in h e r e n t c h a r a c t e r i s t i c s o f each method w i l l le a d m ost e f f i c i e n t l y
t o r e s u l t s w ith in th e d e s ir e d deg ree o f a c c u ra c y .
30
V I.
COMPARISON OF METHODS BY ILLUSTRATIVE PROBLEM
L et ua c o n s id e r th e f i r s t o rd e r n o n - lin e a r d i f f e r e n t i a l eq u atio n
(7 8 )
\f
w ith i n i t i a l c o n d itio n y (0 ) = 0 .
We w ish t o d eterm in e y ( 0 .4 ) and y ( 0 .8 )
c o r r e c t t o th r e e decim al p la c e s .
In o u r p re lim in a ry a n a ly s is we see t h a t f ( x ,y ) = (y + x ) 2 i s co n tin u o u s
th ro u g h o u t th e x y -p la n e , t h e r e f o r e we may s e l e c t any R a b o u t ( 0 , 0 ) .
To
s e c u re a minimum e r r o r by a n a l y t i c m ethods, we s e le c t an R* w ith a sm all
M which w i l l c o n ta in a t l e a s t one o f t h e d e s ir e d p o in ts , o r we s h a ll be
faced w ith more in v o lv e d m ethods o f s u c c e s s iv e a p p ro x im a tio n s.
S e le c tio n o f a l e a s t u p p e r bound f o r f ( x ,y ) c o n ta in in g y ( 0 .4 ) r e s u l t s
i n a re g io n R * ( 0 .5 ,- 0 .5 ,0 .5 ,- 0 .5 ) i n which M = I and K = 2 .
T h is w ill
produce an e r r o r ap p ro x im atio n a t y ( 0 .4 )
( 79)
e M
; 4
f
By m inim izing N th ro u g h u se o f s e v e ra l s te p s o f P ic a r d ’ s Method we can
t h e o r e t i c a l l y o b ta in t h e d e s ir e d a c c u ra c y .
In p ro ceed in g t o y ( 0 .8 ) we
m ust u se two o r more o v e rla p p in g re g io n s beyond R*, in each o f which we
e v a lu a te a new K.
By P ic a r d ’ s Method t h e fo llo w in g r e s u l t s a re o b ta in e d :
31
S in ce R* d o es n o t c o n ta in y ( 0 . 8 ) , we e s t a b l i s h a new R** c o n ta in in g
R* an d , u s in g P ic a r d ’ s M ethod, we o b ta in f o r y ( 0 .8 ) .
y,(0'i) - o .n o iy
(B i)
% / f . f ) - o. xxT-i-9
The e r r o r in t h i s case i s e x trem ely d i f f i c u l t t o e v a lu a te , b u t i t
-
can be seen t h a t
y (0 » 8 ).
Exact v a lu e s f o r y ( 0 .4 ) and y ( 0 .8 ) a r e 0.02279 and 0 .2 2 9 6 4 , r e s p e c t­
iv e ly .
S ince th e maximum e r r o r f o r y ^ ( 0 .4 ) i s to o la r g e t o m eet th e r e q u ir e ­
m ents and f u r t h e r s te p s i n th e method a r e q u i t e in v o lv e d , we co n tin u e
th e method o n ly to t a b u l a t e d a ta f o r com parison p u rp o se s.
U sing t h e form ulae
we o b ta in r e s u l t s a s shown in T able I .
By u se o f y^(x) i n (82), we ex ten d o u r d a ta from y (0 .4 ) t o y(0.8)
by a r e a p p lic a tio n o f P ic a r d ’ s Method o b ta in in g
(TZ's.ir)
It3xi
m oo
Ji CotS) = o. XUitG
32
TABLE I
COMPARISONS OF VALUES BY PICARD'S METHOD
X
n
0 .0
O
0 .1
y3
y
O
O
O
0.00033
0.00033
0.00033
0.00033
0 .2
0026?
00271
00271
00271
0 .3
00900
00932
00933
00933
0 .4
02133
02279
02279
02279
0 .5
04167
04596
04629
04626
0 .6
07200
08281
08401
08414
0 .7
11423
13805
14177
14229
0 .8
17067
22529
22769
22964
?2
33
By t h e method o f T a y lo r 's s e r i e s we d e riv e
(84)
J A;
-gf’+
= y
7
Whereby, u s in g th e f i r s t th r e e term s
jy
CO‘<f) ■=.<?. 0 2-27^
/f/< -
O'OO/V?
By u s in g f o u r t e r n s .
S in c e t h e l a t t e r e r r o r i s v e ry c lo s e t o maximum to le r a n c e , we proceed t o
y ( 0 .8 ) by d i r e c t c a lc u la tio n from (84) o n ly f o r com parative p u rp o se s.
(37)
a n d , a g a in f o r com parative p u rp o se s, we a p p ly an e x te n s io n o f th e method
o v e r a new i n t e r v a l ab o u t y ( 0 .4 ) , w ith t h e r e s u l t s t h a t :
from which i t can be seen t h a t a o p a r e n tly r e a p p lic a tio n o f t h e method
le a d s t o more a c c u ra te r e s u l t s s in c e th e t r u e v a lu e o f y ( 0 .8 ) i s
0.22964,
For th e purpose o f com parison we o b ta in , by use o f th e m odified
method o f T a y lo r 's s e r i e s ( f o u r te rm s ).
-O.ot-%71
(8 9 )
y (o -t)
^
34
S in ce t h e a n a l y t i c m ethods do n o t y i e l d t h e d e s ir e d r e s u l t s w ith o u t
c o n s id e ra b le d i f f i c u l t y , we proceed t o t h e d if f e r e n c e m ethods.
F irs t,
by M iln e 's ( 6 3 ) , u s in g th e d a ta from P i c a r d 's Method f o r y ( O . l ) , y ( 0 .2 ) ,
y ( 0 . 3 ) , we o b ta in ( s e e T able I I ) »
(90)
J (O -V ) =
y (o . r )
-
The e r r o r
(9 1 )
o.oV6f?f
added to
y(e> ty - o o riq t}
y(o.
=
/e j
o. oooa?
/ €^(
o. OOO /O
€" ^ pro d u ces an e r r o r
/ G3I
o. ooO/f
IefI 6
a
ooo z j -
Vihereby we a r e exceeding th e e r r o r t o le r a n c e .
I
6 0 . OOOlB.
zZ 6 a
IeJl
5
C on tin u in g :
ooo34
a o o o t/
To com plete o u r d a ta ,
(92)
Inasmuch a s th e d e s ir e d acc u ra cy was n o t o b ta in e d by M iln e 's R ule,
we proceed t o t h e R unge-K utta Method.
In s o lv in g th e problem by t h e R unge-K utta Method, we have s e v e ra l
o p tio n s :
(a ) Approach y ( 0 .4 ) and y ( 0 .8 ) by sp a c in g o f 0 .4 , ch eck in g y(O.B)
by a double sp acin g 0 .8 .
(b ) Approach y ( 0 .4 ) and y ( 0 .8 ) by sp ac in g o f 0 .2 , ch eck in g w ith
d o u b le sp ac in g 0 .4 .
( c ) Approach y ( 0 .4 ) and y ( 0 .8 ) by sp a c in g o f 0 .1 , ch eck in g w ith
d o u b le sp ac in g 0 .2 .
TABLE I I
i
DIFFERENCE TABLE — MILNE1S RULE
X
7
0 .0
0 .0
f ( x ,y )
0 .0
A 2f U , y )
A 3i ( x , 7 )
A 4r ( x ,y )
0.01007
0.02095
0.00263
0.00219
A f ( x ,y )
0 .1
00033
01007
03102
02358
00482
00314
0 .2
00271
04109
05460
02840
00796
00559
0 .3
00934
09569
08300
03634
01355
00813
0 .4
02272
17869
11964
04989
02163
01554
0 .5
04618
29833
16953
07157
03722
0 .6
08396
46786
24110
10879
0 .7
14197
70896
34989
0 .8
22895
1.05885
36
(d ) C om binations o f t h e above.
By th e f i r s t method we o b ta in , f o r t h i r d o rd e r a c c u ra c y ,
OZZS^tT
(9 3 )
and f o r f o u r th o rd e r a c c u ra c y ,
( 94)
J/CO Y-J = 0 02 . 2- 7 /
S olving f o r y(O .B ) from y ( 0 .4 ) ,
(9 5 )
y (o .$ )
o .z z q s -v
Checking r e s u l t s by o b ta in in g y ( 0 .8 ) In one step s
(9 6 )
y (0 $)^ 0.2.OY7?
which g iv e s an e r r o r o f 0 .0 0 1 6 5 , which when added t o (9 5 ) , g iv e s y (0 .8 )
a c o rre c te d v a lu e o f 0 .2 3125.
S in ce we have n o t sec u re d th e d e s ir e d a c c u ra c y , we p ro ceed t o th e
second method ( b ) .
(9 7 )
yC o .zj
D e term in atio n o f y ( 0 .2 ) from y (0 ) y i e l d s :
= 0 .0 0 2 -7 /
S im ila r ly , we o b ta in y ( 0 .4 ) from y (0 .2 ) s
( 98)
yC o 'tf-O .o Z Z -rcp
Then, in o r d e r t o c o r r e c t o u r r e s u l t s , we u se th e v a lu e o f y ( 0 .4 ) in (9 0 ) ,
w hich i n d i c a t e s an e r r o r o f l e s s th a n 0 . 0(XX)I.
37
P ro ce e d in g i n a s im ila r m anner,
J (o. (,)
(9 9 )
j y C0' &)
ve
o b ta in
o. o M 'r
—^
3-2.963
and c a lc u la tin g y ( 0 .8 ) from y ( 0 .4 ) a s i n ( 9 5 ) , we o b ta in an e r r o r
l e s s th a n 0.0 0 0 0 1 , whereby o u r f i n a l c o rre c te d v a lu e f o r y ( 0 .8 ) i s
0 . 22968.
T h is method o b v io u sly g iv e s th e d e s ir e d a c c u ra c y .
I n o rd e r t o compare th e a c c u ra c y o b ta in e d by o p tio n ( c ) , we l i s t
t h e fo llo w in g d a ta :
(100)
y
( ° -0 - O .O O 0 3 3
y (O-X)-Ooo*?/
y (0.
y
3) = P
oo93V
(0 -* /) - o . o x x ^ o
y (OS) - O-0963/
y (0-6) -O-Oir 9/5~
y
(o .? J Z. O S V 3 3 /
y ( o - s) - o- 3 2 .f6 3
As a more compact com parison o f t h e r e s u l t s o b ta in e d , T ab le I I I
h a s been p re p a re d .
I n c o n c lu s io n , i t i s d e s ir e d t o p o in t o u t t h a t by v a r i a t i o n s o f
th e above d e s c rib e d m ethods, a s o u tlin e d i n S e c tio n V, we can a ls o
o b ta in s im ila r r e s u l t s w ith th e d e s ir e d a c c u ra c y .
TABLE I I I
COMPARISON OF RESULTS
True y
P ic a r d ’ s
y 3 (x )
T a y lo r’ s
(4 te rm s)
0 .1
0.00033
0.00033
0.00033
0 .2
00271
00271
00271
0 .3
00933
00933
00933
0 .4
02279
02279
02279
0 .5
04626
04629
04626
04618
0 .6
08414
08401
08407
08396
0 .7
14229
14197
0 .8
0.22964
14194
0.22952*
0.22820
X
14177
0.22916*
0.22769
*By two a p p l i c a t i o n s .
M odified*
(4 te rm s)
M iln e ’ s
h s 0 .4
R unge-K utta
h=0.2
h=0.1
0.00033
0.00271
.
00271
00934
0.02279
0.22852
0.02279
0.22895
02271
02279
02280
04631
08415
08415
14231
0.23125
0.22968
0.22968
39
V II. CONCLUSION
In t h i s p a p er we have co n fin ed o u rs e lv e s t o th e b a s ic a p p lic a tio n s
o f t h e a n a l y t i c and d if f e r e n c e methods o f n u m e ric al a p p ro x im a tio n .
How­
e v e r , many o f th e s id e ^ a s p e c ts o f th e s e methods which w ere n o t co n sid e re d
h e re in should f u r n is h ample m a te r ia l f o r a f u r t h e r more th o ro u g h d is c u s s io n .
O th e r s u b je c ts f o r f u r t h e r i n v e s t ig a ti o n a r e :
(a )
A n a ly sis o f "ty p e " e q u a tio n s , w ith th e p o s s i b i l i t y o f develop­
in g m ethods o f a t t a c k m ost fa v o ra b ly s u ite d to each .
T h is m ight e a s ily
ta k e th e form o f a compendium s im ila r t o th e D if f e r e n tia lg le lc h u n g e n ,
Losungsmethoden und Losungen
(b )
by E. Kamke.
A n a ly sis o f e r r o r s in h e re n t t o each m ethod, w ith t h e p o s s i b i l i t y
o f f u r t h e r re d u c in g th e maximum e r r o r by more p r e c is e a n a ly s is o f th e i n ­
flu e n c in g f a c t o r s .
T h is would le a d t o g r e a t e r e x te n s io n s o f d a ta w ith in
p re d e s ig n a te d to l e r a n c e s .
I
40
L i t e r a t u r e C onsulted
H ild e b ra n d , F . B. 1950. ADVANCED CALCULUS FOR ENGINEERS.
594 p p ., P r e n ti c e - H a ll, I n c . , New Y ork.
Kamke, E. 1948. DIFFERENTIALGLEICHUNGEN, LOSUNGSMETHODEN UND
LOSUNGEN.
666 p p ., C helsea P u b l. C o ., New Y ork.
Knopp, K onrad. 1928. TH OKY AND APPLICATION OF INFINITE SERIES.
571 p p ., B la c k ie & Son L im ite d , London and Glasgow.
Levy, H. and B a g g o tt, E. A. 1934. NUMERICAL STUDIES IN DIFFERENTIAL
EQUATIONS. (V o l. I )
238 p p ., W atts & C o ., London.
M iln e , W. E. 1949. NUMERICAL CALCULUS.
393 p p ., P rin c e to n U niv. P r e s s , P r in c e to n , N .J .
W idder, David V. 1947- ADVANCED CALCULUS.
432 p p ., P re n tic e -r-H a ll, I n c . , New Y ork.
V a lle e P o u s s in , C h .-J . de l a . 1946. COURS D*ANALYSE INFINITESIMALS (V ol. I ) .
46O p p ., Dover P u b lic a tio n s , New Y ork.
103G 50
E UNIVERSITY LIBRARIES
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3 1762
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.
.. ' y
g
s
r
AJ ' a /
o
103050