Rational approximations to solution of second order nonlinear Ricatti equation, using linear transformations - differential equations and operational calculu

Fair, W.

Luke, Y. L.

English

NASA Technical Reports Server

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RATIONAL APPROXIMATIONS TO THE SOLUTION 
OF THE SECOND ORDER RICATTI EQUATION 
N 6 6  1 3 1 6 9  
01 
ITHRUI 8 (ACCESSION NUMBER)  
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ICODEI - 
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INASA CR OR TMX OR AD NUMBER) 
TECHNICAL REPORT 
15 January - 14 October 1965 
Task Order NASr-63(07) 
NASA Hq. R&D 80X0108(64) 
MRI Project 2760-P 
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ff653 July65 
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Office of Grants and Research Contracts 
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National Aeronautics and Space Administration 
Washington, D. C. 20546 
IDWEST RE CH INSTITUTE 
425 VOLKER BOULEVARD/KANSAS CITY, MISSOURI 64170/AC 816 L O  1-0202 
https://ntrs.nasa.gov/search.jsp?R=19660003880 2020-03-16T21:34:31+00:00Z
a 
I 
WTIONAL APPROXIMATIONS TO THE 
THE SECOND ORDER RICATTI 
\Qman Fa i r  
Yudell Luke 
\ 
TECHNICAL REPORT 
15 January - 14 October 196s 
a s k  Order NASr-63(07) - 
NASA Hq. R&D 80X0108(64) 
- I -  ,. ~ 
- - I _  ,._I -, 
B i R I  P ro jec t  2760-P 
For 
Office of Grants anl! Research Contracts 
Code SC 
National Aeronautics and Space Administration 
Washington, D.  C .  20546 
MIDWEST RESE CH INSTITUTE 
425 VOLKER BOULEVARD/KANSAS CITY, MISSOURI 64110/AC 816 L O  1-0202 
PREFACE 
This repor t  covers research i n i t i a t e d  by Headquarters, National 
Aeronautics and Space Administration on Contract NASA Hq. R&D 80X0108(64), 
10-74-740-124-08-06-11, PR 10-2487, “Nonlinear Dynamics of Thin S h e l l  
S t ruc tu res . ”  The research work upon which t h i s  repor t  i s  based w a s  accom- 
p l i shed  a t  Midwest Research I n s t i t u t e  with M r .  Howard Wolko as p r o j e c t  
monitor. 
This report  covers work conducted from 15 January 1965 t o  14  October 
1965. 
The authors take t h i s  opportunity t o  thank Mrs. Geraldine Cooinbs, 
Mr. John Nelson and Miss Rosemary Moran f o r  t h e i r  a s s i s t ance  with the  ca lcu la-  
ti ons . 
Approved for  : 
MIDIJEST RESEARCH INSTITUTE 
Sheldon L. Levy, b i r e c t o r  
Mathematics and Physics Division 
25 October 1965 
bc 
TABLE: OF CONTENTS 
Page No. -- 
I. I n t r o d u c t i o n .  . . . . . . . . . . . . . . . . . . . . . . . .  1 
11. Development of t he  Rational Approxipations . . . . . . . . . .  1 
I11 . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . .  5 
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  7 
I. INTRODUCTION 
I n  a previous work Merkes and Sco t t  [l] constructed continued f r a c -  
t i o n  so lu t ions  t o  the  first order R i c a t t i  equation by using a sequence of 
l i n e a r  f r a c t i o n a l  transformations. 
paper by Luke [3], t o  develop main diagonal Pad6 approxinations t o  the  solu- 
t i o n  of the  f i rs t  order R i c a t t i  equation wit.h r a t i o n a l  coe f f i c i en t s .  Rational 
approximations are advantageous t o  study the  behavior of t he  so lu t ions  i n  a 
global sense. T h a t  i s ,  they  a r e  u se fu l  f o r  evaluation of func t iona l  values i n  
t h e  complex plane including zeros and poles.  
F a i r  [2] u t i l i z e d  the  7-method, see the  
I n  t h i s  r epor t  we develop continued f r a c t i o n  (and hence r a t i o n a l )  
approxirations t o  the  so lu t ion  of  a second order nonlinear equation which i n -  
cludes as s p e c i a l  cases the  equations t r e a t e d  i n  [l] and [2]. 
t i o n s  are obtained by using a sequence of l i n e a r  transformations which leave 
t h e  d i f f e r e n t i a l  equation invariant,  see Davis [41, and a r e  presented i n  
Section 11. For a n  app l i ca t ion ,  i n  Section 111, t h e  algorithm i s  appl ied  t o  
obta in  approximations t o  Pairilev6's first and second transcendents.  
These approxima- 
11. DEVELOFMXNT OF THE RATIONAL APPROXIMATIONS 
Consider the generalized second order R i c a t t i  equation 
where the  c o e f f i c i e n t s  i n  (1) have Taylor 's  series expansions of t he  form 
A, = x2 a$< , Bo = x2 bkxk , Co = x ckxk , 
k=O k=O k= 0 
Q c, 03 
D, = x d p l ~  , E, = 21 e p k  eo # o , F, = fkx" f, f o , 
k= 0 k=O k=O 
k= 0 k= 0 
- I -  
We f u r t h e r  assume t h a t  the  solution of (1) has a po7wer s e r i e s  expansion of the 
f o m  
m 
k= 1 
Note t h a t  (2) and (3) together with (1) uniquely de te rn ine  a), and f$ . 
also requi re  that t he  coef f ic ien ts  i n  (3) have the  property t h a t  
We 
.... $P CYo B p . .  
B1 62.. .... . B p + l  
% = I :  : / f o  , p =  0,1,2 . . . .  
and 
- r 2p+l - 
....... P, @p+l 
B, Bpr2 ....... . .  
# 0 . p = 0,1,2 ... 
Then ( 4 )  insures  t h a t  y has a continued f r a c t i o n  expansion of the form 
l + .  
( 4 )  
- 2 -  
A transformation of the type 
where m , n p and q a r e  polynomials i n  x nay be necessary t o  bring the  
d i f f e r e n t i a l  equation i n t o  t h e  required form. We suppose t h a t  t h i s  has a l ready  
been done. 
t o  (1). 
See [41 f o r  t h e  r e s u l t s  of applying transformations of t h i s  tyye 
we give a n  example i n  Section III. 
The even approximants of ( 5 )  a r e  the  main diagonal Pad6 approxima- 
t i o n s  t o  y which have t h e  f o l l o i ~ n g  proper t ies .  L e t  
F, 
11 , = n =  
n 
IC 
k= 0 Pn,kx 
n 
k 
qrl,kx 
k=O 
be the  nth order main diagonal Fade approxination t o  y . If Qn i s  formally 
divided i n t o  
representa t ion  of y for  the f irst  (2n+l) terns. The polynomials Pn and 
Qn 
Pn , the  r e su l t i ng  power s e r i e s  agrees with the  power series 
both s a t i s f y  the  recurrence r e l a t i o n  
(7) P = a o ,  p 1 =  cy 0 (l+Cr2x) , Qo = 1 and Q1 = 1 + (el+Q2)x . 0 
Thus, r a t i o n a l  approxinations t o  t h e  so lu t ion  of (1) a r e  immediately f o r t h -  
coming i f  the  values q,CY2 ... can be computed. These values can be obtained 
by u t i l i z i n g  a sequence of l i nea r  f r a c t i o n a l  transformations. Let 
- 3 -  
Repeated app l i ca t ion  of (8) t o  (1) and d iv i s ion  by an:< at each s t e p  y i e lds  
where 
and 
H n + l  = an -1 x 2 E, , n = 0,1>2, .  . . 
En,1 and Fn+l are defined a t  x = 0 and 
Thus the  values @k appearing in  (5) can be computed recursively.  
- 4 -  
. Proof of the convergence of t he  approxinlants i n  (6 )  i n  t h e  general  
case seems e lus ive  espec ia l ly  since the  values of ak  are not i n  genera l  
known i n  closed form. However, i n  t he  sFec ia l  case of the  first order  R i c a t t i  
equation, some convergence proofs a r e  ava i l ab le ,  see [l] and [ 2 ] .  
one can imply convergence of the  approximants i n  (6 )  by comparing the  values of 
t he  nth and (n+l)s t  approximations. 
t he  cases inves t iga ted  the  ac tua l  e r r o r  of t h e  n th  approximnt  i s  the  same 
order of magnitude as the  difference,  yn+l(x)  - yn(x) . 
Heur i s t i ca l ly  
In  p rac t i ce  t h i s  works very w e l l ,  and i n  
111. EXAMPLES 
We consider two examples which exh ib i t  t he  u t i l i t y  of these  approxi- 
mations vhen used t o  approximate both the  funct ion and i t s  poles. 
Painlev6 ' s  first  and second transcendents a r e  defined by the  d i f f e ren -  
t i a l  equations 
and 
v l l  - zV3 - xv - = o , v ( ~ )  = 1 , v y ~ )  = o , 
respec t ive ly .  I n  what follows, A = Ll, = 1.0 . 
To c a s t  (12) and (13) i n t o  the  required form of (1); we s e t  
i n  which case (12) and (13) become u = 1 + 3x2E and v = 1 + 1.5x2V 
2 - 
3~ a'' + 12xE' + ( 6  - 3 6 ~ * ) E  - 5 4 ~ ~ 3  - (6  + X )  = 0 , ~ ( 0 )  = 1 (14) 
and 
- 5 -  
. 
Now u has a pole of the second order a t  x = 1.2067 and v has a simple 
pole a t  x = 1.1577. This behavior manifests i t se l f  i n  Tables I and I1 below 
i n  which E6 and F6 are the s ixth order main diagonal Pad5 approximations 
t o  U and obtained using the algorithm of Section 11. We have 
TABU I TABU I1 
0.0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1.0 
1.1 
1.0000 
1.0305 
1.1264 
1.3015 
2.0228 
2.7212 
3.8909 
6.0383 
10.6226 
23.3936 
87.7732 
1.5831 
1.0000 
1.0305 
1.1264 
1.3015 
1.5831 
2.0228 
2.7212 
3.8909 
6.0383 
10.6223 
23.3860 
a7 3769 
X 
0.0 
0.1 
0.2 
0.3 
0.4 
0.5 
0.6 
0.7 
0.8 
0.9 
1.0 
1.0000 
1.0152 
1.0626 
1.1464 
1.2742 
1.4592 
1.7254 
2.1184 
2.7369 
3.8344 
6.3110 
1.0000 
1.0152 
1.0626 
1.1464 
1.2742 
1.4592 
1.7254 
2.7369 
2.1184 
3. a343 
G ,3104 
The values of u(x) and v(x)  were taken from a paper by Simon E57 who used 
(12) and (13) as exam--les i n  a study of a numerical i n t eg ra t ion  ttchriique for 
t h e  so lu t ion  of  i n i t i a l  value problems i n  ordinary d i f f e r e n t i a l  equations.  
The poles of si.?allest magnitude of u6; and v6 a r e  1.2058 t i.0134 
and 1.1578, respect ively,  s o  tha t  one would expect t h a t  the poles of u and 
v could be computed t o  any desired degree of accuracy using higher order 
approximations. Indeed t h i s  i s  the  case when the  approximations converge. 
- 6 -  
. 
REFEFUiNCES 
I 
1. Merkes, E. P. ,  and Scot t ,  W. T . ,  "Continued Fraction Solutions of -the 
R i c a t t i  Equation," J. Math. Anal. Appl., 4, 309-327 (1962). - 
2. Fa i r ,  W . ,  "Fad& Approximtions t o  the  Solution of t h e  R i c a t t i  Equation," 
Math. Comp., v. 18, No. 88, 627-634 (1964). 
3. Luke, Y. L. ,  "The €ad6 Table and t h e  7-blethod," J. Math. and Fhys., v. 37, 
110-127 (1958). 
4. Davis, H. T . ,  Introduction t o  Nonlinear D i f f e r e n t i a l  and I n t e g r a l  Equations, 
Dover, New York, Ch. 8 (1962). 
5 .  Simon, W. E . ,  "Numerical Technique f o r  Solution and Error  Estimate for the 
I n i t i a l  Value Problem," Math. Comp., v. 18, No. 91, 387-393 (1965). 
- 7 -  


Rational approximations to the solution of the second order Ricatti equation Technical report, 15 Jan. - 14 Oct. 1965

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