A new class of subharmonic solutions to Duffing's equations

CoA R E P O R T AERO N o . 195

bieLiOibt:tK

;i5SEP. tg^

THE COLLEGE OF AERONAUTICS

CRANFIELD

A NEW CLASS O F SUBHARMONIC SOLUTIONS

TO B U F F I N G ' S EQUATIONS

by

THE COLLEGE OF AERONAUTICS CRANFIELD

A New Class of Subharmonic Solutions to Buffing's Equation

by

P . A. T. Christopher. B . C . A e . . A . F . I . M . A .

SUMMARY

In an e a r l i e r paper, Ref. 1, the author established the existence of exact, pure-subharmonic, solutions of a fairly general type of ordinary, nonlinear, differential equation of second-order. A particular, degenerate, form of that equation is Buffing's equation, without damping, i . e . the t e r m In X is absent. The present study is concerned with this equation and it is shown that there is associated with the pure-subharmonic solution a class of exact subharmonic solutions which may be represented by a F o u r i e r s e r i e s

2

-CONTENTS

Page

Introduction 3 Exact Pure-SubharmonicB 3

A Broader Class of Subharmonic Solutions 5

A Proof of Existence 12 Some Norms and Pseudo-norms 16

Conditions for T to be a Contraction Mapping in S* 19 Estimates for a_, b_ b„, I a, - a, .1 , 27

0 0 a X 1 U

. . . . 1 ^ 3 - ^3jj|

An Estimate for lub((u,v) - ol 36 Application of the Proof of Existence When I g, - 11 44

is Small

Regions of Existence in the g, , g„, r Space 47

Conclusion 52 References 53

1. Introduction

In Ref. 1 the author demonstrated that, for particular values of the coefficients, the differential equation

2 3

X + (b + b X )x + c - x + e x = J Q Sin 3ut, ( 1 . 1 )

where x = dx/dt, p o s s e s s e s an exact, pure-subharmonic, solution

X = A Sin ut + B C o s wt, ( 1 . 2 ) i . e . a subharmonic of o r d e r 1 / 3 . A particular degenerate form of ( 1 . 1 )

a r i s e s when b = b = 0, this being Buffing's equation without a damping t e r m , i . e . the t e r m in x i s absent. F o r this equation two different pure subharm-onic solutions e x i s t , the first in which B = 0 and the second in which B Is linearly related to A. F o r given values of c and c t h e s e pure

subharm-X «5

o n l c s a r e generated only by particular values of Q and it i s an obvious, and potentially interesting, question to ask whether subharmonic solutions are generated by other values of Q. The only known broad c l a s s of subharmonic solutions i s that related to the condition c « c- , in which the solution takes the form of a real F o u r i e r s e r i e s whose leading t e r m i s the dominant

subharmonic. See for example Ref. 2, Chapter 4 . F o r the present problem t h e r e i s reason to b e l i e v e that the subharmonic solutions sought m a y have the s a m e form as in the c a s e c « c. , if they exist at a l l . The present study s e t s out to explore the e x i s t e n c e of such solutions, utilizing, and slightly extending the application of, a combined functional analytic, topological, method originally proposed by C e s a r l in Ref. 3 . Alternatively, Cronln g i v e s an account of the method, but with no e x a m p l e s , in Ref. 4 , pp. 180 - 185.

2 . Exact Pure-Subharmonlcs

The real equation to be considered i s 3

X + c - x + e x = Q Sin ut, c f= 0, ( 2 . 1 ) which upon writing

36 = ut ( 2 . 2 ) r e d u c e s ( 2 . 1 ) to

w h e r e

gj = 9 C j / u •^

S3 - 9^=3/"

r = 9Q/w^

(2.4)

and X* = d^lAG. T h e p r o p o s e d p u r e - s u b h a r m o n i c solution of o r d e r 1/3 m a y be w r i t t e n a s X = a Sin 0 + b C o s 0, e e f r o m which and x " = - a Sin 0 - b C o s 0 e e

x^ = | ( a ^ + b^)(a Sin 0 + b Cos 0) e e e e

(2.5)

(2.6)

+ i { a (3b - a ) Sin 30 - b (3a - b ) C o s 30]

e e e e e e -^ (2.7)

Substituting f o r x , x " and x in ( 2 . 3 ) and equating t h e coefficients of t h e d i s t i n c t t e r m s , i . e . t h o s e containing Sin 0, C o s 0, Sin 30 and C o s 30, r e s p e c t i v e l y , to z e r o , gives r i s e to t h e following r e l a t i o n s :

<gl -^K^ ^*H\K ^ ^e) = °

<ei - 1)^ ^ *g3^e<% ^ ^f) = 0

ig^a^(3b2 - a^) = r

( 2 . 8 ) ( 2 . 9 ) ( 2 . 1 0 ) (2.11)

which a r e t o b e satisfied s i m u l t a n e o u s l y . E q u a t i o n ( 2 . 1 1 ) defines two c a s e s 2 2

a s follows: C a s e (i) b = 0 and C a s e (ii) 3a = b . T h e s o l u t i o n s c o r r e s -e -e -e ponding to t h e s e c a s e s a r e then: C a s e (1). b = 0 e % = 3 r / ( g ^ - 1) = {-4(g^ - l ) / 3 g 3 } * and a^ = 0 ( 2 . 1 2 ) o r a^ = 27Q/(9c^ - J^) = ( - 4 ( 9 C j - u)^)/27c } ^ (2.13)

In order that a shall be real it is clear that (g, - 1) and g„ shall be of e 1 3 opposite sign. Re-writing (2.12) in the form

gl = 1 - ïgga^. a^ ^ 0. (2.14) then this equation, with a fixed, defines a family of straight lines in the

gi • go plane, all points of which correspond to the existence of a subharmonic solution X = a Sin 0. See Fig. 1.

Case (ii), 3a^ = b^ e e

ag - 3 r / 2 ( g ^ - 1) - {(g^ - l ) / 3 g 3 J * (2.15) from which

gl = 1 + 3g3aJ (2.16)

and defines another family of straight lines in the g- ,g„ plane. See F i g . 1.

it is clear from this diagram that for every pair of real values g, , g„ there is at least one pure subharmonic solution of order 1/3. The amplitude of these subharmonic oscillations is given by | (a + b ) ^ | which, from (2.12)

e e

and (2.15), is the same for both c a s e s , having zero amplitude at u = ^JcT. Typical amplitude versus frequency diagrams a r e shown in Fig. 2. It is to be noted that for these diagrams Q is not fixed, so these a r e not frequency response diagrams in the usual sense.

3 . A Broader Class of Subharmonic Solutions

Associated with the pure-subharmonic solutions already discussed there would seem to be the possibility of the existence of a broader class of

solutions in the form of a real F o u r i e r s e r i e s 00

X = C ( a , _._ 1 Sin (2n + 1)0 + b„ ^ , Cos (2n + 1)0} (3.1) ^ &n T 1 ^n T" 1

n = 0

Writing the pure-subharmonic solution as

X = a Sin 0 + b Cos 0, T = T , (3.2) e e e

then with g and g fixed and F slightly different from F it is anticipated that there will exist a subharmonic solution of the form of equation (3.1). In order to confirm this, use will be made of the Galerkin procedure whereby truncated Fourier s e r i e s , with an increasing number of t e r m ë , a r e used a s

6

-s u c c e -s -s i v e a p p r o x i m a t i o n -s to the -s o l u t i o n . T h e p r o b l e m of the c o n v e r g e n c e of t h e G a l e r k i n p r o c e d u r e I s , in m a n y a p p l i c a t i o n s , u s u a l l y r e s o l v e d by an a p p e a l to t h e p h y s i c a l m e a n i n g and l i m i t a t i o n s of t h e solution. Such m e a n s of e n s u r i n g c o n v e r g e n c e only apply to a l i m i t e d c l a s s of b o u n d a r y vEilue p r o b l e m and a r e f a r f r o m being s a t i s f a c t o r y . F o r t h e p r e s e n t p r o b l e m t h e f i r s t a p p r o x i m a t i o n to the solution will b e t h e p u r e - s u b h a r m o n i c solution, ( 3 . 2 ) , and t h e second a p p r o x i m a t i o n will b e

x = a Sin 0 + b Cos 0 + a Sin 30 + b Cos 30 ( 3 . 3 )

F o l l o w i n g C e s a r l , t h e information obtained f r o m t h i s second a p p r o x i m a t i o n will b e used to d e n a o n s t r a t e the c o n v e r g e n c e of t h e G a l e r k i n method a n d , t h e r e b y , the e x i s t e n c e of an exact solution ( 3 . 1 ) in v a r i o u s r e g i o n s of t h e g ^ . g g . r s p a c e .

F r o m ( 3 . 3 ) ,

x^ = a Sin 0 + ^^ „ Cos 0 + a Sin 30 + + /3 Cos 90, ( 3 . 4 )

XU XU ó\) oij w h e r e

'^lO = W\ ^ ^l) - |^3<^1 - ^1^ - I ^^^3 ^ I ^<^3 ^ ^3>

3, , 2 , 2 , 3 ^ , 2 ^ 2 . 3 3 , 9 ^ 2 , ^10 = ï\^^l ^ \^ - 4 ^ 3 < ^ - ^ > + 2 ^ ^ ^ 3 + 2 \^^3 ^ ^3) 1 /ou2 2, 3 , 2 ^ 2 , 3 5 2, ^^30 = ï ^ < 3 ^ " ^ > + 2 a 3 < ^ + b j ) + 4^3<^3 + ^ 3 ^

^30 = - WK - ^1) + 1 s^h ^ ^?> ^ K<4" ^3)

^^50 = " ^3<^ • ^\^ + I ^ ^ ^ 3 -^ \\K - ^3> + l^^3^3

^50 '- - K < ^ " ^1^ - I ^ ^ ^ 3 • b&l - ^3> ^ lh^3^3

"70 = • h\K • ^3^ ^ 2 ^^3^3

^70 = - ^1<^3 - ^3) - I ^ V 3

a„„ = —a„(3b„ - a„) 90 4 3 3 3

and

x" = - a Sin 0 - b, Cos 0 - 9a Sin 30 - 9b Cos 30 (3.5) ± L o O

3

Substituting for x, x and x" in (2.3) and equating coefficients of the distinct t e r m s , i . e . Sin 0, Cos 0, Sin 30 and Cos 30 respectively, to zero, gives

rise to the following simultaneous relationships:

(gj - D a j + gga^Q = 0 (3.6) (gl - Dbj + gg/S^Q = 0 (3.7)

(gl - 9)a3 + ggag^ -F (3.8)

<gl - « ^ ^ ^3^30 = ° <3-«> The coefficients a , b , a , b may be expressed In t e r m s of a In the

X X O O 6 following way:

a^ = (1 + e^)a^. b^ = (k^ + ^^)a^. a^ = CgS^, bg = c^a^,

where b = k, a and k, is either zero o r 3*. Also F may be written as

e 1 e 1 •'

r = (1 -f X)r . Equations (3.6) to (3.9) become

(gj - 1)(1 + ej)a^ + h \ ^ ^ "" h^^^^ ^ h^^ -" ^h "• V ^ J -VS^^^ •" ^1^^

- (k^ + e^)^] - I (1 + e^Mkj + €2)^4 + | d + ^i)^^ + ^ ^ ) = ° <3-^°>

(g^ - l)(k^ + e^)a^ + h^lih\ ^ ^2^t<^ ^ ^1^^ "*" ^''l ^ ^2>^^

- I - 4 K I + ^1)' - (\ + eg)'] + I (1 + c,)(k^ + C2)€3 +

+ | ( k j +62) (€3 + e^)} = 0 (3.11)

(g^ - 9)e3a^ + gga^ { i ( l + ej)I3(kj + e^)^ - (1 + e^)']

- 8

and

+ le^l2{l + e^)^ + 2(k^ + Cg)' + ^3 + ^I U = 0 <3.13)

Subtracting (2.8) from (3.10), (2.9) from (3.11), (2.10) from (3.12) and (2.11)

from (3.13) gives, after division by a ^ 0 , the equations

(gl - 1)^1 + fgga^ {(3 + k2)e^ + 2k^e2 + ( k ' - 1)^3 - 2k^e^} + ê^^lo^i^.e^.^^^^)

(3.14) <% - '^'2 ^ k % ( 2 k , e j + (1 + 3k2)e2 + 2k^C3 + ( k ' - l ) e j +

<% - 9)^3 ^ k % ^^^1 - ^)^1 ^ ^\'2 ^ 2(1 + k2)e3] + gga2G3(e^ e^)

= | k 2 ( g j - 1)X (3.16)

(gl - 9)64 + | g 3 a ^ { - 2k^c^ + (k^ - 1)^2 + 2(1 + kJ)e^J + 83^04(^1 ^4)

= 0 (3.17) where

^1<^1 ^4> = h^ + ^1><^1 ^ ^2^ + l < ^ -^ ^1^<^3 + ^4> + K < " l + '^1^2 •

'3 - \'4^ ^ I V ^ ^ 3 - ^4 - ^1^4> ^ h ^ ^ l ^ 2 - ^1>' <3.18)

°2<"l ^4> = l^'^l ^ ^2><"f ^ ^2' -^ I ^''l + "2><^3 ^ "f> + r i < " 2 •" ^^1^3 " ^4^

^ l^2<'^1^2 ^ ^3 ^ ^^1^4 ^ ^1^3) ^ | V ^ 2 - ^1) <3-^^)

^ / V 3 , 2 2. 3 , 2 2, 3 / 2 2. 1 ,„ 2 2.

S<"l "4> = 4<"2 - ^1^ + 2 V"l + ^2> ^ r3<"3 ^ '4> "^ ri^^'2 ' 'l^

•^K"l^2 ^ ^h'3 -" ^'^iVs <3-20>

„ , K 3 , , 2 2, 3 , 2 2, 3 / 2 2. 1 . 2 ^ 2. °4<^r •••^4^ = 4 ^ < " 2 " ^1^ •" 2^4<"l + "2^ ^ ^ ^ 3 ^ ^4> ^ Véh ^ ^h^

and k = 1 or 2. Now from (2.12) and (2.15) 2 1

^3^e "" j ' ^ 3 ^ % ' ^^' ^'^^^^ ^^3 ^ -4 or 1, and, therefore, equations (3.14) - (3.17) become

(gl - DC^I + Jk3l(3 + kj)ej + 2k^e2 + (^J " 1>^3 ' ^^^^^ +

+ j k g G ^ (e^ c^)} = 0 (3.22)

gl - l>{e2 ^ \h^^\h + ^^ + ^*'l^^2 "• '^1^3 "^ *^1 " ^^^4^ ""

+ |-k3G2(ei e^)] = 0 (3.23)

(gj - 9)63 + (g^ - l){^k3t(kj - 1)€^ + 2k^e2 + 2(1 + kj)e3] + jk3G3(ej, . . . , e ^ ) }

= | k 2 ( g j - 1) X (3.24)

(gl - 8)€^ + (gj - 1) {|k3i:-2k^e^ + (kj - 1)€2 + 2(1 + kj)e4] +

+ j k 3 G ^ ( e j £4)} = 0 (3.25) The task of determining e e in t e r m s of X from these four simultaneous

cubic equations is formidable and probably best solved by the use of a digital computer for specific numerical values of X. This, however, is not the present task, which is to prove that (3.1) is a solution of (2.3) for a region of the g, , g „ , r space, close to, and containing, the individual members of the family of curves defined by g, , g , . r . F o r this purpose c, e. may

X »> 6 X 4 be taken to be, at most, of the first order of small quantities, then

G. , . . . , G contain t e r m s of the second and higher o r d e r s of small quantities only. Under these conditions equations (3.22) to (3.25) may be adequately approximated by

10

-- Ik^kgCj + |k3(kj -- l)e2 + {_n --f ^k3(l + k^)} e^ = o,

w h e r e n = (gj - 9)/(g^ - 1) ( 3 . 2 7 ) (3.28) ( 3 . 2 9 ) ( 3 . 3 0 )

T h e Solutions of t h e s e equations will be u s e d a s a guide to t h e s i z e of t h e r e g i o n in which t h e e x i s t e n c e of t h e solution ( 3 . 1 ) i s to be d e m o n s t r a t e d . It i s c l e a r f r o m t h e obvious complexity of t h e e l i m i n a n t of t h e s e equations that t h e g e n e r a l i t y p r e v i o u s l y sought, i . e . in u s i n g k, ,k and k , m u s t b e abandoned

X A o

and v a l u e s of e, , . . . , e . obtained for the two c a s e s s e p a r a t e l y .

C a s e ( i ) . k j = 0. k2 = 1. k3 = -4 T h e equations b e c o m e -2e^ + ^3 = 0

U- '

e^ + (n - 2)e3 = X/3 Cg + ( n - 2)e^ = 0 f r o m which e^ = X/3(2n - 3) = - (gj - l)X/3(g^ + 15)

^2 = °

€3 = 2X/3(2n - 3) = -2(gj - l ) X / 3 ( g j + 15) e. = 0 ( 3 . 3 1 ) C a s e ( i i ) . k j = 3 2 , k2 = 2, kg = 1

^h ^ 3*^2 + ^3 c„ - 3^e. }ie, + 7e„ + 3*€„ + 3^e, + 7^2 + 3^C3 + c^ = O ^1 ^ ^ % ^ 2^" "^ ^^^3 " '*^'^ -3^e^ + ^2 "^ 2 ^ " ^ '^^4 " ° (3.32)

Writing fl = 2(n -f 2), the solution of this s y s t e m of equations i s given by

ejA^ - C2/A2 = C3/A3 = e^/A^ = 1/A,,

where Ao = 12 1 _ q ï 5 32 1 -3= i i 3=* 7 3=» 1 1 3^ «1 -3^ 1 0 8 8x3^ 4 0 i i 3=" 7 3=^ 1 I 3» -32 1 "1 0

0

u^

= -4 0 2x32 1 2 1 7 3 ^ 3 * 0 3' 1 0 -3

i

(3.33) 2x32 1 35 1 1 \ 0 2 1 - 3 ^ + 4S 2x3 2 1 = 4(4 - 12n^ + %^) = 16(1 - n^)(l - 2J^), \ 32 1 -32 0 7 3^ 1 0 3^ n^ 0 -4X/3 1 0 n^ 0 4X/3 1 -3=* 3^ 3 * 1 7 0 n^ 1 . 4X(4 - 4f2^)/3 = 16X(1 - n^)/3

12 -^ 1 -32 o 5 3^ 1 O 3^ n^ O -4X/3 1 O - 3 ^ O ^ 4 X / 3 - 3 ^ 1 5 I 3^ 3^ II O - 3 ' = 16X(1 - fi^)/3^ ^ 5 3 i 3* 7 - 3 ^ O "^ ^* 1 O O -4X/3 1 32 - 3 ^ 1 n^ O 4 X / 3 - 3 ^ 5 32 1 3^ 7 n. - 3 ^ 1 = -128X(1 - n^)/3 Thus 5 3 = 3 * 7 3 * i •4X/3 1 32 O O - 3 * 1 O = - 4 X / 3 5 3^ 1 3^ 7 3^ -3' 1 O and €j = X/3(l - 2n^) = (g^ - l)X/3(43 - l l g ^ ) , €2 = X/3^(l - 2^^) = (g^ - l)X/3^(43 - U g ^ ) , 8e^. ^4 = 0 4 . A Proof of E x i s t e n c e

In the previous section it has been shown how the Galerkin technique of using truncated F o u r i e r s e r i e s may be used to obtain an approximate solution to equation ( 2 . 3 ) . However, it i s implicit in this p r o c e s s that if ( 3 . 3 ) i s to be an approximate solution to ( 2 . 3 ) then the difference between ( 3 . 3 ) , with a , . . . , b evaluated by m e a n s of ( 3 . 6 ) to ( 3 . 9 ) , suid the proposed

exact solution ( 3 . 1 ) m u s t be s m a l l in c o m p a r i s o n with ( 3 . 3 ) o r ( 3 . 1 ) . In t h e following t h e m e c h a n i s m of an e x i s t e n c e proof of ( 3 . 1 ) a s a solution to ( 2 . 3 ) will be e s t a b l i s h e d , b a s e d on the u s e of ( 3 . 3 ) a s an a p p r o x i m a t i o n .

T h e motivation behind the proof i s a s f o l l o w s . C o n s i d e r t h e function s p a c e S of a l l r e a l p e r i o d i c functions defined by F o u r i e r s e r i e s of t h e f o r m 2 ( 3 . 1 ) £md having a n o r m v(x) defined a s t h e L n o r m , i . e . 2v y(x) = {(2n)-^ ƒ x^{d)de]' ( 4 . 1 ) O A p r o j e c t i o n o p e r a t o r P naay be defined in S by t h e r e l a t i o n CO

P x = PE l a „ , , Sin (2n + 1)0 + b„ , Cos (2n -i- 1)0 } - Zn + 1 Zxi + 1 n = 0 m = E { a „ , Sin (2n + 1) 0 + b„ , Cos (2n + 1)01 , m<<», ( 4 . 2 ) „ 2n + 1 ^n + 1 •• n = 0

and in the p r e s e n t p r o b l e m m i s c h o s e n to be unity so that

Px = a Sin 0 + b Cos 0 + a Sin 30 + b Cos 30 ( 4 . 3 )

T h i s m e a n s t h a t if x given by ( 3 . 1 ) i s an exact solution to ( 2 . 3 ) then P x i s 2 t h e a p p r o x i m a t e s o l u t i o n . A l s o , by definition, P = P . C o n s i d e r now t h e s u b s p a c e S of S defined by S = ( x : x e S , P x = 0 } ( 4 . 4 ) T h u s if xeS t h e n eo

X = E f a„ ^ , Sin (2n + 1)0 + b„ . Cos (2n + 1 ) 0 } ( 4 . 5 ) . , '• Zn+ I 2 n + l

n = m + 1

Beflne the o p e r a t o r H on S by the r e l a t i o n

Hx = E (2n + 1 ) ' 2 {-a„ , Sin (2n + 1)0 - b„ , Cos (2n + 1 ) 0 ] ,

, Zn + 1 Zn + I ' n = m + 1

( 4 . 6 ) which c o r r e s p o n d s to a d o u b l e i n t e g r a t i o n of xeS with t h e c o n s t a n t s of i n t e -g r a t i o n taken to be z e r o .

14 -W r i t e ( 2 . 3 ) in t h e f o r m 3 x " = qx = - g X - g X + r Sin 30, ( 4 . 7 ) X Ó w h e r e q i s an o p e r a t o r on S, and c o n s i d e r t h e o p e r a t o r s f, F and T on S defined by 3 3 fx = qx - P q x = -g^x - g^x + P(gjX + g3X ) , ( 4 . 8 ) F x = Hfx = H { - g ^ x - ggX^ + P(g^x + g^xh] ( 4 . 9 ) and y = T x = P x -f F x = P x + H t - g ^ x - g3X^ + P(gj^x + g3X^)} ( 4 . 1 0 )

By p l a c i n g c e r t a i n bounds on i/(x), j x l , v(x - Px) and |x - r*x | i t i s p o s s i b l e t o define a s u b s p a c e S* a n d , provided c e r t a i n i n e q u a l i t i e s a r e s a t i s f i e d , it

will b e shown t h a t T : S -»^S and i s a l s o a c o n t r a c t i o n m a p p i n g . B e c a u s e T i s a c o n t r a c t i o n in S , B a n a c h ' s fixed point t h e o r e m (See Ref. 4 , p . 141) m a y b e invoked to conclude t h a t y(0) e x i s t s uniquely in S and i s continuously dependent on t h e a p p r o x i m a t i o n ( 3 . 3 ) . T h i s m e a n s that a , b , a . . . , a r e uniquely d e t e r m i n e d b y and continuously dependent on a b .

X o

If y(0) i s t h e fixed e l e m e n t of T in S , then y = x and f r o m ( 4 . 1 0 )

O Q y - P y = H { - g ^ y - g^j + P ( g j y + ggy )}

B i f f e r e n t i a t l n g t h i s e x p r e s s i o n t w i c e with r e s p e c t t o 0 gives

3 3 y " - P y " = -g^y - g^y + p ( g ^ y + g^y )

or

3 3 y " + g i y + ggy = P ( y " + g i y + ggy )

= r Sin 30 + P ( y " + g^y + ggy^ - F Sin 30)

T h u s y(0) will satisfy ( 2 . 3 ) p r o v i d e d

W r i t i n g 3

y (0) = a. Sin 0 + ^, Cos b + a^ Sin 36 + ^^ C o s 30 + a^ Sin 56 + , (4.12) 1 1 o o o

t h e n t h e condition ( 4 . 1 1 ) c o r r e s p o n d s to t h e s e t of d e t e r m i n i n g equations ( 3 . 6 ) to ( 3 . 9 ) in which a^. ^ ^ , a^ and ^3 r e p l a c e a^^, ^^^, a^^ and ^ 3 ^ ,

r e s p e c t i v e l y . T h i s exact s e t of d e t e r m i n i n g equations m a y be w r i t t e n in t h e f o r m

^ 1 = %

-

^^h ^

h^'l

= 0

r = 0

V„ = (g, - 9)a„ + go„ 3 '^1 3 ^3 3 ^ 3 = ^ % - ^ ^ ^ ^3^3 0. ( 4 . 1 3 ) whilst t h e c o r r e s p o n d i n g a p p r o x i m a t e s e t of d e t e r m i n i n g e q u a t i o n s ( 3 . 6 ) t o ( 3 . 9 ) m a y b e r e - w r i t t e n in a s i m i l a r f o r m ^ 1 = < % " 1 = < % ^ 3 = ^ ^ 1 u „ = ( g , - 1 ) ^ ^ g3'^10 - 1 ^ ^ ^ 3 ^ 1 0 - « ) ^ 3 ^ ^3*^30 • - 9 ) b „ + g„i3„„ = 0 = 0

- r = 0

= 0 30 ( 4 . 1 4 ) y

B e n o t e by A the f o u r - c e l l defined by I a | 4n | a | , jbj^ I"^^2 1^ I ' 1^3 l ^ ' " 3 l % l ' I'^S'•^'"4 I % ! • '\' ^' '^' ^4 ^ ° ' In t h e E u c l i d e a n f o u r - s p a c e of C a r t e s i a n c o - o r d i n a t e s a , b , a b . L e t Mand Mo be m a p p i n g s , d e s c r i b e d by equations ( 4 . 1 3 ) and ( 4 . 1 4 ) , r e s p e c t i v e l y , f r o m t h e v e c t o r s p a c e of c o m p o n e n t s (a , b a , b ) to t h e s p a c e of c o m p o n e n t s (Vi , U , V , U ) and (v, , u, , v , u ). T h e s e m a p p i n g s a r e s i n g l e - v a l u e d 1 1 3 3 l l i j 3

and c o n t i n u o u s . Befitie C and C a s the c l o s e d t h r e e - c e l l s d e s c r i b e d by MA and M A^ r e s p e c t i v e l y , w h e r e A^ i s t h e b o u n d a r y of A. It m a y b e v e r i f i e d d i r e c t l y w h e t h e r , o r n o t , t h e o r i g i n of t h e i m a g e f o u r - s p a c e l i e s in C , and w h e t h e r C h a s n o n - z e r o o r d e r v (C , 0), with r e s p e c t to the

0 0 o

o r i g i n (See Ref. 4 , p . 1 5 and p . 3 0 ) . If (u, v) r e p r e s e n t s a g e n e r a l point in C t h e n i t s d i s t a n c e f r o m t h e o r i g i n i s

16

-| ( u , v ) - OJ = -| { v ^ + u j + v^ + u ^ j ^ l (4.15)

S i m i l a r l y , if (U,V) i s a g e n e r a l point in C then the d i s t a n c e between (U,V) and ( u , v ) i s

| ( U , V ) - ( u , v ) | = | [ ( V j - v^)2 + (Uj - u ^ + (Vg - Vg)^ + (U3 - ^^f}^\

- I g 3 [ ( « i - « 1 0 ) ' + (^1 - ^ 1 0 ) ' + (-3 - - 3 0 ) ' ' <^3 • ^ 3 0 > ' i ' l •

(4.16)

which m a y b e computed using c e r t a i n e s t i m a t e s for o- " ''^i« • l^i ~ ^ i n ' ' I a„ - a I and j ^ - ^OQI • ^ i* c a n b e e s t a b l i s h e d that

| ( U , V ) - ( u . v ) | < | ( u . v ) - 0 | ( 4 . 1 7 )

f o r a l l points in C and C , o r o

g i b | ( U , V ) - ( u , v ) | < l u b | ( u , v ) - o | , (4.18)

t h e n by R o u c h e ' s t h e o r e m (See Ref. 5, V o l . 3 , p . 103) it follows that

v(C ,0) = v(C ,0) ^ 0, (4.19) o

o r t h a t

7 ( M , A , 0 ) = 7(M , A , 0 ) ^ 0, ( 4 . 2 0 ) w h e r e ^ ( M . A . O ) i s t h e l o c a l topological d e g r e e of M at t h e o r i g i n r e l a t i v e to

A. It then follows f r o m Ref. 4 , p . 3 2 , T h e o r e m 6 . 6 t h a t t h e r e i s a point in t h e i n t e r i o r of A for which v. = u = v = u = 0, and a n o t h e r point in t h e

± X O O

i n t e r i o r of A for which V^ = U^ = V- = U„ = 0. T h i s i m p l i e s that t h e exact s y s t e m of d e t e r m i n i n g equations ( 4 . 1 3 ) a r e s a t i s f i e d for c e r t a i n v a l u e s of a. , b . , a , b contained in t h e c e l l A a n d , t h e r e f o r e , y = x , a s given b y equation ( 3 . 1 ) , I s an exact solution of ( 4 . 7 ) a n d , t h e r e b y , equation ( 2 . 3 ) , f o r c e r t a i n v a l u e s a , b , , a , b contained in A,

5 . S o m e N o r m s and P s e u d o - n o r m s

2jr

o i i/(x) = [(27r) ' / x^(0)d0] ^

V

_o

2jr

I(27r)' ƒ (a^ Sin 0 + b^ Cos 9 + a^ Sin 30 + h^ C o s 30 + . . . . ) ^ d 0 ] ^

o Now 2jr * n ^ m ^"^ " ^ Sin m 0 d0 = O, m ^ n o 2;r a b Sin n0 C o s m 0 d0 = O, n m 23r b b C o s n0 Cos m 0 d6 = O, m /t n n m whilst 2ir

al Sin n0d0 = a ^ i 0 / 2 - i - Sin 2n0]^'' = na^

n n 4n ^o n and 2 T hl Cos^ n0d0 = b^(0/2+ i - Sin 2n0]^'' = wh^, Il n 4n o n "o therefore v(x) = I2'^(a]^ + ""l + ^3 + ''s + ^ 5 + • • • • ) ] " (5.1) - 1 . 2 , 2 2 , 2 . 2 . i V F r o m (4.3)

(Px) = 12"\af + bj + a^ + b^)]*

18 and i/(x - Px) = 1 2 ' ^ a ^ + bg + a^ + . . . . ) f . t h e r e f o r e i/(Px) .$ i/(x) and (* ( 5 . 2 ) v(x - Px) ^ v(x) f o r a l l xe S . Now

X - P x = a Sin 50 + b Cos 50 + a Sin 70 + b Cos 70 +

and f r o m ( 4 . 6 )

H(x Px) = 5 ' ^ a Sin 50 s ' ^ b Cos 50 7~^a Sin 70

-T h u s

- 1 - 4 2 -4 9 -4. 2 i i^H(x - Px) = [2 (5 ag •+ 5 bg + 7 a^ + )]2

-2 -2

^ 5 ^(x - Px) <: 5 i/(x) ( 5 . 3 ) In addition to t h e n o r m v, c e r t a i n r e s t r i c t i o n s will be put on x by m e a n s

of t h e p s e u d o n o r m

| x ( 0 ) | , 0 ^ 0 < 2jr

and t h e r e s u l t t h a t

I X I = I a Sin 0 + b C o s 0 + a Sin 30 + b Cos 30 + a Sin 50 +

' X X o o O

^ 1^1 1+ I M + h g l + Ibg 1+ l a j + . . . . (5.4)

will b e u s e d . T h u s | H ( X - P X ) I = | - 5 ' ^ a Sin 50 - 5 ' \ C o s 50 - 7 " \ Sin 70 - |

^5-2 | a J + 5 - 2 ( b J + 7 2 | a J + . . . .

^ (5 + 7 + . . . ) (a + b + a + . . . ) = ' 5 5 7

Now i -4 -4 i 4 2^(5 + 7 + ) ^ i / ( x - Px)

c<:2^(5'^ + 7'^ +

....)KM 1 + 3 " * + 5 ' ^ + 7"* + = »r*/96, (See Ref. 6 p . 167, E x a m p l e 3 , and s u b s t i t u t e x = 0), t h e r e f o r e | H ( X - P x ) | 4:2^(7r^/96 - 1 - 3"*)^ i/(x) ;$ 0 . 0 6 8 3 i/(x) ( 5 . 5 ) In a s i m i l a r m a n n e r , if h i s an o p e r a t o r in S, i/H[hx - P ( h x ) ] ^ 5 ' \ Ih - P ( h x ) ] ^ 5'^i/(hx) ( 5 . 6 ) and | H [ h x - P(hx)]|<: 0. 06831/fhx - P(hx)] <: 0.0683f(hx) ( 5 . 7 ) 6. Conditions f o r T t o b e a C o n t r a c t i o n Mapping in S

C o n s i d e r t h e four c e l l A defined in Section 4 . Then x* i s defined a s

X* = a Sin 0 + b C o s 0 + a Sin 30 + b C o s 3 0 , ( 6 . 1 ) with a , b , a , b contained in A. It follows t h a t

i/(x*) = [2-\al + bj + a^ 4- b^)]*^ [2"^^^ +4^4^ 4^)^K I = '^

and | x * | < | a j + | b ^ | + l a g ! + Ibg I < (MJ + '^2 ^ ' ' S "^ V % l = "" o r i / ( x * ) < c = a(/u) | a ^ I , ( 6 . 2 ) and | x * | ^ r = T(ui) j a ^ | , ( 6 . 3 ) w h e r e CT(JU) = { 2 ' (MJ + ^^ + H^ + ^ 4 ) } "

20

-and

T (ju) = *"! + ^"2 "*" '"3 "*" '^4 *

Befine S to b e the set of a l l x(6) a s given by ( 3 . 1 ) which satisfy t h e conditions P(x) = X* ^ v{x) 4 d | x | ,^ R

v(x -

Px) ^ *l% I

| x - P x j < p | a j

w h e r e d, R, 6 and p will b e chosen b e l o w . T h e n

i/(Px) = v(x*) < c and | P x | = | x * | < r for e v e r y x in S C o n s i d e r t h e m a p p i n g T : S -* S defined in ( 4 . 1 0 ) , t h e n P y = P T x = P ( P x + Fx) = P P x + P F x . Now f o r t h e p r o j e c t i o n P , P P x = P x and P F x = PHfx = 0, t h u s P y = Px = X* and f r o m (4.10) ( 6 . 4 ) ( 6 . 5 ) y - P y = H { -g^x - g3X + P(gjX + g3X )] * *

It i s r e q u i r e d now to obtain conditions f o r T : S -^^ S . F o r t h i s p u r p o s e i/(y - Py) and | y - r>y| will b e evaluated in t e r m s of

fJt., . . . . A«4, I a |, 6 and p. W r i t e

( 6 . 6 )

t h e n

X = Px + (x - Px)

Thus

viy - Py) = i/H [g^(x - Px) + g3(x^ - Px^) ]

= I/H {g^(x - Px) + g3[(Px)^ - P(Px)^]

+ 3 g 3 [ ( P x ) V - Px) - P ( P x ) ^ x - Px)]

+ 3g3[(Px)(x - Px)^ - P(Px)(x - Px)^]

+ g 3 [ ( x - Px)^ - P(x - Px)^] j

« | g j yH(x - Px) + | g 3 | { v H [ ( P x ) ^ - P(Px)^]

+ 3i'H [ ] + 3i/H [ ] + i/H [ ]} (6.7) and s i m i l a r l y

ly - P y U |gi I

|H(X

- Px)| + jgglflHKPx)^ - P(Px)^] I +

3 | H [ . . . . ) |

+ 3 | H [ ] | + | H | , . . . ] | ] ( 6 . 8 ) Consider the t e r m s in t h e s e e x p r e s s i o n s in o r d e r , with xeS .

Then from ( 5 . 3 ) , ( 5 . 4 ) and ( 6 . 4 )

I/H(X - PX)4 5'\(X - P X ) <: 5"^6 la I ") ' e '

and

|H(X - P X ) | ^ 0.0683 i/(x - Px)4 0 . 0 6 8 3 6 | a Now

(Px)^ - P(Px)^ = «gQ Sin 50 + 0gQ Cos 50 + a^^ Sin 70 + ^^^ Cos 70 + agQ Sin 90 + /Sg^ Cos 90,

therefore

. ((Px)' - P(P,)'l . l2-\.^„ * S^„ * .?„ . S5„ * al^ . 4„)1*

22

-and

|(PX)2 - P(PX)^\4 | a g j + l ^ g j + . . . . + l ^ g j .

In o r d e r to evaluate these, estimates for I a,.n , etc. a r e required. These ou

may be obtained from the expressions for a , etc. in Section 3 and the definition of A . Thus

| | ^ | | 3 ( 3 , 2 2. 3 3 , 2 2, 3 . l**5ol ^ ' % l fl'^S^^l + ^2> "" 2 ' ' l V 4 ^ 4^1<^3 + ^4> "^ 2 ''2*'3''4^ '

It will be seen later that the values of M. , ..,,tJi. will be expressed in t e r m s of X alone, thus l a ^ n l , . . . . , | ) 3 I may be expressed in the form

l"5o'-< ^0<^>'%l'

1^50 ' -< ^50<^>l%l'

u,oi ^ ^ o < ^ ) ' ^ l '

• N y (6.10) where

^50<^^ = h^3^4 + ^2^ "^ '^l^'^S + ^^4^^ ^ I V4<''l ^ ^3^

^50<^> = | t ^ 4 < ^ f + ^2^ + *'2^^3 "• ^4>^ •" l ^ l ' ' 3 < ^ 2 "" ^4>

A7o(^> = K ( ^ 3 + ^f) + l^2^3^4

^70<^> =K^^3 + ' ' ? ^ I ^1*^3^4

Ago(X) = ^^y^ + 3^2)

«90<^) " K<^''3 ^ ^J

-\ (6.11) y

T h u s i/[(Px)^ - P ( P x ) ^ ] < </>(X)|a3|^ (6.12) and ((Px)^ - P ( P x ) ^ | ; ^ , ^ ( X ) | a ^ | ^ , (6.13) w h e r e ^X) = [ 2 " ' ( A 2 ^ + B 2 ^ + . . . . .BI^)]" (6.14) and *<^) = ^ 0 ^ ^ 5 0 ^ • • • • ^ ^ 9 0 - < « - ' ^ )

It will be noted that

AgQ B Q Q , (^(X), ^ ( X ) ^ 0 .

F r o m ( 5 . 6 ) and ( 5 . 7 ) it now follows that

v H [ ( P x ) ^ - P(Px)^] 4 : 5 " ^ ^ ( X ) | a ^ | ^ ( 6 . 1 6 ) and | H [ ( P X ) ^ - P(Px)^] I ^ 0 . 0 6 8 3 ^ ( X ) l a ^ l ^ (6.17) Now f r o m ( 5 . 6 ) i/H[(Px)^(x - Px) - P(Px)^(x - Px)]4 5"^i/[(Px)^(x - Px)] < 5 " ^ | P x | ^ v ( x - Px) < 5 " ^ T V ) . « l a ^ l ^ (6.18) F r o m ( 6 . 2 ) and ( 6 . 3 ) and t h e p r e v i o u s r e m a r k s c o n c e r n i n g t h e e x p r e s s i o n of Ml . . . . iM. in t e r m s of X, it I s c l e a r that a and T a r e functions of X.

S i m i l a r l y

24 -Also, i/H[(Px)(x - Px)^ - P(Px)(x - P x ) ^ ] ^ 5'^i/[(Px)(x - Px)^] ^ 5 " ^ | P x | | x - Pxli/(x - Px)

;$ 5"^T(X). pója If (6.20)

e JH KPx)(x - Px)^ - P(Px)(x - Px)^] \4 0.0683 T(X).p6|a | ? (6.21) i/H[(x - Px)^ - P(x - Px)^] < 5~K{x - Px)^ < 5"^|x - Px|^i/(x - Px)

;$; S'^p^óla 1^ (6.22)

and | H [ ( X - Px)^ - P(x - P x ) ^ ] | < 0.0683 p^6 |a | ? (6.23)

Substituting these inequalities into (6.7) and (6.8) then gives

v(y - Fy)4 5 " ^ | a ^ | { | g j 6 + jg3||a^|^[0(X) + 3T^(X) 6 + 3 T ( X ) . p 6 + p^6] } and

|y - Pyl < 0 . 0 6 8 3 | a ^ | { | g j 6 + ^ 3 ! | ag|^I0(X) + 3 T W + 3 T ( X ) P 6 + P^6 ] }

Now from (2.12) and (2.15)

% =

We, -

l)/g3

or

Ki'-iigig^-ii/u,!.

therefore v(y - P y ) < 5 ' ^ | a ^ ( ( | g j 6 + | l k 3 | | g j - lI[<^(X) + 3T V ) 6 + 3T(X)p6 + p^ó] 1 (6.24) and ly - Pyj ^ 0 . 0 6 8 3 l a ^ | { | g j 6 + |-|k3| | g^ - ll [ ^X) + 3T ^X)6 + 3T(X)p6 + p^6]} (6.25)

The conditions for T : S * - S* "^^y now be established. F i r s t it is required to ask whether, for xeS ,

"^(y - P y ) ^ i^(x - P X ) ^ 6 | a I

and

|y - Pyl ^ |x - P x U pUgl

o r , upon using (6.24) and (6.25) and dividing throughout by a ^ 0 ,

5 " ^ { l g j ó + 3 | k g t t g ^ - lI[^X) + 3T (X)Ó + 3 T ( X ) P Ó + p^6]J4: 6 (6.26)

and

0 . 0 6 8 3 { | g j 6 + | - | k g | | g ^ - l | [ ^ X ) + 3T^(X).6-f 3T(X)P6 + p^è]]^ p. (6.27)

Now for xeS , Py = Px = Px* = x* and, therefore, j/(x) = 1/ [Px + (x - P x ) ] ^ i/(Px) + v(x - Px) 4 [o(X) + 6 ] | a | ^ d and | x | = JPX + (x - Px)| .$: |px| + |x - P x | < [ T(X) + P l j a ^ l ^ R. (6.28) (6.29) Similarly

v ( y ) ^ i/(Py) + i/(y - Py) < o(X)|a^| + i/(y - Py) (6.30) and

y ^ Py + l y - Py| ^ T(X)|a^| + ly - P y l . ( 6 . s i ) If the Inequalities (6.26) and (6.27) hold then from equations (6.28) to (6.31)

viy) 4 [oW + 6] l a ^ j < d | y U IrM + P ] | % U K

and hence yeS . Choosing d = [a(X) + 6]| a^l and

R = [T(X) + p ] | a I

with X, 5 and

26

-p satisfying (6.26) and (6.27), then T : Conditions for T to be a contraction mapping in in the following way. With x , x in S*,

EUld Also h = ^ 1 ^2 = P^2 Px^ therefore and »'(yi yi - yg =

- V ^ 5'

^ 5 ' < 5" ^ 5 ' < 5 ' Substituting for i ' ( y i _{- y2> ^ 5 '} - H[g^(Xj - Px^) + g^ixl - PxJ)] - H[g^(x2 - PX2) + g3(x| - Px^)] = X* = P X g , • - H { g ^ [ ( x ^ - P x ^ ) - (Xg - P X 2 ) ] + g 3 [ (

•^v[g^(x^ - =^2^ ^ h^4 ' 4^^

• ^ ' ' [ g l ( X i - X2) + g3(x^ - X2)(x^ + XjXg •2 r 2 2 i/{(x^ - X 2 ) [ g j + g3(x^ + x^Xg + X 2 ) ] }

s*.

S* 3 ^1 * S * . may be

- P . J ,

^4)]

' ' { . | [ g l + gg(Xi + X^X2 + X2)] (Xj - X2)} ' ^ g j + aigg K^' (xj - X2). R and then a gives

^ t l g j + [T(X) + p f ^ 3 ! |g^ - ij} v{x^

which means that T is a contraction in S provided

g j + [T When the mappint in S , theorem, Ref.

(X) + p ] ^ | k 3 | | g j - l | < 2 5

above conditions a r e satisfied for T to then it may be concluded, on the basis 4, p . 141, that the fixed element

-be of Xg). established - < ' ^ a contraction Banach' s fixed Px^)]] (6.33) point

y(0) = a, Sin 0 + b , C o s 0 + a^ Sin 30 + b , Cos 30 + a^ Sin 50 + 1 1 o o o

e x i s t s , i s unique in S and i s continuously dependent on x*. T h u s a_, b _ , a_, 5 5 7 b , . . . . , a r e uniquely d e t e r m i n e d by and continuously dependent on a , a b , ,

I X O X b for a , . . . . , b in A.

7 . E s t i m a t e s for a^. b ^ b g , \a^ - a^^l ^ - ^3^)

In o r d e r to b e a b l e to obtain t h e E u c l i d e a n d i s t a n c e between t h e c e l l s C^ and C , e s t i m a t e s of a^, b ^ b ^ , U^ - Q^J, IjSg - ^3^]

will be r e q u i r e d . F o r x in S , P y = P x = x* and it follows that

y - P y = a Sin 50 + b C o s 50 + a Sin 70 + , ( 7 . 1 )

a n d , t h e r e f o r e , t h a t a_, b_ e t c . a r e t h e F o u r i e r coefficients of (y - P y ) . o 0

C o n s i d e r now a p e r i o d i c function G(0) of p e r i o d 27r and t w i c e d i f f e r e n t i a b l e . A s s u m i n g t h a t G(0) h a s a F o u r i e r s e r i e s r e p r e s e n t a t i o n In both Sin n0 and

Cos n 0 , t h e n t h e F o u r i e r coefficients will b e given by

2ir 2ir a = TT" / G(0) Sin n0d0, h = w~ I G(0) C o s n 0 d 0 . o o E v a l u a t i n g t h e f i r s t of t h e s e i n t e g r a l s by p a r t s g i v e s 2v -1 f -1 -1 2^ a = ff / G(0) Sin n0d0 = ^ [-n G(0) C o s n 0 ]

J

o 2ff 27r + w'^ / n''^ G\9) C O S n0d0 = w'^ / n'^G* (0) C o s n0d0 o o which upon i n t e g r a t i n g by p a r t s a g a i n g i v e s 2ff a = -rr'^ / n ' ^ G " ( 0 ) Sin n0d0, ( 7 . 2 )

J

o

- 28 and in a s i m i l a r way 2ff b ^ = -ff"^ / n"^G"(0) C o s n0d0 (7.3) Now n ' 2n -1 f -2 -TT / n G"(0) Sin n0d0 27r « ir -1 | G " ( 0 ) n ' Sin n 0 | d0 - 2 which by t h e S c h w a r z i n e q u a l i t y i s 2jr 2ff 4 ^'^n'^[ I | G " ( 0 ) | ^ d 0 ] ^ [ / | s i n n 0 | ^ d 0 ] * 2w 2ff < 2^n'^[(27r)"^ / | G " ( 0 ) | ^ d 0 ] ^ [TT"^ ƒ j s i n n 0 | ^ d 0 ] ^ = 2*n"^i/(G") S i m i l a r l y j ^ » . " 2 „ / r < i i \ |b I < 2 ^ n ' V G " ) . I n'

Identifying (y - Py) with G(0) then y i e l d s

2»n v [ ( y - P y ) " ] ,

1^1

<

(7.4)

B i f f e r e n t i a t l n g ( 6 . 6 ) with r e s p e c t to 0 g i v e s

which m e a n s t h a t vHy - P y ) " ] = i/[g^(x - Px) + g3(x^ - P x ^ ) ] < | g j i / ( x - Px) + \g^liv[{Px)^ - P(Px)^] + 3 i / [ . . . J + 3i/[ ] + v[...]} = M a in a c l o s e l y s i m i l a r m a n n e r to ( 6 . 7 ) , o r f r o m ( 6 . 2 4 ) M = I g ^ j ó + i l k g l jg^ - l | [ ^ X ) + 3 T ^ X ) 6 + 3 T ( X ) P 6 + p^ó ] ( 7 . 5 ) T h u s

'^J '^ i -2 I I

2^n ^ M J a , ( 7 . 6 )

b I <

' n' and in p a r t i c u l a r

U J . |bj « 2^5-2

M U J

| a , ^ | , j b j < 2 * 7 " ^ M | a ^ |

\aj , Ibgl < 2 ^ 9 ' ^ M | a J

As a p r e l i m i n a r y to d e t e r m i n i n g \a^ - a. _| , e t c . t h e following r e l a t i o n s will b e r e q u i r e d : 2

(a Sin 0 + b j C o s 0 + a Sin 30 + b C o s 30) Sin 0

= 7 Sin 0 + ^y^ C o s 0 + Y Sin 36 + + y C o s 70 ( 7 . 7 )

(a Sin 0 + b C o s 0 + a Sin 30 + b C o s 30)^ C o s 0

= ?, Sin 0 + ?, C o s 0 + 5„ Sin 30 + + ?„ C o s 70 ( 7 . 8 ) s i c 1 s 3 c 7

2

(a Sin 0 + b Cos 0 + a Sin 30 -f b Cos 30) Sin 30

30

-and

2

(a Sin 0 + b Cos 0 + a Sin 30 + b Cos 30) Cos 30

?, Sin 0 + f, Cos 0 + r Sin 30 + + ?„ Cos 9 0 , ( 7 . 1 0 ) s 1 c 1 s 3 c 9 where 3T1 = | ( 3 a 2 + bf + 2a2 + 2b2 - 2a^a3 - 2b^b3) cT'l = l < ^ ^ ^ ^ ^ 3 - ^ ^ 3 > 1 / 2 , 2 , s^3 = ^ ^ 3 - 4 ^ - ^ >

c^3 = -ri<^ - 2^3)

s'^5 = | f - ^ ^ 3 ^ V 3 ^ l < 4 "^3>J

c'>'5 = " bh^3 ^\^3 ^ V3>

1 / 2 . 2 , s^7 = - 4 <^3 • ""S^ 1 . y » —^a b c^7 2 3 3 s^l = l < ^ ^ ^ ^ ^ 3 - ^''3>

Jl - h4 ^ 3bJ + 2a32 + 2b32 + 2a^a3 + 2b^b3)

s^3 = K < ^ ^ 2a3)

c^3" w - h4 - 4^

sh ' I^^^'S ^ ^ ^ 3 ^ ^3*^3^

c^5 ' l l - ^ l ^ S ^ - V s -|^^3 -^3>1

s^7 = |-^3^3 e I7 2 , 2.

0^7 '-t^3 - V

L 2 ,2,

^ ^ 3 " 4 ^ ^ - ^ >

c'^ ' K S ^^ag)

s'^ c'^ s ^ c*^

s'V

c'V

s"^ c'^ s^l c^^l s^3 c^3 s^5 c 5 8^7 c^7 s^9 c^9

'

hK

- K - K -

4^

-

1 ^ 3

=

| i v 3 ^

Vs

-V4

-4^^

- -

kW

^

^^3 -

h^3>

4<-^^3^^v

= - | < ^ ^ 3 ^ ^ ^ 3 >

L 2 , 2 ,

= - 4<^3 - ^ )

= - k^3

= J ^ ( 2 b 3 - b,)

=

b,b3

-

|(a2

- 4)

= i^3^3

' J ( 2 a j + 2b2 + a^ + Sb^) = | ( a ^ b ^ + bj^ag - a^bg)

=

|fV3^

Vs - K '

-4^^

' hhh ^ ^^3^

= | ( - a ^ a 3 + b^bg)

= l^3^3

1/ 2 ,2,

= -4<^3 - V

- 32 F r o m t h e s e , the following i n e q u a l i t i e s m a y be d e t e r m i n e d :

(s'^5l'W^< | f ^ ^ ^ ^ ^ 4 ^ <^'^^^J|%|'

ic^sl'ls^^^ h^\^ + ^ ^ +^^4^1^'^

Is^l 'lc^7l<

I M •''^^4^I%I^

Ic'^l'IsM^ I V 4 ^ ^ ^ ^ l % l '

l s ^ 9 l - l c ^ 9 l < I < ^ ' - - 4 ) | % l '

Ic'^l ' I s ^ ^ 1^4!^

2 el 3 F r o m ( 4 . 1 2 ) t h e F o u r i e r coefficients of y ( 0) m a y b e obtained by t h e r e l a t i o n s 2w 2-n - 1 / 3 - 1 / 3

a " Ti I y (0) Sin n0d0 and ^ = w / y (0) Cos n 0 d 0 .

*

F o r X in S , P y = P x = x*, and f r o m ( 3 . 4 ) t h e F o u r i e r coefficients of (x*)^ = (Py)^ a r e

27r 2ff

a = ff"'^ / (Py)^ Sin n0d0 and ^ = ff''^ / (Py)^ Cos n0d0

T h u s 27r

- 1 / 3 3

a - a = IT I [y - (Py) ] Sin n0d0

o

2n

= TT'^ / (y - P y ) [ ( y - Py)^ + 3(y - Py)(Py) + 3 ( P y ) ^ ] S i n n0d0

= J , + 3 J „ + 3 J „ , ( 7 . 1 1 ) n 1 n 2 n 3

w h e r e

2ir 2K

o o

= 2ly - PyJ^ax.^^^y • ^"^f < 2p6^|aJ?

2v 2it

l ^ j j =

I T T ' M

(y - Py)^Py) Sin n0d0| < ^1 Py| ^^^^ (27r)"^ / (y - Py)^d0

= 2lpylj^^^_ [v{y - Py)f < 2T(X)62|aJ?

27

I^Jgl = Iff'^ I (y - Py)(Py)^ Sin 6 d0l,

which f r o m ( 7 . 7 ) b e c o m e s

27

l.^al - !•"' ƒ

(a Sin 50 + b C o s 50 + )( 7 , Sin 0 + . . . . + 7 , C o s 70)d0l

D O B X C I

a_ 7c + b_ 7_ + a_ 7r, + b_ 7„ ,

- 34

and upon substituting for a 7 etc. this becomes

D S O

I1J3I ^ l i J g l U j '

where

I1J3I = 2 ^ M { 5 " 2 [ | ( M ^ ^ + ^^^i^) + 1 ( / | + M4) +1(^1^4 + A^/^ + ^^^^)]

similarly

+ 7"'[i(A| +/|) ^I^M,]};

2v

-if, „ w„ >2

3J3I = I T / (y - Py)(Py)^ Sin 30 do o 2]r " TT / (^5 ^^^ ^^ + bg C o s 50 + ...)(gr)^ Sin 0 + . . . + ^r^ C o s 90)d0 o = ' ^ 5 s % + ^5c^5 ^ ^ s ^ ^ N c ' V ^ ^ 9 s ^ -^ ^ c o r

3'^3l

-^

13^3" %'^'

where

igjgl - 2W5-2[|(M^^ + ^^^) +1(^2 ^ ^ j ^ 1 ^ ^ ^ ^ ^ ^ ^ ^^^^^

+ ^ ' ^ l | < ^ ' ^ +'^^4> + | ^ ^ 4 ^ ^ ^ ^ J

-

9 - ' [ ^ A |

. M^) 4^M4]}

It follows that a^ - a ^ J < [2p6^ + 6 T ( X ) 6 2 + 3 | ^ J 3 | ] | a ^ | ^ '*3 " ' ' 3 0 ' ^ f^pó^ + 6 T ( X ) 6 2 + 3I3J3I ] la^l^

A l s o , 2ir ^n • ^ n o " ' ^ ' ^y^ ' ^ ^ ^ ^ J ^ ° ^ "®**® = L + 3 L + 3 L ( 7 . 1 2 ) n 1 n z n o where

LsU<2p6 2laJ?

|^L2|<2T(X)62|aJ?

2 7

'.^1 • ! ' • ' ƒ <

a_ Sin 50 + b . C o s 50 + . . . ) ( f, Sin 0 + . . . + ?„ C o s 70)d0l 5 5 s i c 7 ' ^ s « 5 ^ ^ c ^ 5 ^ ^ s ^ ^ ^ c ^ ' ' 2 7

, - L , | = 1 7 " ^ / (a Sin 50 + b_ Cos 5 0 + . . . ) ( f, Sin 0 + . . . + r C o s 90)d0|

0 0 f O 5 S i c 9 » | a _ r_ + b_ L + a_ ? , + b„ 5„ + a„ C„ + b„ r | . * 5s 5 5c ^ 7s 7 7c 7 9s 9 9c 9 and, thereby,

l i S l ^ l i ^ J ' s l '

3^1 < I3^3"S''

where 1^13! = l^jgj and [glgj = \^i^\. T h u s

'**! " ' ' 1 0 '

< / [ 2 P + 6 T ( X ) ] 6 ^ + 3 t ^ J 3 | } | a ^ ( 3 ( 7 . 1 3 )

1^1 "^in I

1 ' ^ l O '

36

-and

"^3 " '*30'

1^3 - ^ 3 0 l

< { [ 2 p + 6T(X)]Ó^ + 3 | 3 J 3 J } j a ^ | ? (7.14)

8. An Estimate for lub I (u,v) - ol

Since n. , . . . . , u , define the cell A, and hence S , it is clear that If S is to contain the proposed exact solution then A must contain the point (a ,b ,a ,b ) defined by the leading coefficients of the exact solution and

X X O 0

H. u must be chosen so as to make this possible. Now l a j = 1(1 + e ^ ) a j « d + U i l ) | a g |

j b j = [(k^ + e 2 ) a j < ( | k j + | e 2 l ) l a j

' ^ 3 ' = I V e l

^ 3 " % l

' V e ' ^1^4! I % l '

and if ju. ^ a r e chosen so that

( 8 . 1 ) 1 + l^ll^ l\

j k j + |e2l< ^

^ 3 ! ^ ^

1^4! < ^4

(8.2) then

| a j < M j a ^ j . IhJl^H^l^J. Ugl 4n^\aJ and [bgl KU^JsiJ

a s required. The four-cell A is then defined a s the set of points correspond-ing to all combinations of e^ e. in the intervals

1 - ^^ < e^ < M^ - 1. k^ - ^2 ^ ^2 *= ''2 " ^ 1 ' " ^ ^ S -^ ^ 3 ' • ''4 -^ ^4 ^ ^'4 (8.3)

The set of points defining the boundary three-cell A^ are obtained by taking e. e., in turn, to be their extreme values in the above definition. Thus Ap is made up of the following collection of three-cells:

e^ = 1 - ^ 1 . k^ - ^2 ^ ^2 ^ ^2 • ^ 1 ' "'^3 ^ ^3 ^ ^ 3 ' '^4 ^U^ ^ 4 '

Cj = ti^ - 1, k^ " ^"2 ^ ^2 ^ '^2 ' '^l • '^3 ^ h ^ ^^3' '^4 ^ ^4 "^ '^4'

N

> (8.4)

1 - y^ 4€^ 4n^ - 1, kj - jU2 < ^2 ^ ^ 2 ' '^l' "'^3 ^ ^3 ^ ^ 3 ' ^4 "" ' ^ 4 ' ^ where e. e take on all values over their respective intervals.

From equations (2.8), (2.9), (2.10), (2.11) and (4.14) it foUows that Vj - (g^ - l)(aj - a^) + g3[«^^ - | a ^ ( a 2 + b^)}

Ui = ( g i - l ) ( b i - b ^ ) + g 3 f ^ i o - | V % ^ ^ e > >

-3 = <% - ^ ^ " g3^"30 - WK - 4^^ - <^ - ^e)

U3 = (g, - 9)^3 + gat^ao 4 ^ ( 3 % - b ^ ) 5 .

which, from equations (3.22), (3.23), (3.24) and (3.25), reduce to

vi = (gj - DCe^ +i-k3 [(3 + 4)e^ + 2k^e2 + (kj - Dcg - 2k^eJ + l-kgG^l a e (8.5) Uj = (g^ - 1){£2 +4k3[2kjCj + (1 + 3kJ)€2 + 2k^e3 + (kj - l)c^] + gkgGg'i a

Vg = [(gj - 9)eg + (g^ - l ) { | k 3 [ ( k j - De^ + 2k^e2 + 2(1 + kj)e3]

Ug = [(gl - ^ \ + (gl - l)[\k^[-2k^€^ + (kj - 1)€2 + 2(1 -f kj)c^]

e (8.6)

(8.7)

38

w h e r e G, , . . . , G . a r e given by equations ( 3 . 1 8 ) , ( 3 . 1 9 ) , ( 3 . 2 0 ) and ( 3 . 2 1 ) .

T h u s C i s given by M A _ , w h e r e M i s defined by equations ( 8 . 5 ) , ( 8 . 6 ) , o o B O

( 8 . 7 ) and ( 8 . 8 ) , and A^ i s t h e t h r e e - c e l l defined by ( 8 . 4 ) . T h e d i s t a n c e | ( u , v ) - o | Is then given by ( 4 . 1 5 ) .

In o r d e r to employ t h e Inequality ( 4 . 1 8 ) it i s r e q u i r e d to d e t e r m i n e l u b | ( u , v ) - o | . F o r m a l l y it i s not a difficult p r o b l e m t o d e t e r m i n e t h e m i n i m a of | ( u , v ) - o | . T o do t h i s , h o w e v e r , r e q u i r e s t h e solution of a s e t of s i m u l t a n e o u s equations of an equal d e g r e e of complexity a s ( 3 . 2 2 ) , ( 3 . 2 3 ) , ( 3 . 2 4 ) and ( 3 . 2 5 ) , a t a s k which h a s a l r e a d y been avoided. T h i s difficulty m a y , to s o m e e x t e n t , b e o v e r c o m e by u s e of t h e inequality

l u b | ( u , v ) - o| > | { ( l u b v ^ ) ^ 4 ( l u b u j ) ^ + (lubv3)^ + ( l u b u 3 ) ^ J ^ j , ( 8 . 9 )

which, when o n e of c o m p o n e n t s i s d o m i n a n t , usefully r e d u c e s t o

f V, I I I " l l u b | ( u , v ) - 0 | > l u b ^ ( 8 . 1 0 ) T h e u s e of t h e right hand s i d e of ( 8 . 9 ) , in p l a c e of l u b | ( u , v ) - 0 I, in ( 4 . 1 8 ) will u s u a l l y u n d e r e s t i m a t e t h e s i z e of t h e r e g i o n in t h e g. . g o . l ^ s p a c e f o r which t h e e x i s t e n c e t h e o r e m can b e p r o v e d t o b e v a l i d . N e v e r t h e l e s s , v a l u a b l e r e s u l t s c a n s t i l l b e obtained by i t s u s e . C o n s i d e r t h e evaluation of lubv^ . T h e f i r s t s t e p i s to d e t e r m i n e w h e t h e r any m i n i m a e x i s t in v- a s e^ , . . . , e. v a r y o v e r Ap.. B i f f e r e n t i a t l n g equation ( 8 . 5 ) g i v e s 9v d€ dv dê dv 8? 8v

^ =(gi -1){1 4 k 3 ( 3 + k 2 ) 4 k 3 ^ ) a ^

BG

'- =(gi - I J ^ K ^ ^ K B^^%

1 1 9 1 ^^1 N ^ 8v- , . dG

_ i =(g, - D f - h k g . l k g — )a

4 4 ( 8 . 1 1 ) y

w h e r e dG 8

8-^

=

l^^^l

-

^l<-2

-

^4)

- ^^

^'M ^ 4 -

2<^3

-

^4)^

^

= F ^ < ^ 1 -^ ^3> •" "2 • "4 -^ ^1<^2 - ^4> •" *^lV3i

1 8G ^^2 8G^ ^^3 and

• | ' - ' l * V 2 * ^V * 'V3 * I'V2

«f>

8G^ 3 8 ^ = 2 ^-"^l^l 4 ( 8 . 1 2 ) y -2 ^ 2e4 + e^(2e^ _{^ 2 «} , u . will b e c h o s e n so Now in t h e s u b s e q u e n t a n a l y s i s t h e v a l u e s of u. , t h a t t h e v a l u e s of e. , . . . , e . on A a r e , at m o s t , of t h e f i r s t o r d e r of s m a l l q u a n t i t i e s . T h u s , f r o m ( 8 . 1 2 ) , t h e d e r i v a t i v e s 8 G . / 8 c . , , 8G^/8e^ a r e , at m o s t , of t h e f i r s t o r d e r of s m a l l q u a n t i t i e s . T h e c o n s t a n t t e r m s In t h e t^ b r a c k e t s in ( 8 . 1 1 ) a r e e i t h e r z e r o o r of o r d e r unity, it follows, t h e r e -f o r e , t h a t it i s p o s s i b l e -f o r 8 v ^ / 8 c , , . . . . , 8 v . / 8 e . t o b e z e r o only when t h e s e c o n s t a n t t e r m s a r e z e r o . T a k i n g t h e s e in t u r n

| l + k3(3 + kj)/4| > 0. Ik^k3/2| ^ 0, |k3(kj - 1)/4| > 0,

thu s it Is p o s s i b l e f o r 8 v . / 8 e and 8 v , / 8 e . to b e z e r o in t h i s r a n g e of C j , . . . , e , but not 8v-/8e- o r 8v / 8 e . Now with e ,

on A t h e l e a s t condition f o r a m i n i m u m i s 1 - ju, o r e,

Ü1

8e., 8 € , 8v^ 8e. = 0 .

But 8 v , / 8 e ^ 0 in t h i s r a n g e , so t h e r e can be no m i n i m u m on the p a r t of X Ó

' = H. - 1 . S i m i l a r l y with e „ = k. - / ^ o r

A defined by c , = 1

B 1 o r

40

-But 8v-/8e- h 0 in this range, so there can be no minimum on the part of A defined by c„ » ± (k^ - nJ). Similar arguments apply to the parts of A defined by e„ = ± M, and e » * H.- Since, then, there a r e no minima In v.(e, e ) for e , . . . , e in A , it may be concluded that lub v^ is the value of V. at one of the sixteen combinations of extreme values of e-, . . . , e . The values of | v J at these combinations of extreme values may conveniently be called "corner" values, (v^) , and these a r e readily evaluated. Birect conaparison of these corner ' values then yields the least value

min (v,) ] . • lubv^

1 c Ag (8.13)

Consider now the evaluation of lubu^ . Bifferentiatlng (8.6) gives

'2i

~ = ( g l - 1) ^ ^ ^ ^ 3 ^ ^ ^ 3 8 - ^ 1

du

^ = ( g , - I ) { l . l k 3 ( 1 . 3 k f ) . l k 3 ^ ) a ^

3^=(g, -l){-k,k3 4 k 3 3-^}

^ '3 864 * e where

^ S 3

^ '- 2 ^'^l^l ^ h ^ Sh - '4 ^ ^1^2 ^ ^2^3 - ^l"4>

y = I <^1 ^ '^^2 ^ ^3 ^ ^^4^ n<^l ^ '4 ^ 2-3 •*• 2^4>

(8.14) \ , >(8.15) ^ 2 <^1^3 ^ V4>

^ S 3

i ^ = 2 ^Sh ^ ^2 ^ ^kjCg + e^e2 + 2^263)

87-' = l < - ^ ^ V 2 ^ 2 k ^ V ^ H - ^ l > ^ 3 V 4

₄

In C a s e (11), and u n d e r t h e s a m e conditions a s w e r e a s s u m e d in t h e d i s c u s s i o n of l u b V j , t h e p a r t i a l d e r i v a t i v e s 8 u . / 8 e , 8 u . / 8 € , 8Uj/8e and 8 u - / 8 e . a r e not z e r o . It follows t h a t lubu^ c o r r e s p o n d s to t h e l e a s t c o r n e r v a l u e . In C a s e (1)

|-k^k3 = 0, l+|-k3(l + 3kJ) = 0. Jk3(kj - 1) > 1

and only 8 u . / 8 e . can b e g u a r a n t e e d t o b e o t h e r t h a n z e r o , without f u r t h e r exanninatlon of t h e p a r t i a l d e r i v a t i v e s of G , F o r t h e p a r t s of A defined b y

z a

c- = ± (1 " /"i^i -o ~ * (ki " A*J and c = ± M, , r e s p e c t i v e l y , it i s c l e a r t h a t

t h e l e a s t condition f o r a m i n i m u m cannot b e s a t i s f i e d , b e c a u s e 8 u ^ / 9 e ^ 0. When e . = ± ^ t h e l e a s t condition f o r a m i n i m u m in u, on t h e a p p r o p r i a t e 4 4 1 part of M A _ i s o B

®"l 4 » S

3 ^ = - 3 (gl - l ) a ^ . ^ -0 bM 8G

8 ^ = - 1

<^i

- 'K- ^ - °

8u 4 8G

E x c l u d i n g t h e condition g, = 1, a = 0, t h e conditions for a m i n i m u m on A b e c o m e B 8G2/8€i = 8G2/8€2 = 8 0 2 / 8 6 3 = 0 o r f r o m ( 8 . 1 5 )

l<^2 - ^ 4 ^ V 2 * - V 3 - V4> ' ° <«-^«>

I <^i - ^3)' M ^ '4 ^ ^4' '4^ ^ I <^i^3 ^ V4) = ° <«-^'>

| e 2 ( l + -1 + 263) = 0 ( 8 . 1 8 ) Now 1 + e, + 2e_ A 0, t h u s f r o m ( 8 . 1 8 ) , e_ = 0 at t h e m i n i m a . X 0 A

- 42

- f e ^ d ^ c ^ ) = 0.

But 1 + e, ^ O, thus e. = O at the m i n i m a . However, the value of e. on this part of A h a s been fixed at ^ or - ju.. where u^ ^ 0. It follows that

ti 4 4 4

there can be no m i n i m a in u (e. , . . . , e . ) for e e in A defined by

VI e = ± A». ^ 0. Summarizing, lubu

_{1 = " ^ ^ < " l ) c U}

B' F r o m ( 8 . 7 ) , 8v ^1 8v

^"^3 rl 2 1 ^ S

8G„ J- = (g^ - 1) {2kik3 + 3 k 3 9-^ _{\ ^} N ^""3 1 2 1 ® S

^ = [(gl - 9) + (g, - 1) {|k3 (1 . k^) . ik3 ^ ) ] a ^

""3 1 , , n ' ° 3

4 4

y ( 8 . 1 9 ) (8.20)

Now In Case (11) both

W4 - l ) > 0 a n d | k ^ k 3 > 0 ,

so that lubv corresponds to the least corner value. In Case (i)

^^y - 1) > 0 and |k^k3 = 0 .

Thus 8v„/8e, ^ 0 and there can be no minima in v„ for the parts of A

3 1 o o defined by Cg = ± (k^ - A^), Cg = ± ^ and e^ = ± ^«4. r e s p e c t i v e l y .

When e = ± (1 " K ) the l e a s t condition for a minimum in v on the appro-priate part of M A i s

^ 3 4 , , , ^ 3

« - 3 «^3 «^3 Now 8G„ = {gl - 9

= - 1

(gl

o ( g , -- ^ ) %

4 ^ S

_{) j a = O (8.22)} e = O ( 8 . 2 3 ) B^ - I ^2<^ -^ ^1 ^ 2C3). and t h e condition ( 8 . 2 1 ) i s s a t i s f i e d by €„ = 0. A l s o

! i -3

8e^ 2 V 4 ' 4

and condition ( 8 . 2 3 ) i s satisfied by t a k i n g e = O o r e = 0.

Upon s u b s t i t u t i n g f o r 8 G „ / 8 € in ( 8 . 2 2 ) t h i s b e c o m e s , a f t e r s o m e r e a r r a n g e

-«3 O

m e n t with e„ = 0,

(-1 - 4e^ - 2^1 - 3el - €^)(gi - 1) - 8 = 0,

a condition which can b e s a t i s f i e d f o r a v a l u e of g^ a p p r o x i m a t e l y equal to - 7 . If, h o w e v e r , g i s r e s t r i c t e d to the i n t e r v a l - 6 < g^ ;g:oo ( 8 . 2 4 ) t h e n t h e m i n i m u m cannot e x i s t and lubVg = m i n (V3)^]^ ( 8 . 2 5 ) B A c l o s e l y p a r a l l e l a r g u m e n t shows that p r o v i d e d g i s r e s t r i c t e d t o t h e i n t e r v a l ( 8 . 2 4 ) then l u b u , = m i n ( u . ) ^ ] ^ ( 8 . 2 6 ) 3 3 c ' A „ i 3 . T h i s r e s t r i c t i o n on t h e v a l u e of g^ m a y s e e m u n d e s i r a b l e , h o w e v e r , it will b e s e e n l a t e r t h a t t h e ( 6 . 3 3 ) i s m o r e s e v e r e . b e s e e n l a t e r t h a t t h e r e s t r i c t i o n on g. inaposed by t h e c o n t r a c t i o n condition

44

-S u m m a r i z i n g t h e s e r e s u l t s , it follows t h a t ü H. - l , M , ~ k , K, and u a r e c h o s e n to b e , at m o s t , of the f i r s t o r d e r of s m a l l q u a n t i t i e s and g. i s r e s t r i c t e d to t h e i n t e r v a l - 6 - ^ g . ^ so, g ^ 0, t h e n

l u b l ( u , v ) - ol 5. M[min ( v . ) f + [ m i n ( u , ) f + [ m i n ( v . ) f+ [ m i n (u„) fy\

1 ' I " I c I c 3 c j c - * '

( 8 . 2 7 )

T h i s e s t i m a t e for l u b l ( u - v ) - o | will be used in Sections 9 and 1 0 .

9 . Application of t h e Proof of E x i s t e n c e When | g^ - II i s S m a l l .

The technique of the e x i s t e n c e proof h a s been given in Section 4 , and it i s now intended to apply t h i s method to d e m o n s t r a t e the e x i s t e n c e of s o l u t i o n s , of the f o r m ( 3 . 1 ) , s t e m m i n g f r o m the two f a m i l i e s of p u r e - s u b h a r m o n i c

s o l u t i o n s r e p r e s e n t e d in F i g . 1. T h e r e q u i r e d inequality ( 4 . 1 8 ) i s m o s t r e a d i l y e s t a b l i s h e d when |g. - 1 | i s s m a l l , and the r e s u l t s for t h i s condition will b e obtained f i r s t . In o r d e r to e s t a b l i s h t h e proof in a l a r g e r r e g i o n of t h e g, , g , r s p a c e f a i r l y e x t e n s i v e computation i s r e q u i r e d . E x a m p l e s of

X o

such extended r e s u l t s will b e obtained in Section 10.

Some guidance to t h e c h o i c e of M. 1 • • • . / " . ïnay b e obtained f r o m t h e l i n e a r i z e d second a p p r o x i m a t i o n s obtained in Section 3 . It can b e s e e n f r o m t h e s e t h a t , provided g i s t a k e n to be sufficiently c l o s e to unity, e , . . . . , € will a l w a y s b e s m a l l f o r a l l finite X. F o r any finite X and c h o s e n i n t e r v a l 1 " " y ^ g , < 1 + T i with 0 < 7 < < 1, t h e a s s o c i a t e d v a l u e s of e , . . . ,e , given b y the l i n e a r i z e d second a p p r o x i m a t i o n , define a f o u r - c e l l A . A s a t i s f a c t o r y choice f o r A i s then a c e l l slightly l a r g e r than A such that A i s contained in t h e i n t e r i o r of A. T h u s when t h e l i n e a r i z e d second

e

a p p r o x i m a t i o n p r e d i c t s z e r o v a l u e s f o r any of t h e e. , . . . , e then t h e

c o r r e s p o n d i n g n would b e c h o s e n t o b e n o n - z e r o , but s m a l l in r e l a t i o n t o t h e o t h e r c o m p o n e n t s . T h e v a l u e s of R M, defining A m a y t h e n be used to e v a l u a t e T , A . „ B . . and é. It will b e o b s e r v e d that u, , . . . . ,M. c a n

50 go 1 4

a l w a y s b e chosen so that T-»^1 + k and ^-»0 a s | g - l | - > 0 .

T h e s i m u l t a n e o u s i n e q u a l i t i e s (6.26) and ( 6 . 2 7 ) b e c o m e

and

I g j ö + i l k g l j g ^ - l\{.4> + S T ^ Ó + 3 T P 6 + p^6] ^ 1 4 . 6 4 1 p . ( 9 . 2 )

With g fixed, t h e s e i n e q u a l i t i e s will yield the s m a l l e s t value of 6 if they a r e t a k e n to b e e q u a t i o n s . T h u s

25 Ó = 1 4 . 6 4 1 p o r

p = 1.7075 6 ( 9 . 3 )

T a k i n g ( 9 . 1 ) to b e an equation and s u b s t i t u t i n g for p f r o m ( 9 . 3 ) t h e n gives

1.7075^0^ + 5. 1 2 2 5 T 6 ^ + { 3 ( | g J - 2 5 ) / | k311 g^^ - l | +3r^)6 + <l)=0 ( 9 . 4 )

T h e c o n t r a c t i o n condition ( 6 . 3 3 ) i s

I g J + IkgUr +P)^|gi - l| < 2 5 ,

which, a l t e r n a t i v e l y , m a y be w r i t t e n

IgJ - 25 + | k 3 | k ^ ( T + p ) ^ | g i - ll < 0 , ( 9 . 5 )

w h e r e k > 1, T h i s p a r a m e t e r k m a y be looked upon a s c o n t r o l l i n g the amount of c o n t r a c t i o n of t h e m a p p i n g T . F o r given v a l u e s of k , T and 6 t h e l a r g e s t a s s o c i a t e d v a l u e of | g . - I j will b e obtained by t a k i n g ( 9 . 5 ) to b e a n e q u a t i o n . Thus | g - l | m a y t a k e a l l v a l u e s in t h e i n t e r v a l f r o m z e r o to

Igl - l | = - ( | g j - 2 5 ) / | k 3 | k ^ ( T + p ) 2 . ( 9 . 6 )

It i s now r e q u i r e d t o d e t e r m i n e 6 f r o m ( 9 . 4 ) u n d e r t h e conditions t h a t

k , T and tfi a r e fixed and Ig. - l | h a s i t s g r e a t e s t v a l u e . T h i s c o r r e s p o n d s t o r e - w r i t i n g ( 9 . 6 ) a s

( I g J - 25)/|k3||gi - l | = -k4(T +P)^

and s u b s t i t u t i n g into (9.4), giving

1 . 7 0 7 5 ^ 1 - 3k^)63 + 3 x 1.7075(1 - 2k^)T62 + 3(1 - \ii^)T^6 + 0 » 0 ( 9 . 7 )

3 2

Since k > 1 and T > 1 + k. > 0, t h e coefficients of 6 , 6 and ó in ( 9 . 7 ) are a l l n e g a t i v e , whilst 0 > 0. It follows that t h e equation h a s only one p o s i t i v e r e a l r o o t . (Note. T h e r e l e v a n t v a l u e of 6 m u s t , by definition, b e p o s i t i v e . )

46

T h e v a l u e of t h i s r o o t can a l w a y s b e m a d e a s s m a l l a s d e s i r e d by t a k i n g k sufficiently l a r g e ; t h e a s s o c i a t e d g r e a t e s t v a l u e of | g . - l | w i l l , h o w e v e r , b e r e d u c e d .

T h e v a l u e of 6 obtained f r o m ( 9 . 7 ) can b e used to e v a l u a t e M = 256, l l j g l - 13^31' 10^1 - «*iol' 1^3 " ^^301 ^ ^ ° " ^ *^® r e l a t i o n s given in Section 7 . F r o m t h e s e c a l c u l a t i o n s will e m e r g e e x p r e s s i o n s of t h e f o r m

I"! " "lo'

^ C , 6 l a [ 3 ( 9 . 8 ) e and l'*3 " ""30 ^ C „ 6 | a I ^, ( 9 . 9 )

w h e r e C. and C h a v e finite u p p e r b o u n d s . Substitution into ( 4 . 1 6 ) will t h e n X «3 give | ( U , V ) - ( u , v ) | ^ I 2 ^ g 3 ( c j -f C^) ^ I j a J ^ Ó , But

'%iKl'4i^3iu.-'I

t h e r e f o r e 1| A.^2 . ^ 2 i | o r | ( U , V ) - (u,v)l ^ j | 2 ^ ( C ^ + C ^ ) ^ | | k 3 | | g j - l l l a j a g l b | ( U , V ) - ( u , v ) ( = j | k 3 | l 2 ^ ( c j + C 3 ) ^ l | a ^ | | g j - l | Ó ( 9 . 1 0 ) T h i s quantity can b e m a d e a s s m a l l a s d e s i r e d by m a k i n g |g - l | o r 6 sufficiently s m a l l . F o r t h e p r e s e n t p u r p o s e it i s sufficient to e s t i m a t e l u b l ( u , v ) - 0 f r o m ( 8 , 1 0 ) . Thus f r o m ( 8 . 8 ) l u b l ( u , v ) - ol > l u b l u I ,

w h e r e

lub U3 = lub I [(g^ - 9)e^ + (g^ - 1){ k 3 [ - 2 k ^ e i + (k^ - 1)62 + 2 (1 + k^)e_j]

+ k g G ^ i J a ^ l ( 9 . 1 1 )

When \g^ - l l ^ O , g l b | ( U , V ) - ( u , v ) | - ^ 0 , whilst

lublugl-^tg^ - 9[iU4ra^l.

w h e r e li > 0. It i s c l e a r that for | g - l| sufficiently s m a l l

g l b | ( U , V ) - ( u , v ) | < l u b j u g U l u b | ( u , v ) - 0 |

a n d , f r o m Section 4 , t h e d e s i r e d r e s u l t f o l l o w s .

It i s i m p o r t a n t to note that t h e inequality i m m e d i a t e l y above i s not s a t i s f i e d when a = 0 . T h i s condition c o r r e s p o n d s to g. = 1 . It follows that in t h e g. = 1 , g , F plane t h e a m p l i t u d e of t h e p u r e s u b h a r m o n i c , equation ( 2 . 5 ) , i s z e r o and t h e e x i s t e n c e of an a s s o c i a t e d s u b h a r m o n i c s o l u t i o n , equation ( 3 . 1 ) , h a s not been e s t a b l i s h e d .

1 0 . Regions of E x i s t e n c e in the g^g , F S p a c e

It h a s been shown in t h e p r e v i o u s s e c t i o n t h a t t h e r e i s a n a r r o w r e g i o n of t h e g,.g„, r s p a c e , g in the i n t e r v a l 1 - 7 < g , - $ 1 + 7 , 0 < 7 < < 1 , for which t h e solution ( 3 . 1 ) e x i s t s . T h e v a l u e of 7 defining t h i s r e g i o n , which m a y be looked upon a s t h e n u c l e u s of t h e c o m p l e t e r e g i o n of e x i s t e n c e , i s n o t , h o w e v e r , known. To d e t e r m i n e t h e c o m p l e t e r e g i o n would, a l m o s t c e r t a i n l y , r e q u i r e e x t e n s i v e n u m e r i c a l c o m p u t a t i o n . N e v e r t h e l e s s , it i s useful to obtain s o m e i n c o m p l e t e e s t i m a t e s for t h e r e g i o n , p a r t i c u l a r l y t h e s i z e of t h e g. I n t e r v a l and t h i s will now b e done f o r C a s e (1).

F r o m the l i n e a r i z e d second a p p r o x i m a t i o n obtained in Section 3 ,

^1 = -(gl - l ) ^ / 3 ( g i + 15), £3 = 2 e j , 62 = £4 = 0 ( 1 0 . 1 )

C o n s i d e r an i n t e r v a l 0 ^ g. 4 2, then from ( 1 0 . 1 )

- 48

C h o o s e K A* to b e

A^ = 1 + | x | / 4 0 , A^ = | x | / 2 0 , *^ = M4 = l x l / 1 0 0 ( 1 0 . 2 )

Take Ixl = 1/5, then

Ai = 1.005, A^ = 0.01 and t^j = ^\ = 0.002 (10.3)

Substituting these values of A<, t ...,^i. into the expressions for a, T , etc. gives

a = 0.7107, T = 1.019, T ^ = 1.038

\o

^ o

^ 0 = 0 = 0. = 0 007575 000075 000000 ^50 ^70 ^90 = 0.001575, = 0.000030, = 0.000000, (j, = 0.005471.

Using these values, equation (9.7) becomes

1.7075^(1 - 3 k J ó ^ + 5.2198(1 » 2k^)6^ + 3.114(1 - k^)6 -t- 0.005471 = 0

(10.4) It is now convenient to make the estimate for lub|(u,v) - o| . This information will then facilitate the choice of k which will yield a value of glbi(U,V) - (u,v)| which just satisfies the inequality (4.18). The maximum value of Ig - l| associated with this value of k will then be given by (9.6). The interval so defined can then be compared with the assumed interval 0 < g 42, and the smaller of these two intervals, which will be contained in the l a r g e r , will be the interval for which the existence theorem is valid. This can be maximized by iterating until the two intervals are the same.

Using the values of A*, I\ in (10.3) tables of corner values (v ) , . . . , ( u ) were prepared, and it was found that the value

•J c

min(u„) < {0..OI6 - 0.00406 |g, - ll} l a | (10.5) 3 c 1 e

was much larger than any of the other components. A value of

lub|(u,v) - o| > 0.011941a [, (10.6) e

obtained by putting | g - 11 = 1 in (10.5), may, therefore, be adopted. Now M = 256, therefore, from Section 7,

l^jgl = 0.0086206 , I3J3I = 0.37166 (2p + 6 T ) 6 ^ + 3I j I = 3.41506^ + 6.1146^ + 0.025866 and (2p + 6 T ) 6 ^ + 3I J3I = 3.41506^ + 6.1146^ + 1.11486 Thus, from (4.16), |(U,V) - (u,v)| < 2 ^ X I {(3.41506^ + 6.1146 + 0.0259)^ + (3.41506^ + 6.1146 + 1.1148)^}" Ig, - ll | a U (10.7) X e

Selecting a value 6 = 0.0055 and substituting into (10.7) gives |(U,V) - (u,v)l < 0 . 0 1 1 9 2 Ig^ - i l i a |

Now the greatest value of | g - H in this expression has to be consistent with the value assumed in arriving at the values of M. Mi ^ (10.2), i . e . |g - ll < 1.0. Thus

gib I (U,V) - (u,v)| = 0.011921a | < lub | (u,v) - o |

for |g^ - ll 4 I or 0 4 g^ 4 2.

The value of k. required to give 6 = 0.0055 may be obtained by substituting for 6 in (10.4) and solving for k , giving

k^ = 1.3 04 ₄

The extreme values of g associated with this value of k a r e given by (9.6), which becomes

5.52 Igj - ll - 25 -f IgJ = 0 (10.8)

Solving this equation by iteration then gives an associated interval for g of