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Green's functions for intial value problems Trumpler, Donald Alastair 1953

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GREEN'S FUNCTIONS FOR INITIAL VALUE PROBLEMS by DONALD ALASTAIR TRUMFLER  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE" DEGREE: OF MASTER OF ARTS i n the Department of Mathematics  We accept t h i s t h e s i s as conforming t o the standard r e q u i r e d  from candidates f o r the.  degree, o f MASTER OF ARTS'  Members, o f the Department o f Mathematics  THE UNIVERSITY: OF BRITISH COLUMBIA A p r i l , , 1953  Abstract A method i s g i v e n by which a - d i f f e r e n t i a l equation i n i t i a l conditions: can be converted  into- an i n t e g r a l  with  equation.  This, procedure; i s used t o derive; the M u l t i p l i c a t i o n Theorems f o r B e s s e l functions,, and t o o b t a i n an expansion o f t h e c o n f l u e n t hypergeometric f u n c t i o n i n terms, o f B e s s e l f u n c t i o n s . The method! is- adapted t o f i n d approximate eigenvalues: and eigenftmctions  o f bounded quantum mechanical problems,, and  to o b t a i n an approximate s o l u t i o n o f a non-linear- d i f f e r e n t i a l equation.  Acknowledgements; The author wishes t.o> express, h i s thanks t o B r . T.E. H u l l f o r suggesting  the topic, o f t h i s thesis; and f o r i n v a l u a b l e  a s s i s t a n c e i n i t s ; p r e p a r a t i o n , and a l s o t o Dr„ G.E. L a t t a f o r his: h e l p f u l suggestions.. He g l a d l y acknowledges his: i n d e b t e d ness t o the N a t i o n a l Res ear eh C o u n c i l o f Canada and to; t h e Department of Mathematics of the U n i v e r s i t y of B r i t i s h Columbia, for/ the f i n a n c i a l a s s i s t a n c e which made t h i s study p o s s i b l e .  Contents I n t r o d u c t i on Chapter 1  (i) The I n t e g r a l E q u a t i o n  (1)  1. D e r i v a t i o n of the I n t e g r a l E q u a t i o n  (1)  2. A. c o n v e n i e n t form f o r the; D i f f e r e n t i a l Equation Chapter. 2.  (3;)  E x a c t Solutions;  (5)  3;. General S o l u t i o n  (5)  4-.. M u l t i p l i c a t i o n Theorems f o r B e s s e l F u n c t i o n s  (8)  %  Expansion o f the C o n f l u e n t Hypergeometric Function  Chapter 3>  (11) Oh)  Approximate: S o l u t i o n s  6„ Bounded Quantum Mechanical Problems 7» Non-Linear E q u a t i o n s Bibliography  (I *) 1  -  (17) (19)  CD  Introduction There: are several, ways; In which a d i f f e r e n t i a l  equation  w i t h I n i t i a l , or. boundary c o n d i t i o n s , can be; transformed into; an i n t e g r a l e q u a t i o n . One a r i s e s from the use of Green's f u n c t i o n s for  boundary v a l u e problems. Another occurs: i n the* use  of  Laplace,, Mellin,, or other transforms,, i n which case, the; i n t e g r a l obtained i s an I n v e r s i o n or. c o n v o l u t i o n i n t e g r a l . The. method to, be- e x p l o i t e d here can be c a l l e d the method of Green's f u n c t i o n s f o r i n i t i a l v a l u e problems;. The i d e a was the; s p e c i a l  f i r s t used by L i o u v i l l e [ 8 J , who  considered  equation  as; a, non-homogeneous; d i f f e r e n t i a l equation, which c o u l d then be w r i t t e n  This, i n t e g r a l equation was  then used t o study t h e  behavior of the s o l u t i o n s f o r l a r g e j>.  asymptotic  Incidentally, this  was  t h e f i r s t appearance of an i n t e g r a l equation of the second k i n d . More r e c e n t authors, p r i n c i p a l l y Ikeda[5],. Fubini[3]> and Tricomi[l6,;17] ,> have- c o n s i d e r e d the problem r a t h e r as a comparison of the given equation w i t h a s i m i l a r one whose s o l u t i o n s a r e known. I t i s . this, l a t t e r p o i n t of view which w i l l be. adopted  here.  The; c o n v e r s i o n of a. d i f f e r e n t i a l system i n t o an i n t e g r a l e q u a t i o n i s a n a t u r a l procedure whenever the d i f f e r e n t i a l equation i s . not e a s i l y solved,, s i n c e i n t e g r a l equation  theory  (ii)  i s : the b a s i s for. most approximation methods. Even i n c a s e s where e x a c t s o l u t i o n s can be obtained,, the i n t e g r a l e q u a t i o n w i l l  lead  t o an expansion o f the solutions- i n terms of other,, b e t t e r known functions.  (1)  Chapter' 1 The; I n t e g r a l E q u a t i o n fn  t h i s chapter,; the procedure for. c o n v e r t i n g a d i f f e r e n t i a l  system Into- an i n t e g r a l e q u a t i o n i s d e r i v e d , and c e r t a i n g e n e r a l r e s u l t s are. o b t a i n e d which w i l l  other  be. o f use i n l a t e r  chapters:. The method o f d e r i v a t i o n u s i n g o p e r a t o r s  ( e f f e c t i v e l y the  method o f Ikeda J5J) i s . c e r t a i n l y n o t t h e o n l y one p o s s i b l e . The i n t e g r a l e q u a t i o n can a l s o be d e r i v e d by v a r i a t i o n o f parameters (see F u b i n i [3] , T r i c o m i [16,17) , Sv/anson [13) ) and, i n some cases, by L a p l a c e transform methods. The method used here was chosen because i t a l s o o f f e r s a means o f e v a l u a t i n g various, i n t e g r a l s which a r i s e i n the s o l u t i o n o f p a r t i c u l a r 1. D e r i v a t i o n o f the; I n t e g r a l  problems.  Equation  Let  be a second  order, l i n e a r d i f f e r e n t i a l o p e r a t o r whose c o e f f i c -  i e n t s ftC*.^. S&)  V  i n an i n t e r v a l (<*-J4-) except f o r  a r e continuous  p o s s i b l e s i n g u l a r i t i e s a t n--o- Jr  , w i t h the s i n g u l a r i t y a t  i  a r e g u l a r one. The e q u a t i o n L  (1.2)  will  i  c  ^  ^  =  0  then have two l i n e a r l y independent  which w i l l  be continuous  i n t e g r a l s /*7<*J , AT^CVL.)  i n the; open i n t e r v a l  C&-j4-) • I n a l l  cases c o n s i d e r e d , one o f the s o l u t i o n s , say nr, , can be chosen to  remain f i n i t e a t x = c t .  R  (2)  Let  , defined by  (1.3)  /  ^  ( a 0  ^_^  M / / 7  /<x> r  ^  >  be; a l i n e a r i n t e g r a l operator; here and i n what follows o- must be replaced by cx*£  i f :x=a. i s a s i n g u l a r i t y . The expression  /^ist)/»^^>--i^c*>^''«-;  i s the Wronskian of the solutions AT, , AT  /  /  l  U  and w i l l be denoted WO*) • Theorem 1 . 1 (l.lf)  =  &)-*p,/i*Cz)  For the proof, integrate appropriate terms by parts, and use ( 1 . 2 ) to simplify the r e s u l t s . Let the equation to be solved be  with given i n i t i a l values:  u.c^),  The; comparison equation same s i n g u l a r i t i e s as  ufca.).  ( 1 . 2 ) w i l l be chosen to have the  ( 1 . 5 ) .  ( 1 . 5 ) may be; rewritten as L^UC^c) = [fZCx)-PC*)]j^~f{SL^)~ Operate on both sides with M^, (1.6)  U  S  W  ^  (  «  QWJa  and apply  (1.4-)  to obtain  . /«!S^asS{t«-«^«-M«JA.  Integrate one term by parts, and c o l l e c t terms to obtain „2-  (1.7) where (1.8)  UC*)**ft4riC*)+.liA&(Z) +  KC-»;. 2>x^  £j<»-«oo?  Jft£t,2)UOc)o^  ^saizssisa vvo.)  J  (3)  and  , W(<x)  (1.9)  Thus any  s o l u t i o n of (1.5)  The converse w i l l now Theorem  be  i s a l s o a s o l u t i o n .,of (1.7).  shown.  1.2  L n j-t~)~ic*)  (1.10)  z  z  F o r the proof, d i f f e r e n t i a t e under the; i n t e g r a l s i g n , and use  (1.2)  to  simplify.  L e t UC2) be any  s o l u t i o n of (1.7)  ( o r o f (1.6)  since  (1,6)  i s e q u i v a l e n t t o ( 1 . 7 ) ) . Operate on both s i d e s of (1.6) w i t h and a p p l y  and  (1.10). Then  U.CZ) s a t i s f i e s  (1.5).  Thus; the d i f f e r e n t i a l system and are  the i n t e g r a l  form f o r the D i f f e r e n t i a l  The; expressions the constants ft6*0H. P&.).  ,y3  (1,8, 1.9)  Equation  f o r the k e r n e l  KCX,JS)  9  and f o r  a r e seen to s i m p l i f y c o n s i d e r a b l y i f  T h i s can always be arranged  by the f o l l o w i n g change  o f variable;: L e t the g i v e n equation  Put  equation  equivalent.  2... A, convenient  ( 2 , 1 )  L  be i n the form  + F^~j*+2fcO  O  2  Then  c*^  and; (2.1)  j-^cF-A^f-i/C'-aV*}  u  becomes;  a«"-o  (2.3)  where- <?(*.) i s a c o m p l i c a t e d f u n c t i o n of ?  ,«  ,  , and  their,  d e r i v a t i v e s . I n t h i s ; case;  C2-JO  [ S « - ^ ]  (2.5)  J  *  Our' equations w i l l always be taken i n the form t h a t (2.4„  2.5)  r e p l a c e (1.8,  1.9).  A p a r t i c u l a r case occurs i f P L ^ ^ o e a s i l y be shown t h a t  (2.6) (2.7)  WC*-)=C '  *  L  *  9  C  and  « ^ f & i l *  from which  <  •  (2.3),  1  / i^^jj  „ Then i t can  so  (5)  Chapter 2 Exact S o l u t i o n s The i n t e g r a l , equation w i l l be s o l v e d by/ the method o f s u c c e s s i v e s u b s t i t u t i o n s * Since; the k e r n e l I s bounded i n the; range o f i n t e g r a t i o n , t h e Liouville-Neumann  theorem[9}assures  the a b s o l u t e and uniform convergence o f t h e r e s u l t i n g  series  t o the; unique s o l u t i o n * In this: chapter, a g e n e r a l s o l u t i o n i s obtained, and two examples from the theory o f s p e c i a l f u n c t i o n s a r e t r e a t e d . The method I s a p p l i c a b l e t o many other problems o f a s i m i l a r nature, but I n most o f these,- there i s a p r a c t i c a l d i f f i c u l t y i n evaluating Integrals. 3- General S o l u t i o n The; comparison o f two. d i f f e r e n t i a l , equations has l e d t o the i n t e g r a l equation  (1*7)» "Which can now b e r e w r i t t e n i n t h e  form  (3.1)  U(z) = M W ^  f  e  ^ ^  (iSto-Qtylue*.)')  To s o l v e , we a p p l y the method of s u c c e s s i v e s u b s t i t u t i o n s *  Let  u' (a) 0>  a  and so on f o r h i g h e r  approximations*.  (6)  Thus -we a r e l e d t o c o n s i d e r two sequences of functions: {Get*)} d e f i n e d hy Ac^!(^t  M (f5*>-«G.)f 2  w h e r e ; a r e  ^ . Fc+, (*)  constants determined  as f o l l o w s :  Operating on both s i d e s o f (3.2) w i t h a n d  u s i n g (1.10),.  we o b t a i n  Z. T h e ^ ^ ; , >t  £7  a  a r e determined from the p a r t i c u l a r i n t e g r a l , o f a  non-homogeneous l i n e a r d i f f e r e n t i a l equation, fii, & , are: c o n s t a n t s determined  , Di„  from  S o l v i n g the r e s u l t a n t a l g e b r a i c equations, we o b t a i n  (3-3)  f  and s i m i l a r e x p r e s s i o n s f o r d o f J*i  9  Oc  but w i t h X ^ - M i n place;  •  Thus, i f t h e v a r i o u s elements of the terms o f the s e r i e s are: rearranged, the s o l u t i o n w i l l be o f the form (3»  w F-^^* a  %°%l cG  The exact s e r i e s w i l l be obtained f o r m a l l y , and each p a r t i c u l a r case must be examined f o r convergence..  (7)  We c o n s i d e r the two s e r i e s o f (3. !-) s e p a r a t e l y ,  and show  1  t h a t , w i t h suitable; i n i t i a l v a l u e s , each s e r i e s w i l l (3.1). L e t u,c*>- %-p< /\-C»)  satisfy  jfc Cj^(s) .  vUiQi^j?.  4 . ' " 0 > J  F i r s t , l e t the i n i t i a l v a l u e s be u.c<o =• u., co-j , u'c*-l=-«/co.;. Substitute  U/C2) i n t o (3.1), and use (3*2.) t o o b t a i n  (3.5)  ^p F {.i)*UAr,a)ifir ix\+^?iKF +S*U^ L  c  x  c  This; equation i s s a t i s f i e d i f (3.6.) _  "pc^t-y^iji L _ ^  Pif  ~  \I A*  and  from which  p-  MA.  o  i f the terms i n /tr cz) v a n i s h ; but x  V 5 / J  " ,  —^771)  £22.£  1  fr^i-  (  pc„ FJ,^)  f  r  o  as r e q u i r e d .  ( 3  (from  wo*-) * -  m  3 ) )  (3.6))  '  (from (2.5) & i n i t i a l  O  -  values)  Similarly  JLfcAi^Jo  (3-7)  and we can choose "p.* /  without l o s s o f g e n e r a l i t y . above, u o&)  Next, l e t t h e : - i n i t i a l v a l u e s be u c*), uJ&j.-As x  x  is; a s o l u t i o n i f  (3.8)  fe+."T TA r  ,  f>»/  Now l e t a r b i t r a r y i n i t i a l c o n d i t i o n s  be w r i t t e n I n the  form u.ca)sp*y.,c*w ^. u ca), u/feo =p,«.>-) + p.«a'at), where p„, ^ 4  d  , are  a r b i t r a r y constants;. I n this, case; (3. *-) is; a s o l u t i o n i f the Y J 1  satisfy (3.6),  (3.8) r e s p e c t i v e l y .  c  (3.^) i s then a. s o l u t i o n  of ( 3 . 1 ) s a t i s f y i n g a r b i t r a r y I n i t i a l c o n d i t i o n s , s o l u t i o n s a r e o f t h i s form, w i t h appropriate;  so a l l  p© , <^<> .  r  (8)  4. M u l t i p l i c a t i o n Theorems f o r B e s s e l  Functions  As an example of the/ procedure o f t h e p r e v i o u s  section  we o b t a i n the; s o - c a l l e d M u l t i p l i c a t i o n Theorems f o r B e s s e l functions.. T h i s r e s u l t has been obtained by d i r e c t expansion methods [l8, p.14-2], and a l s o by the method used here by Ikeda.  but the l a t t e r d e r i v a t i o n c o n t a i n s  several errors.  L e t usi compare the equations  For.  , and n o t an i n t e g e r , we can take w,= J^Cx.)  r  v*- J . ^ t j ,  so t h a t the i n t e g r a l e q u a t i o n (3*1) becomes  We f i n d  that  u s i n g a w e l l known r e c u r r e n c e r e l a t i o n [ l 8 , p.4-5^. Thus fi(z)~  (4.4)  Z J>,+iCz) c  { and  Similarly  2(± + >)  Therefore,  (4.5)  and  by  I  (3.6,,, 3.8)  '  7  V  (>2-)= f>*U,C2)+ ^ U ^ 3 )  (9)  F O E ; s m a l l 2,.  s.o> t h a t  ^ = o 6  We  now  ,  p . *• V  and  show t h a t the s e r i e s (4.6)  converges:.. From  Watson [l8, p.,44/ ;  If/''  ) Jj,(*>)^  (4.7)  r  ^  l ?  6 where H for  -^fl^r  for  M iir^  itr-z  i s , an upper, bound f o r a l l I 9cI . T h i s s e r i e s converges  a l l finite; H  independently  Since; the c o e f f i c i e n t s ; i n (4.6)  of.(provided  only  v->o)^  are; c e r t a i n l y bounded, t h e  s e r i e s i s a b s o l u t e l y and u n i f o r m l y convergent f o r a l l 2- i n any finite  interval. The f u n c t i o n s  (4.:6;) i s  J > ( 2 ) are: continuous; i n V , and the; s e r i e s  i s a l s o v a l i d f o r integer, v a l u e s of V We  (4.6)  u n i f o r m l y convergent w i t h r e s p e c t to V , so t h a t .  a l s o have;  Compare c o e f f i c i e n t s , of  ; s i n c e v i s not an i n t e g e r ,  does; not appear i n e i t h e r ' X „ Lte)  Near: "2 = © ,  1  or  u 0/. 2  %  v  (10)  Thus  V We  see t h a t the: series? w-a. does not  unless  converge; near 2-=  / ; b u t since,, under t h i s r e s t r i c t i o n ,  o  the. s e r i e s  does c o n v e r g e - f o r a r b i t r a r i l y small- *  y  finite: z  i n v o l v e d a rearrangement  » T h i s e v a l u a t i o n of ^  o f terms of power, series,; but Weierstrass  ihjs) We  double-series  ••.  <x> ( * » >  now  consider  C e r t a i n l y , for, v  d  has  i t w i l l converge f o r  t h i s i s j u s t i f i e d by  p.83\J)  theorem (see, e..g», [7>  y £  ^  W  L*y*.*-c  r  11 - ^ i <\  <*).  the case of i n t e g e r v a l u e s  not an  the  of v ,  integer  V7T  „ -yjr  (<  ~*^ ( f / " C  ) C  J  "^^ " ^ t  <  ^  L e t t i n g y-*v,  on both s i d e s , we  (We J  If  obtain  T h i s l a s t expansion i s a g a i n v a l i d o n l y for. \i~V-\ <  any  (II)  5*. Expansion  of t h e C o n f l u e n t Hypergeometric F u n c t i o n  In the: c o n f l u e n t hypergeometric.  jfr  (5.D  <  *  *  equation  J  *~  put: (5.2)  V-^»w ,  ac = >t , a  The e q u a t i o n then becomes: (5.3)  ol* 2  *  C  x  We; compare t h i s w i t h  (5J+)  ~  c  whose s o l u t i o n s  are  ,. /vi? X / * )  I n t e g e r . The i n t e g r a l equation  (5.5)  u  ~ *  ;  >;  f o r V;>o, n o t an  becomes  <*>^» 01,)•+ ^» f jx^-)JlZ(*>€ oo - TJ#<r, c~)j3c»c*) oV v  As i n the p r e v i o u s  section,;  so:> we a g a i n c o n s i d e r the s e r i e s  r  u.,  J L  p";^«'  J»+ (*) c  By a r g u i n g e x a c t l y as b e f o r e ,  (5.7)  J  pt  ** ~ > ( " ~ ^ T p: -  i  7  p >  °  I s a s o l u t i o n o f (5.5) I f  " f" 5 c ^ o ) • p,  (12)  and u (s) i s a. s o l u t i o n i f a  —, ,  I  (5.8)  _  -  i  v+\  The: s o l u t i o n s of. (5.1) a r e £19, p-337j|  ^ a ) , ft^ & )  when 2^=.!/ i s n o t an i n t e g e r . A f t e r the changes these s o l u t i o n s a r e M^  ?y  (5.2) a r e made,,  (^-) ,, ^ , - v ^ »  Thus, we have F O E s m a l l 2 ,, so; t h a t  (5.9)  ^  o  .•.  ,  ^ -  2  y  ^  * A"* .rr(>f) <t jc  E x c e p t f o r a. f a c t o r  ^T*-"  >  <r . cz) M  £  f o r l a r g e ^ „ from (5»7)y  so; t h e c o e f f i c i e n t s are- bounded and the s e r i e s converges as b e f o r e . The f a c t o r i n /\ can be: absorbed i n t o the power o f 2 , We a l s o have: A g a i n by comparing c o e f f i c i e n t s , o f 2 % ^ , = 0 ,. F i n a l l y  ^  0  is  determined from  (5.10): but  we cannot e v a l u a t e p„  know the; p  explicitly.  i n a c l o s e d form since/ we do n o t  T h i s e v a l u a t i o n w i l l also; g i v e the  c o n d i t i o n on TV. t o ensure the convergence o f the With s u i t a b l e : changes  series.  o f v a r i a b l e : and c h o i c e s o f the  parameters,; i t i s p o s s i b l e t o o b t a i n expansions o f L a g u e r r e polynomials,; Hermite polynomials,, e t c . , t h e s e being o n l y s p e c i a l cases o f the above r e s u l t s .  (13) Expansions: o f t h i s type have:, two u s e s : (1) T h e i r v a l u e as a computational a i d has: been discussed: by Karlin[6],, who states, that,,, i n some cases, a. few terms o f a s e r i e s : o f the above; form g i v e an accuracy obtainable: o n l y from s e v e r a l hundred  terms o f the power s e r i e s .  (2) They give; a- convenient, form f o r the study o f the asymptotic behavior o f t h e sum f u n c t i o n by u s i n g the known asymptotic ;  form for: t h e B e s s e l functions.. T r i c o m i [ l ^ , 15> 16,l?] has used s i m i l a r s e r i e s t o t h i s  purpose.  CI*)  Chapter  %  Approximate S o l u t i o n s We  w i l l now  c o n s i d e r equations  f o r which i t i s not  p o s s i b l e t o f i n d t h e form of t h e L1ouville-Neumann s e r i e s . The o n l y cases c o n s i d e r e d w i l l be those f o r which the; d i f f e r e n c e between t h e given equation and  the comparison  equation i s s m a l l , so t h a t o n l y a., few terms of the: s o l u t i o n ar.e needed. The- examples used below are chosen because of t h e i r i n t e r e s t to t h e o r e t i c a l physicists;. 6» Bounded Quantum Mechanical I n the past  15  Problems  years,; a number of authors [ l , , * , 10,.11,12] 1  have c o n s i d e r e d t h e behavior, o f a quantum mechanical system c o n f i n e d In an e n c l o s u r e . The u s u a l d i f f e r e n t i a l  equations:  s t i l l a p p l y * but the; boundary c o n d i t i o n s ar.e not the normal ones.. The o n l y g e n e r a l method, which i s a p p l i c a b l e t o more than one  of t h e s e problems, i s a g r a p h i c a l one [ i l l . We  a p p l y the method developed  above t o o b t a i n approximate a n a l y t i c  expressions; f o r t h e e i g e n f u n c t i o n s and eigenvalues eigenvalue  will  of a g e n e r a l  problem.  The unnatural, boundary c o n d i t i o n i n each case; i s t h a t the; wave f u n c t i o n v a n i s h on the f i n i t e s u r f a c e bounding the r e g i o n i n which the; system i s e n c l o s e d . We Schrodinger cr,,.. a-x  Tl  assume t h a t the  equation i s seperable i n c u r v i l i n e a r :  = t j , such t h a t the f i n i t e  s e p a r a t i o n and  s o l u t i o n o f the ^  coordinates  surface i s ;x, = C . A f t e r and ^  3  p a r t s of the wave;  (15)  equation,, t h e r e remains an e q u a t i o n o f the: form  I f the s i n g u l a r i t i e s a r e a t * ' ^ , ^ ,  t h e normal boundary  c o n d i t i o n s : a r e t h a t u.t<*-) , IA.C-6) be. a t l e a s t f i n i t e . I n t h i s case the e i g e n f u n c t i o n s are,  ;  say  AfiLx^  s  n\ W ^ ' ^ J " ] ^ * .  t  h  a  t  w i t h eigenvalues  X'X:>  C-6-3")  o  C*»A*,  )  We d e s i r e a s o l u t i o n o f (6.1) s u b j e c t t o the c o n d i t i o n s  U.CO,) f i n i t e ,  uto~c>  with  c  i n the i n t e r v a l ( A , ^ ) . The  i n t e g r a l equation i s : then  (6.4) MM  wfcxhrJ*  (xrl  I n the cases: considered,, the c o n d i t i o n *UA-) f i n i t e givesys Put  *  (6.5)  u. >(»W  ^  6  '  ^  j  j  ^  ^  f  ^  ^  The c o n d i t i o n u.u)* o g i v e s , i n general  (6.6)  >=  - »  -c  '  a.  *-  By s i m i l a r means h i g h e r approximations The constant oL i s determined  may be o b t a i n e d .  by the n o r m a l i z a t i o n .  The problem o f f i n d i n g the eigenvalues has thus been reduced  t o one o f q u a d r a t u r e s . Even i f these: cannot be; c a r r i e d  out e x p l i c i t l y , ; the problem o f e v a l u a t i n g i n t e g r a l s n u m e r i c a l l y i s much e a s i e r than t h a t o f o b t a i n i n g eigenvalues: from a d i f f e r e n t i a l e q u a t i o n by n u m e r i c a l means.  (16) As a. s p e c i f i c example, o f the above method, we the of  consider  problem o f a bounded hydrogen atom. The p h y s i c a l importance t h i s problem was  discussed  by Michels,, de Boer and  Bijl[loJ,  and s o l u t i o n s were o b t a i n e d by them and extended by de Groot and ten S e l dam [4-], The= bounding s u r f a c e i s the sphere » » C . The r a d i a l of  part  the e q u a t i o n i s then . X*-  (6.7) with; the u s u a l  1 t  A)"'~  ( / a n i n t e g e r >/o )  c >  solution  ~ ~ "H*-  and e i g e n v a l u e s  (*\ an i n t e g e r I )  c o r r e s p o n d i n g t o the boundary c o n d i t i o n s as  f i n i t e , ; ui*) —» ©  . For  / , J~o,  =•  *  (6.8)  i  f5  The; n u m e r i c a l v a l u e s l i s t e d below were c a l c u l a t e d u s i n g the  summation procedure g i v e n by de Groot and ten B e l d a m ^ ] ,  and t h e i r v a l u e s a r e quoted f o r comparison. The u n i t o f length i s — ~  ,, of energy  —-p,_  .  (17)  TV  — . i tr  3 X*' - 9<V (a)  ¥  - . 96 »c) — . 99 3 0 )  (,.  -  For. v> - £ „  . 79  <f  o „ £/cxj = «T * * c * - * ;  (6.10) \  I For  x° - - - -+ {  ^ -J^.  w a „ ^= / ,, :  tf,/c~>«  ^  (6.11)  S i m i l a r e x p r e s s i o n s can be o b t a i n e d f o r other v a l u e s of the  parameters.  7. Non-Linear  Equations  In most problems l e a d i n g t o n o n - l i n e a r d i f f e r e n t i a l equations we  seek p e r i o d i c s o l u t i o n s (or l i m i t c y c l e s ) . Such  equations can be transformed i n t o i n t e g r a l equations, but the; most convenient method i s t o use a Green's f u n c t i o n t o ensure the p e r i o d i c i t y of the s o l u t i o n . Such i n t e g r a l e q u a t i o n s are not of the type c o n s i d e r e d h e r e . There a r e , however, some nonl i n e a r problems where l i m i t c y c l e s are not r e q u i r e d , and method can be a p p l i e d t o these. One example i s g i v e n .  our  (18)  One o f the experimental checks o f the g e n e r a l theory of  r e l a t i v i t y i s the measurement  of  l i g h t p a s s i n g near the sun. I n s o l v i n g the f i e l d  for  o f the d e f l e c t i o n of a: team equations  t h i s case,, i t . i s n e c e s s a r y t o s o l v e [2, Ch. 7, a l s o p..2.07 J  (7.1)  d**  w i t h the; i n i t i a l c o n d i t i o n s  (7.2)  LLC*)  -  Comparison of- (7.1)  ,  -O  m  with  (7.3)  J^*"'*  lea#s. t o t h e i n t e g r a l (7.1+)  (AC2-) -  Put; u. " C'0» t  (7.5)  VU'(*>  5  ,  0 1 1  equation  ^ ^ ~  h  3 v y x  j  ^i^-Z)U, L^cl^ X  s i m p l i f i c a t i o n we o b t a i n '  ^ <:ar)' tr>  which i s ; the;; r e q u i r e d f i r s t  approximation.  (19)  Bibliography 1., F.C. Auluck and: D.S. Kothari, Proc. Camb. P h i l . S o c ,  41,, 175 (1945). 2'.. A.S. Eddington, Space. Time, and Gravitation., Cambridge. University Press (1923). 3> P.. Fubini, Rend. L i n c e i (6), . 26, 253 (1937). ;  4. S.R. de Groot and C.A. ten Seldam, Physica, 12, 669 (1946). 5. I . Ikeda, Math. Z e i t . , 22, 16 (1925). 6. M. K a r l i n , J . Math. Phys., 28,, 43 (1949). 7. K. Knopp, Theory of Functions I, Dover Publications (1945). a.. J.. L i o u v i l l e , J . de Math., 2, 24 (1837). 9. W.V. Lovltt,, Linear Integral Equations. McGraw-Hill (1924). 10.. A. Miehels, J . de Boer and A. B i j l , Physica, 4, 98l (1937). 11. A. Sommerfeld and Hi. Hartmann, Ann. Physik (5), 2Z, 333 (1940). 12.. A. Sommerfeld and H. Welker, Ann. Physik (5), 32, 56 (193.8). 13. C.A. Swanson, M.A. Thesis, University of B.C. (1953). l4„ F., Tricomi, Glorn. 1st. I t a l . Attuari, 12, 14, (1942). ;  15. F. Tricomi, Ann. Mat. Pur., App. (4),. 26, l 4 l (194.7). 16.. F» Tricomi,. Univ. e. Politecnico Torino. Rend. Sean.. Mat.,  8, 7 (1949). 17. F. Tricomi, Comment. Math. Helv., 22,. 150 (1949). 18. G.N. Watson, Theory of Bessel Functions. Cambridge University Press, 2nd E d i t i o n (1952). 19. E.T. Whit taker, and G.N. Watson, Modern Analysis. Cambridge University Press, 4th E d i t i o n (1952).  

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