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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 to the standard required from candidates f o r the. degree, of MASTER OF ARTS' Members, of the Department of Mathematics THE UNIVERSITY: OF BRITISH COLUMBIA April,, 1953 Abstract A method i s given by which a - d i f f e r e n t i a l equation with i n i t i a l conditions: can be converted into- an i n t e g r a l equation. This, procedure; i s used to derive; the M u l t i p l i c a t i o n Theorems for Bessel functions,, and to obtain an expansion of the confluent hypergeometric function i n terms, of Bessel functions. The method! is- adapted to f i n d approximate eigenvalues: and eigenftmctions of bounded quantum mechanical problems,, and to obtain an approximate solution of 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 to Br. T.E. H u l l f o r suggesting the topic, of t h i s thesis; and f o r invaluable assistance i n it s ; preparation, and also to Dr„ G.E. Latta f o r his: h e l p f u l suggestions.. He gladly acknowledges his: indebted-ness to the National Res ear eh Council of Canada and to; the Department of Mathematics of the University of B r i t i s h Columbia, for/ the f i n a n c i a l assistance which made t h i s study possible. Contents Introducti on ( i ) Chapter 1 The Integral Equation (1) 1. Derivation of the Integral Equation (1) 2. A. convenient form for the; D i f f e r e n t i a l Equation (3;) Chapter. 2. Exact Solutions; (5) 3;. General Solution (5) 4-.. M u l t i p l i c a t i o n Theorems f o r Bessel Functions (8) % Expansion of the Confluent Hypergeometric Function (11) Chapter 3> Approximate: Solutions Oh) 6„ Bounded Quantum Mechanical Problems (I 1*) 7» Non-Linear Equations - (17) Bibliography (19) CD Introduction There: are several, ways; In which a d i f f e r e n t i a l equation with I n i t i a l , or. boundary conditions, can be; transformed into; an i n t e g r a l equation. One a r i s e s from the use of Green's functions f o r boundary value 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 Inversion or. convolution i n t e g r a l . The. method to, be- exploited here can be c a l l e d the method of Green's functions f o r i n i t i a l value problems;. The idea was f i r s t used by L i o u v i l l e [ 8 J , who considered the; s p e c i a l equation as; a, non-homogeneous; d i f f e r e n t i a l equation, which could then be written This, i n t e g r a l equation was then used to study the asymptotic behavior of the solutions f o r large j>. Incidentally, t h i s was the f i r s t appearance of an i n t e g r a l equation of the second kind. comparison of the given equation with a similar one whose solutions are known. I t is. this, l a t t e r point of view which w i l l be. adopted here. The; conversion 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 equation i s a natural procedure whenever the d i f f e r e n t i a l equation is. not e a s i l y solved,, since i n t e g r a l equation theory More recent authors, p r i n c i p a l l y Ikeda[5],. Fubini[3]> and Tricomi[l6,;17] ,> have- considered the problem rather as a ( i i ) is: the basis for. most approximation methods. Even i n cases where exact solutions can be obtained,, the i n t e g r a l equation w i l l lead to an expansion of the solutions- i n terms of other,, better known functions. (1) Chapter' 1 The; Integral Equation fn t h i s chapter,; the procedure for. converting a d i f f e r e n t i a l system Into- an i n t e g r a l equation i s derived, and certa i n other general r e s u l t s are. obtained which w i l l be. of use i n l a t e r chapters:. The method of derivation using operators ( e f f e c t i v e l y the method of Ikeda J5J) is. c e r t a i n l y not the only one possible. The i n t e g r a l equation can also be derived by v a r i a t i o n of parameters (see Fubini [3] , Tricomi [16,17) , Sv/anson [13) ) and, i n some cases, by Laplace transform methods. The method used here was chosen because i t also offers a means of evaluating various, inte g r a l s which a r i s e i n the solution of p a r t i c u l a r problems. 1. Derivation of the; Integral Equation L e t be a second order, l i n e a r d i f f e r e n t i a l operator whose c o e f f i c -i e n t s ftC*.^. S&)V are continuous i n an i n t e r v a l (<*-J4-) except f o r possible s i n g u l a r i t i e s at n--o-iJr , with the s i n g u l a r i t y at a regular one. The equation (1.2) L i c ^ ^ =  0 w i l l then have two l i n e a r l y independent i n t e g r a l s /*7<*J , AT^CVL.)R which w i l l be continuous i n the; open i n t e r v a l C&-j4-) • In a l l cases considered, one of the solutions, say nr, , can be chosen to remain f i n i t e at x=ct. ( 2 ) Let , defined by ( 1 . 3 ) / ^ ( a 0 ^ _ ^ M / / 7 / < x > r ^ > be; a linear integral operator; here and in what follows o- must be replaced by cx*£ i f :x=a. i s a singularity. The expression /^ ist)/»^ / >^--/i^ lc*>^ ''«-; i s the Wronskian of the solutions AT, , ATU 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 results. Let the equation to be solved be with given i n i t i a l values: u.c^), ufca.). The; comparison equation ( 1 . 2 ) w i l l be chosen to have the same singularities as ( 1 . 5 ) . ( 1 . 5 ) may be; rewritten as L^UC^c) = [fZCx)-PC*)]j^~f{SL^)~ Q W J a Operate on both sides with M^, and apply ( 1 . 4 - ) to obtain ( 1 . 6 ) U S W ^ ( « . / « ! S ^ a s S { t « - « ^ « - M « J A . Integrate one term by parts, and collect terms to obtain „2-( 1 . 7 ) UC*)**ft4riC*)+.liA&(Z) + Jft£t,2)UOc)o^ where ( 1 . 8 ) K C - » ; . £j<»-«oo? ^ s a i z s s i s a 2>x^ vvo.) J (3) and , W(<x) (1.9) Thus any solution of (1.5) i s also a solution .,of (1.7). The converse w i l l now be shown. Theorem 1.2 (1.10) Lznzj-t~)~ic*) For 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 sign, and use (1.2) to s i m p l i f y . Let UC2) be any solution of (1.7) (or of (1.6) since (1,6) i s equivalent to (1.7)). Operate on both sides of (1.6) with L 2 and apply (1.10). Then and 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 the i n t e g r a l equation are equivalent. 2... A, convenient form f o r the D i f f e r e n t i a l Equation The; expressions (1,8, 1.9) f o r the kernel KCX,JS)9 and f o r the constants ,y3 are seen to si m p l i f y considerably i f ft6*0H. P&.). This can always be arranged by the following change of variable;: Let the given equation be i n the form ( 2 , 1 ) + F^~j*+2fcO O P u t Then c*^ u j - ^ c F - A ^ f - i / C ' - a V * } and; (2.1) becomes; (2.3) a « " - o where- <?(*.) i s a complicated function of ? , « , , and their, d erivatives. In this; case; C2-JO [ S « - ^ ] (2.5) J * Our' equations w i l l always be taken i n the form (2.3), so that (2.4„ 2.5) replace (1.8, 1.9). A p a r t i c u l a r case occurs i f P L ^ ^ o „ Then i t can e a s i l y be shown that (2.6) WC*-)=C and (2.7) ' * L * 9 C « ^ f & i l * from which < • 1 / i ^ ^ j j (5) Chapter 2 Exact Solutions The integral, equation w i l l be solved by/ the method of successive substitutions* Since; the kernel Is bounded i n the; range of integration, the Liouville-Neumann theorem[9}assures the absolute and uniform convergence of the r e s u l t i n g series to the; unique solution* In this: chapter, a general solution i s obtained, and two examples from the theory of spe c i a l functions are treated. The method Is applicable to many other problems of a sim i l a r nature, but In most of 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 Solution The; comparison of two. d i f f e r e n t i a l , equations has l e d to the i n t e g r a l equation (1*7)» "Which can now be rewritten i n the form (3.1) U(z) = M W ^ f e ^ ^ (iSto-Qtylue*.)') To solve, we apply the method of successive substitutions* Let u'0>(a)a and so on for higher approximations*. (6) Thus -we are l e d to consider two sequences of functions: {Get*)} defined hy M2(f5*>-«G.)f Ac^!(^t ^ . Fc+, (*) w h e r e ; a r e constants determined as follows: Operating on both sides of (3.2) w i t h a n d using (1.10),. we obtain Z.a £7 The^^; , >t are determined from the p a r t i c u l a r integral, of a non-homogeneous l i n e a r d i f f e r e n t i a l equation, fii, & , , Di„ are: constants determined from Solving the resultant algebraic equations, we obtain (3-3) f and si m i l a r expressions f o r d 9 Oc but with X ^ - M i n place; of J*i • Thus, i f the various elements of the terms of the series are: rearranged, the solution w i l l be of the form ( 3 » waF-^^* %°%lcG-The exact s e r i e s w i l l be obtained formally, and each p a r t i c u l a r case must be examined f o r convergence.. (7) We consider the two series of (3.1!-) separately, and show that, with suitable; i n i t i a l values, each series w i l l s a t i s f y (3.1). Let u,c*>- %-p< /\-C») vUiQi^j?. jfc Cj^(s) . 4.'"0> J F i r s t , l e t the i n i t i a l values be u.c<o =• u., co-j , u'c*-l=-«/co.;. Substitute U/C2) i n t o (3.1), and use (3*2.) to obtain (3.5) ^pLFc{.i)*UAr,a)ifirxix\+^?iKFc+S*U^ This; equation i s s a t i s f i e d i f (3.6.) "pc^t-y^iji from which _ L _ ^ Pif ~ \ I MA. p-A* o and i f the terms i n /trxcz) vanish; but V " — ^ 7 7 1 ) 1 ( f r o m ( 3 - 3 ) ) 5 / J , £ 2 2 . £ fr^i- pc„ FJ,^) (from (3.6)) wo*-) * ' - O (from (2.5) & i n i t i a l values) as required. S i m i l a r l y (3-7) JLfcAi^Jo and we can choose "p.* / without loss of generality. Next, l e t th e : - i n i t i a l values be uxc*), uJ&j.-As above, uxo&) is; a solution i f (3.8) fe+."TrTA , f>»/ Now l e t a r b i t r a r y i n i t i a l conditions be written In the form u.ca)sp*y.,c*w ^ . u4ca), u/feo =p,«.>-) + p.«a'at), where p„, ^ d , are ar b i t r a r y constants;. In this, case; (3.1*-) is; a solution i f the Yc J s a t i s f y (3 .6) , (3.8) respectively. (3.^) i s then a. solution 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 conditions, so a l l solutions are of thi s form, with appropriate; p© , <^<> . r (8) 4. M u l t i p l i c a t i o n Theorems f o r Bessel Functions As an example of the/ procedure of the previous section we obtain the; so-called M u l t i p l i c a t i o n Theorems f o r Bessel functions.. This r e s u l t has been obtained by d i r e c t expansion methods [l8, p.14-2], and also by the method used here by Ikeda. but the l a t t e r derivation contains several errors. Let usi compare the equations For. , and not an integer, we can take w,= J^Cx.) r v*- J . ^ t j , so that the i n t e g r a l equation (3*1) becomes We f i n d that using a well known recurrence relation[l8, p.4-5^. Thus fi(z)~ ZcJ>,+iCz) and (4.4) { S i m i l a r l y and 2(± + >) Therefore, by (3.6,,, 3.8) (4.5) I ' 7 V (>2-)= f>*U,C2)+ ^ U ^ 3 ) (9) F O E ; small 2,. s.o> that ^ 6 = o , p. *• V and We now show that the s e r i e s (4.6) converges:.. From Watson [l8, ; p.,44/ I f / ' ' (4.7) ) Jj,(*>)^ r ^ l ? -^fl^r f o r 6 M i i r ^ itr-z where H i s , an upper, bound f o r a l l I 9cI . This s e r i e s converges fo r a l l f i n i t e ; H independently o f . ( p r o v i d e d only v->o)^ Since; the coefficients; i n (4.6) are; c e r t a i n l y bounded, the series i s absolutely and uniformly convergent f o r a l l 2- i n any f i n i t e i n t e r v a l . The functions J > ( 2 ) are: continuous; i n V , and the; series (4.:6;) i s uniformly convergent with respect to V , so that (4.6) i s also v a l i d f o r integer, values of V . We also have; Compare coefficients, of ; since 1 v i s not an integer, % v does; not appear i n either' X„ Lte) or u 2 0 / . Near: "2 = © , (10) Thus V We see that the: series? w-a. does not converge; near 2-= o unless / ; but since,, under this r e s t r i c t i o n , the. series does converge-for a r b i t r a r i l y small- * y i t w i l l converge for any f i n i t e : z » This evaluation of ^ d has involved a rearrangement of terms of power, series,; but t h i s i s j u s t i f i e d by the Weierstrass double-series theorem (see, e..g», [7> p.83\J)r ihjs) ••. <x> ( * » > y £ ^ W L*y*.*-c <*). 11 - ^ i <\ We now consider the case of integer values of v , Certainly, for, v not an integer V7T „ - y j r ( <~*^ ( f / C " ) C J " ^ ^ t " < ^ ^ (We J If Letting y-*v, on both sides, we obtain This l a s t expansion i s again v a l i d only for. \i~V-\ < (II) 5*. Expansion of the Confluent Hypergeometric Function In the: confluent hypergeometric. equation ( 5 . D jfr < * * * ~ J put: (5.2) V-^»w , aca= >t , The equation then becomes: (5.3) ol*2- C * x We; compare t h i s with (5J+) c ~ ; whose solutions are ,. /vi? X / * ) > ; f o r V;>o, not an Integer. The i n t e g r a l equation becomes (5.5) u ~ * <*>^ » 01,)•+ ^» f jx^-)JlZ(*>€voo - TJ#<r, c~)j3c»c*) oV As i n the previous section,; so:> we again consider the se r i e s r u., J L p";^ «' J»+c(*) By arguing exactly as before, Is a solution of (5.5) I f (5.7) J p t** ~ > ( " ~ ^ T " f" 5c^o ) p : - i 7 p > ° • p , (12) and u a (s) i s a. solution i f ( 5 . 8 ) I — , , _ - i v+\ The: solutions of. (5.1) are £19, p-337j| ^ a) , ft^ & ) when 2^ =.!/ i s not an integer. After the changes (5.2) are made,, these solutions are M^?y (^-) ,, ^ , - v ^ » Thus, we have F O E small 2 ,, so; that ^ o , ^ - 2 y ^ > (5.9) .•. * A"* .rr(>f) <t jc <rM.£ cz) Except for a. factor ^T*-" f o r large ^ „ from (5»7)y so; the c o e f f i c i e n t s are- bounded and the series converges as before. The factor i n /\ can be: absorbed in t o the power of 2 , We also have: Again by comparing coefficients, of 2 % ^ , = 0 ,. F i n a l l y ^ 0 i s determined from (5.10): but we cannot evaluate p„ i n a closed form since/ we do not know the; p e x p l i c i t l y . This evaluation w i l l also; give the condition on TV. to ensure the convergence of the s e r i e s . With suitable: changes of variable: and choices of the parameters,; i t i s possible to obtain expansions of Laguerre polynomials,; Hermite polynomials,, etc., these being only special cases of the above r e s u l t s . (13) Expansions: of t h i s type have:, two uses: (1) Their value as a computational a i d has: been discussed: by Karlin[6],, who states, that,,, i n some cases, a. few terms of a series: of the above; form give an accuracy obtainable: only from several hundred terms of the power s e r i e s . (2) They give; a- convenient, form for the study of the asymptotic behavior of the sum function by using the ; known asymptotic form for: the Bessel functions.. T r i c o m i [ l ^ , 15> 16,l?] has used s i m i l a r series to t h i s purpose. CI*) Chapter % Approximate Solutions We w i l l now consider equations f o r which i t i s not possible to f i n d the form of the L1ouville-Neumann s e r i e s . The only cases considered w i l l be those for which the; difference between the given equation and the comparison equation i s small, so that only a., few terms of the: solution 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 Problems In the past 15 years,; a number of authors [l,, 1*, 10,.11,12] have considered the behavior, of a quantum mechanical system confined In an enclosure. The usual d i f f e r e n t i a l equations: s t i l l apply* but the; boundary conditions ar.e not the normal ones.. The only general method, which i s applicable to more than one of these problems, i s a graphical one [ i l l . We w i l l apply the method developed above to obtain approximate a n a l y t i c expressions; f o r the eigenfunctions and eigenvalues of a general eigenvalue problem. The unnatural, boundary condition i n each case; i s that the; wave function vanish on the f i n i t e surface bounding the region i n which the; system i s enclosed. We assume that the Schrodinger equation i s : seperable i n c u r v i l i n e a r coordinates cr,,.. a-x Tl = t j , such that the f i n i t e surface i s ;x , = C . After separation and solution of the ^ and ^ 3 parts of the wave; (15) equation,, there remains an equation of the: form I f the s i n g u l a r i t i e s are a t * ' ^ , ^ , the normal boundary conditions: are that u.t<*-) , IA.C-6) be. at l e a s t f i n i t e . In t h i s case the eigenfunctions are, ; say AfiLx^ s o t h a t n\ W ^ ' ^ J " ] ^ * . with eigenvalues C-6-3") X ' X : > C*»A*, ) We desire a solution of (6.1) subject to the conditions 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 is: then (6.4) MM (xrl- wfcxhrJ* In the cases: considered,, the condition *UA-) f i n i t e givesys P u t * (6.5) u. 6>(»W ^ ' ^ j j ^ ^ f ^ ^ The condition u.u)* o gives, i n general (6.6) >= - » - c ' a. *-By s i m i l a r means higher approximations may be obtained. The constant oL i s determined by the normalization. The problem of finding the eigenvalues has thus been reduced to one of quadratures. Even i f these: cannot be; ca r r i e d out e x p l i c i t l y , ; the problem of evaluating integrals numerically i s much easier than that of obtaining eigenvalues: from a d i f f e r e n t i a l equation by numerical means. (16) As a. s p e c i f i c example, of the above method, we consider the problem of a bounded hydrogen atom. The physical importance of t h i s problem was discussed by Michels,, de Boer and Bijl [ loJ, and solutions were obtained by them and extended by de Groot and ten S e l dam [4-], The= bounding surface i s the sphere »»C . The r a d i a l part of the equation i s then (6.7) . X*- 1 t A ) " ' ~ c > ( / a n integer >/o ) with; the usual solution and eigenvalues ~ ~ "H*- (*\ an i n t e g e r I ) corresponding to the boundary conditions f i n i t e , ; ui*) —» © as . For / , J~o, =• * (6.8) f5 i The; numerical values l i s t e d below were calculated using the summation procedure given by de Groot and ten Beldam^], and t h e i r values are quoted for comparison. The unit of length i s — ~ ,, of energy —-p,_ . (17) TV 3 — . i tr ¥ X*' - 9<V (a) - . 96 »c) — . 99 3 0) (,. - . 79 <f For. v> - £ „ o „ £/cxj = «T * *c*-*; (6.10) \ I x{° - - - -+ ^ -J^. For w a„ ^= / ,:, tf,/c~>« ^ (6.11) Similar expressions can be obtained f o r other values of the parameters. 7. Non-Linear Equations In most problems leading to non-linear d i f f e r e n t i a l equations we seek periodic solutions (or l i m i t c y c l e s ) . Such equations can be transformed in t o i n t e g r a l equations, but the; most convenient method i s to use a Green's function to 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 equations are not of the type considered here. There are, however, some non-l i n e a r problems where l i m i t cycles are not required, and our method can be applied to these. One example i s given. (18) One of the experimental checks of the general theory of r e l a t i v i t y i s the measurement of the d e f l e c t i o n of a: team of l i g h t passing near the sun. In solving the f i e l d equations for t h i s case,, i t . i s necessary to solve [2, Ch. 7, also p..2.07 J (7.1) d** with the; i n i t i a l conditions (7.2) LLC*) - , VU'(*> - O m Comparison of- (7.1) with (7.3) J^*"'* lea#s. to the i n t e g r a l equation (7.1+) ( A C 2 - ) - ^ ^ ~ h 3 v y x j ^i^-Z)U,XL^cl^ Put; u.t",C'0» 5 0 1 1 s i m p l i f i c a t i o n we obtain' (7.5) ^tr><:ar)' which is; the;; required f i r s t approximation. (19) Bibliography 1., F.C. Auluck and: D.S. Kothari, Proc. Camb. Phil . S o c , 41,, 175 (1945). 2'.. A.S. Eddington, Space. Time, and Gravitation., Cambridge. University Press (1923). 3> P.. Fubini, Rend. Lincei (6),;. 26, 253 (1937). 4. S.R. de Groot and C.A. ten Seldam, Physica, 12, 669 (1946). 5. I. Ikeda, Math. Zeit., 22, 16 (1925). 6. M. Karlin, J . Math. Phys., 28,, 43 (1949). 7. K. Knopp, Theory of Functions I, Dover Publications (1945). a.. J.. Liouville, 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 Edition (1952). 19. E.T. Whit taker, and G.N. Watson, Modern Analysis. Cambridge University Press, 4th Edition (1952). 


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