July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 ON THE EIGENFUNCTION EXPANSION METHOD FOR THE CALCULATION OF GREEN S FUNCTIONS Roberto Toscao Couto Departameto de Matemática Aplicada, Uiversidade Federal Flumiese, Brazil toscao@imuffbr ABSTRACT I this wor, we discuss some aspects of the eigefuctio expasio method for the calculatio of Gree's fuctios To this ed, we completely solve Poisso's equatio i a bouded domai uder the Dirichlet boudary coditios The problem geometry ad boudary data were chose so as to show up the features ivestigated The systematic ad detailed calculatio preseted here is a attempt to fill up a gap i the literature Keywords: Eigefuctio expasio, Gree's fuctio, Poisso s equatio INTRODUCTION Gree's fuctios are ofte determied as a expasio i eigefuctios However, depedig o the umber of spatial dimesios, two or more sets of orthogoal eigefuctios may become available Therefore, a issue that surely arises is that of decidig which of the sets is the more appropriate This wor addresses this questio To this ed, we cosider a specific problem We solve Poisso's equatio i a half-dis uder ohomogeeous Dirichlet boudary coditios employig, as a atural choice, the plae polar coordiates We will see that the choice of the eigefuctios is of special importace for problems with o-homogeeous boudary coditios That particular problem also eables the discussio of three special features that the eigefuctio expasio method may exhibit The first special feature is a ucommo eigevalue problem, whose spectrum is cotiuous, i spite of the bouded domai cosidered The secod oe refers to the possibility of expressig the resultig Gree's fuctio expasio i a closed form (that is, i terms of the elemetary fuctios) The third feature is the possibility of performig the itegral that yields Gree's fuctio as a expasio i the cotiuous-spectrum eigefuctios I the literature, it is hard to fid the Gree's fuctio method applied to problems havig o-homogeeous boudary coditios As a attempt to fill up this gap, the calculatios are preseted systematically ad i detail Sectio cotais the formulatio of the problem cosidered ad of its solutio i terms of Gree's fuctio Sectio 3 presets the calculatio of Gree's fuctio as expasios i two ids of eigefuctios Sectio 4 describes the determiatio of the problem solutio i terms of the Gree's fuctios calculated i Sectio 3 Sectio 5 cocludes the body of the paper with a discussio of the results
July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 FORMULATION OF THE PROBLEM AND OF THE SOLUTION It is well established (Refereces [, Chap 5], [, Sec 8], [3, Sec 9], [4, Sec ] ad [5, Sec 7]) that the solutio of Poisso's equatio uder the Dirichlet boudary coditio, ( r ) 4 ( r ) [ r ], () f A where A is a domai of give by r r where G( ) r r r ( ) g( ) [ A ], () ad A is its boudary, is r r r r ( ) dag( ) f ( ) A G r ds ( ) g( ) 4 Ñ A r r, (3) r [ A ] is the solutio of r r r r r G( ) 4 ) [ A ], (4) r r r G( ) [ A ], (5) r beig G/ the ormal derivative, equal to G, with the uit ormal vector r directed outward from A at a poit of A Let A be the half-dis show i Figure I this geometry, the plae polar coordiates ad are the most suitable For the boudary coditios give i that figure, () ad () become (, ) 4 f (, ), (6) (, ) g ( ), (7) (,), (, ) ( ), (8) g with ad Liewise, Equatios (4) ad (5) ow read G G G (,, ) (4 / ) ( ) ( ), (9) G(,, ), () G(,, ) G(,, ), () with ad varyig as ad vary The solutio of our problem [the problem defied by (6) to (8)] is, i accordace with (3), (, ) (, ) (, ) (, ), () where the source term is give by ds d f f G(,, ) f (, ) d d, (3) ad the boudary terms, by G (, ) (,, ) g( ) d, (4) 4 g ( ) ( ) g ds d x Figure Specificatio of the domai A ad of the boudary data g() for the problem defied by () ad () y 3
July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 G d (, ) (,, ) g( ) (5) 4 3 CALCULATION OF GREEN S FUNC- TION To calculate G, we cosider two subregios of A, those obtaied with either a radial divisio, ad, or a sectorial divisio, ad (Figure below) Let us cosider the radial divisio first I each subregio, we have that ad, therefore, ( ) vaishes; that is, the PDE for G give by (9) is homogeeous ad ca be solved by meas of the method of separatio of variables Thus, substitutig G R( ) F( ), we obtai R R F, (6) R { F Figure The two ways the problem domai is divided i two subregios: radially (, ) ad sectorially (, ) where we have recogized that the secod term must be a costat, The separated ODE F F( ) is to be solved i each subregio uder the boudary coditios F() F( ) [derived from ()] This problem (with homogeeous ODE ad boudary coditios) is a eigevalue problem, whose solutios are easily foud to cosist of the eigevalues (,,3 L ) ad the eigefuctios F ( ) si (Referece [, Sec 8]) [Notice that the radial problem, formed with the separated ODE for R( ), is ot a eigevalue problem; i fact, o the commo boudary at of both subregios, homogeeous coditios caot be derived!] The resultig eigefuctios ca be used to express Gree's fuctio as a liear superpositio of terms of the type R ( ) F ( ) R ( )si : G(,, ) R ( )si (7) To determie the fuctios R ( ) (whose depedece o ad is implicit), we substitute the above expasio ito the PDE give by (9), obtaiig R R ( / ) R ( ) si 4 ( ) ( ), (8) from which we ifer that the term eclosed by bracets are the coefficiets of the Fourier sie series of the fuctio o the right-had side over the iterval (, ), that is, R R R ( ) 4 ( ) ( ) si d 8 si ( ) (9) This is a Euler equatio, which is homogeeous i each subregio (where ) Its solutio is (Referece [3, Sec 6]) 4
July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 A B ( ) R ( ) A B ( ) () The four costats above are determied by imposig the followig four coditios (Referece [, Sec ]): (i) Fiiteess at the origi, which is achieved by settig B (ii) The coditio R () A B which follows from () (iii) The cotiuity coditio R( ) R( ) at, sice G R F is a potetial ad, therefore, must be a cotiuous fuctio Observe the otatio, with (iv) The jump discotiuity coditio for the derivative of R( ), Let us calculate Gree's fuctio agai, this time cosiderig the regio A divided i the two sectors ad I both of them, it is ow for R( ) that a eigevalue problem arises with the separatio of variables G R( ) F( ), because of the homogeeous coditio R() [deduced from ()] o the boudary at of both sectors We thus separate the ODE R R R( ) by equatig the first term i (6) to the costat The eigevalue problem so obtaied ca be coverted to a familiar oe by chagig the idepedet variable to u l It becomes R R( u), with R() u ad u, where R( u) R[ ( u)] ad ( u) e The well-ow eigevalues ad eigefuctios are ad R ( u) si u [or R ( ) si (l ) ], with (a cotiuous spectrum: cf Referece [, Sec 87]) I may istaces, it is better to wor with the ew variable u, i terms of which (9) reads 8 R( ) R( ) si, obtaied by itegratig (9) i the eighborhood of, from to Oce the calculatio of () is completed, its substitutio ito (7) yields G (,, ) 4 ( ) si si, () where ( ) is the smaller (larger) of ad The symbol resemblig the radial divisio of A is used to idicate the calculatio of Gree's fuctio cosiderig this divisio of A G G 4 ( uu ) ( ), () u where G( u, u, ) G[ ( u), ( u), ] The calculatio of G proceeds i the same maer described above We substitute the expasio ito () to obtai G( u, u, ) F ( )si u du (3) F F ( ) si u du 4 ( uu) ( ) 5
July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 The, by usig the Fourier sie itegral formula (Referece [6, Sec 64]), we calculate the term i the itegrad which is eclosed by bracets, obtaiig the equatio F F ( ) 4 ( u u) ( ) si u du 8si u( ) (4) Next, we solve it separately i each sector, Acosh Bsih ( ) F ( ) Acosh Bsih ( ) ad determie the four costats by imposig the four coditios: (i) F () ad (ii) F ( ) [both from the boudary coditio ()]; (iii) F ( ) F ( ) (cotiuity at ); (iv) F ( ) F ( ) 8siu (jump discotiuity of F ( ) at, derived by itegratig (4) i the eighborhood of, from to ) Fially, we substitute the F ( ) so determied ito (3) to obtai G ( u, u, ) 8siu siusihsih ( ) d, (5) sih CALCULATED GREEN S FUNCTION We develop below oly the boudary terms (, ) ad (, ) appearig i (); the source term f (, ) is pretty well discussed i the literature (eg, Referece [4]) Looig at (4) ad (5), we see that we eed to calculate G / at ad G / at Usig () first, we obtai ad G (,, ) 4 ( )si si 8 si si G (,, ) 4 ( )si si 4 ( ) ( )si Substitutig these results ito (4) ad (5), we obtai (, ) si, (6) where ( ) is the smaller (larger) of ad The symbol (lie a sector) idicates that G is calculated cosiderig the problem domai A divided ito two sectors 4 THE SOLUTION IN TERMS OF THE where (, ) ( ) I ( )si, (7) g( ) si d, (8) 6
July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 I d ( ) g( ) ( ) Now usig (5) to calculate G / at ad G / at, we get I ( ) d g( )sih sih ( ), ( ) dug ( u) siu (3) ad G u (,, ) e d u 8si u si usih sih ( ) sih u si u sih sih ( ) 8 d sih G (,, ) d 8si u si usih sih ( ) sih 5 DISCUSSION OF THE RESULTS a) The solutio (, ) of our problem [defied by (6), (7) ad (8)] is split i three terms, accordig to () We discuss here the expressios obtaied for the boudary terms (, ) ad (, ) More specifically, we verify the ability of these terms to reproduce the boudary data g ( ) ad g ( ) Let us first cosider the expasios obtaied for (, ) ad (, ) i terms of the eigefuctios si (derived with the radial divisio of the problem domai), give by (6) ad (7) These equatios furish g (, ) si ( ), si u si usih 8 d sih sice, i accordace with (8), are the coefficiets of the Fourier sie series for g ( ), as well as Substitutio ito (4) ad (5) gives (, ) ( ) I( )si { where si( l ) (, ) d I ( ), (9) sih sih si( l ) (, ) d ( ), (3) sih Therefore, the expasio i the eigefuctios si reproduces the boudary data g ( ), but it fails do yield g ( ) Cosider ow (9) ad (3), the expasios of the boudary terms i terms of the cotiuous-spectrum eigefuctios si ( l ) (derived with the sectorial divisio of the problem domai); they furish 7
July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 ad si( l) (, ) d I ( ) sih b) Gree's fuctio G (,, ) ca be expressed i closed form I fact, with the defiitios p, q /, d ad s, we ca develop () as follows: (, ) d ( ) si u g ( u) g ( ) This last result is explaied by the fact that, i view of (3), g ( u) is the iverse Fourier sie trasform of ( ) Therefore, the expasio i the cotiuous-spectrum eigefuctios si ( l ) reproduces the boudary data g ( ), but if fails to yield g ( ) We thus coclude the followig: If the boudary data is a fuctio of some variable, the correspodig Gree's fuctio boudary term "is better" expaded i terms of the eigefuctios which are fuctios of that variable We say "is better" rather tha "has to be" for the followig reaso Sice both expressios ad for the solutio of the problem coverge everywhere f i the (ope) domai A (cf Referece [7]), ad, o the boudary A, is ow, we could cosider uimportat the fact that ad do ot reproduce the boudary data g ( ) ad ( ), respectively; but we g ca ot! Ideed, it is a corollary of this fact that the covergece of ad i A will be more difficult to achieve tha that of ad, respectively G (,, ) 4 cos d cos s ( p q ) p q cosd cosd p q coss coss (3) However, otice that (cf Eq (53) i Referece [8]) where r cos Re z Re log ( z) l z l ( r cos r ), (33) i z r e, ad the well-ow Taylor's series of log ( z) was used (i the above, we distiguish betwee the complex logarithmic fuctio ad the real oe by employig the otatios log ad l, respectively) Therefore, we ca use the formula i (33) to replace each series i (3) by a logarithmic term, thus accomplishig our itet of expressig Gree's fuctio i a closed form: G (,, ) l ( pcos d p ) l ( qcos d q ) l ( pcos s p ) l ( qcos s q ) c) The itegral which furishes the Gree's fuctio G ( u, u, ), i (5), ca be evaluated by cosiderig it alog the closed cotour of the -plae show i 8
July 3 Vol 4, No Iteratioal Joural of Egieerig ad Applied Scieces EAAS & ARF All rights reserved wwweaas-jouralorg ISSN35-869 Figure 3 (where the radius teds to ifiite: R ) This is possible because the itegrad is a eve fuctio of, allowig switchig to half the itegral from to The, sice the poles of the itegrad are the zeros of sih, except the removable sigularity, that is, m i ( m ) show, by usig the residue theorem, that m, it is a simple matter to G ( u, u, ) i Res( mi) m 8( ) sih mu sih mu si m si m( ) m 4i 3i i i i Im * m mi ( m ) are the poles of the itegrad of G ( u, u, ) R Re [] Butov, E (973) Mathematical Physics Readig, Massachusetts: Addiso-Wesley Publishig Compay [3] Hildebrad, F B (976) Advaced Calculus for Applicatios (d ed) Eglewood Cliffs, New Jersey: Pretice-Hall [4] Jacso, J D (975) Classical Electrodyamics (d ed) New Yor: Joh Wiley & Sos [5] Morse, P M, & Feshbach, H (953) Method of Theoretical Physics New Yor: McGraw-Hill Boo Compay [6] Churchill, R V, & Brow, J W (978) Fourier Series ad Boudary Value Problems (Iteratioal Studet Editio, 3rd ed) Sigapore: McGraw-Hill Boo Compay [7] Titchmarsh, E C (96) Eigefuctio Expasios (d ed): Oxford Uiversity Press [8] Duffy, D G () Gree's Fuctios with Applicatios (Studies i Advaced Mathematics) Boca Rato, Florida: Chapma & Hall/CRC Press Figure 3 The closed cotour used to evaluate the real itegral i (5) with the help of the residue theorem REFERENCES [] Barto, G (989) Elemets of Gree s Fuctios ad Propagatio: Potetials, Diffusio, ad Waves: Oxford Uiversity Press 9