Integral Equation Approach for Computing Green s Function on Doubly Connected Regions via the Generalized Neumann Kernel

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1 Jura Teoog Fu paper Iegra Equao Approach for Compug Gree s Fuco o Douby Coeced Regos va he Geerazed Neuma Kere S Zuaha Aspo a* A Hassa ohamed urd ab ohamed S Nasser c Hamsa Rahma a a Deparme of ahemaca Sceces Facuy of Scece UT 83 UT Johor Bahru Johor aaysa b UT Cere for Idusra ad Apped ahemacs (UT-CIA) 83 UT Johor Bahru Johor aaysa c Deparme of ahemacs Facuy of Scece Kg Khad Uversy PO Box 94 Abha Saud Araba *Correspodg auhor: szuahaaspo@yahoocom Arce hsory Receved : February 4 Receved revsed form : 3 Augus 4 Acceped :5 Ocober 4 Graphca absrac Absrac Ths research s abou compug he Gree s fuco o douby coeced regos by usg he mehod of boudary egra equao The mehod depeds o sovg a Drche probem The Drche probem s he soved usg a uquey sovabe Fredhom egra equao o he boudary of he rego The ere of hs egra equao s he geerazed Neuma ere The mehod for sovg hs egra equao s by usg he Nysrӧm mehod wh rapezoda rue o dscreze o a ear sysem The ear sysem s he soved by he Gauss emao mehod ahemaca pos of Gree s fucos for severa es regos are aso preseed Keywords Gree s Fuco; Drche Probem; Iegra Equao; Geerazed Neuma Kere Absra Kaa beraa dega pegraa fugs Gree pada raau bera gada dua erbaas dega megguaa aedah persamaa amra sempada Kaedah bergaug epada peyeesaa masaah Drche asaah Drche emudaya dseesaa megguaa persamaa amra Fredhom berpeyeesaa u pada sempada raau I persamaa amra adaah Neuma era Kaedah uu meyeesaa persamaa amra aah dega megguaa aedah Nysrӧm dega peraura rapezod uu meghasa sebuah ssem ear Ssem ear emuda dseesaa dega aedah peghapusa Gauss Po ahemaca bag fugs Gree uu beberapa raau ua uga dpersembaha Kaa uc: Fugs Gree; asaah Drche; Persamaa Kamra; I Neuma Tera 4 Peerb UT Press A rghs reserved INTRODUCTION Gree s fucos are mpora sce hey provde a powerfu oo sovg dfferea equaos They are very usefu severa feds such as sod mechacs apped physcs apped mahemacs mechaca egeerg maeras scece ad quaum fed heory Herc shows hree dffere mehods for compug Gree s fuco for douby coeced regos whch eads o hree dffere aayca represeaos for he Gree s fuco The mehods are he Fourer seres mehod fe produc mehod ad Thea seres mehod Crowdy ad arsha have preseed a aayca formua for Gree s fuco for Lapace s equao mupy crcuar domas The mehod s cosrucve ad depeds o Schoy-Ke prme fuco assocaed wh mupy coeced crcuar doma 3 Wegma ad Nasser have suded Fredhom egra equao assocaed wh he ear Rema-Hber probems o mupy coeced regos wh smooh boudary curves The ere of hese egra equaos s he geerazed Neuma ere They vesgaed he exsece ad uqueess of souos of he egra equaos by deermg he exac umber of ear depede souos ad her ados 4 Based o Wegma ad Nasser Nasser e a have proposed a ew boudary egra mehod for he souo of Lapace s equao o mupy coeced regos usg eher Drche boudary codo or he Neuma boudary codo The mehod s based o wo uquey sovabe Fredhom egra equaos of he secod d wh he geerazed Neuma ere 5 Recey Aagee has proposed a ew mehod for compug he Gree s fuco o smpy coeced rego by usg he mehod of boudary egra equao whch depeds o he souo of a Drche probem 6 Based o paper by Wegma ad Nasser ad Nasser e a he Drche probem s soved usg a uquey sovabe Fredhom egra o he equao boudary of he rego 45 Ths 7: (4) wwwuraeoogummy eissn 8 37

2 5 S Zuaha Aspo e a / Jura Teoog (Sceces & Egeerg) 7: (4) paper wsh o exed Aagee s wor o compue Gree s fuco for bouded douby coeced regos usg egra equao wh geerazed Neuma ere 6 ( ) J ( ) J AUXILIARY ATERIALS Le Ω be a bouded douby coeced rego he compex pae (Fgure ) The ouer boudary has a couer cocwse dreco ad surrouds he boudary whch has cocwse oreao So we have The eror Drche probem (3) s uquey sovabe ad ca be regarded as a rea par of a aayc fuco F whch s o ecessary sge-vaued 5 The fuco F ca be wre as F( z) f ( z) a z z (4) f s a sge-vaued aayc fuco z s a fxed po ad a s rea cosa uquey deermed by 5 We assume for bouded ha Im f ( ) The cosa a s chose o esure ha Fgure Bouded douby coeced rego We assume ha each boudary has a parameerzao ( ) J whch s a compex perodc fuco wh perod J [ ] s he The parameerzao () paramerc erva for each aso eed o be wce couousy dffereabe such ha d() ( ) d () Therefore he parameerzao of he whoe boudary ca be wre as : ( ) J [ ] : ( ) J [ ] () Le u be a rea fuco defed he rego ad e z x y I our research for smpcy we wre u(z) sead of u(xy) Le H be he space of a rea Höder couous fuco wh expoe o he boudary Γ The eror Drche probem s defed as foows: Ieror Drche probem: Le H be a gve fuco Fd he fuco u harmoc Hoder couous o ad sasfes he boudary codo u( ) (3) f '( ) d I geera he Gree s fuco for ca be expressed by 8 G( z z) u( z) z z z z (5) u s he uque souo of he eror Drche probem u( z) z u( ) z By compug F gve by (4) wh (6) z ad z (7) he uque souo of he eror Drche probem (6) s gve by u( z) Re F( z) (8) 3 INTEGRAL EQUATION FOR THE INTERIOR DIRICHLET PROBLE A be couousy dffereabe π-perodc fucos Le () for a J We cosder wo rea fucos 9 N A( s) ( ) ( s ) Im A ( ) ( ) ( s) (3)

3 5 S Zuaha Aspo e a / Jura Teoog (Sceces & Egeerg) 7: (4) A( s) ( ) ( s ) Re A ( ) ( ) ( s) (3) The ere N ( s ) s caed he geerazed Neuma ere formed wh compex-vaued fuco A () ad ( 4 ) Whe A he ere N s he cassca Neuma ere whch arse frequey he egra equaos for poea heory ad coforma mappg Theorem 3 4 a) The ere N ( s ) s couous whch aes o he dagoa he vaues N ( ) A( ) ( ) Im ( ) A( ) (33) b) The ere ( s ) s couous for he ere ( s ) has he represeao Whe s ( s ) co ( s ) (34) wh a couous ere whch aes o he dagoa he vaues ( ) A( ) ( ) Re ( ) A( ) (35) To fd he fuco Fz () gve by (4) we eed o fd he fuco f() z ad he rea cosa a We defe rea fucos ad [] z sasfy (7) I foows ha 5 for (36) f ( ( )) ( ) h ( ) ( )for p (37) are boudary vaues of aayc fuco f s he uque souo of he egra equao ( s) N ( s ) ( ) d N ( s ) ( ) d ( s ) ( ) d ( s ) ( ) d s J p (38) ad [ ] p h ( s ) ( ) d N ( s ) ( ) d (39) wh h [ p ] [ p ] h h h I foows from Nasser e a 5 ha he uow cosa s he souo of he equao The h [] [] a h h h [] [] [] [] h h a Hece he boudary vaues of he fuco f s gve by 5 f ( ( )) ( ) a ( ) z ( ) ( ) ( ) a ( ) [] [] a (3) (3) By hs resu we ca compue he eror vaues of f() z over he whoe rego by usg he Cauchy egra formua f( w) f ( z) dw w z (3) We he compue he fuco Fz () from (4) ad uz () from (8) 4 NUERICAL IPLEENTATION Deog he rgh-had sde of he Equao (38) by () s we ge [ p ] [ p ] [ p ] [ p ] ( s) N ( s ) ( ) d N ( s ) ( ) d ( s) (4)

4 5 S Zuaha Aspo e a / Jura Teoog (Sceces & Egeerg) 7: (4) Sce he fucos A ad are - perodc he egras are dscrezed by he Nysrӧm mehod wh rapezoda rue 7 Le be a gve eger ad defe he equdsa coocao pos by ( ) (4) The usg he Nysrӧm mehod for (4) we oba he ear sysem ( ) ( ) ( ) ( ) ( ) ( ) N N s s N s a approxmao o ad ad (43) A( ) ( ) Im or A ( ) ( ) ( ) ( ) ( ) A ( ) Im Im ( ) A ( ) P [ P ] Q [ Q ] R [ R ] S [ S ] ad vecors x [ x ] ad y [ y ] by P N ( ( ) ( )) Q N R N ( ( ) ( )) S N ( ( ) ( )) ( ( ) ( )) x ( ) y ( ) Hece he Equao (43) ca be wre as a by sysem ( I P) x Qx y Rx ( I S) x y (46) To sove he sysem (46) we use he mehod of Gaussa emao Sce (4) has a uque souo he for a wde cass of quadraure formua he sysem (46) aso has a uque souo as og as s suffcey arge 7 Afer we ge he uque souo x ( ) he we cacuae f ( ( )) ad f ( ( )) by usg he foowg formua: d ( ) co ( ) ( ) ( ) f ( ( )) ( ) a ( ) z ( ) f ( ( )) ( ) a ( ) z ( ) (47) p ad ( ) ( ) (44) whch represes he boudary vaues of f (z) o We ca compue he eror vaues of f (z) over he whoe rego by usg he Cauchy egra formua gve (3) e A( ) ( ) ( ) Re A ( ) ( ) ( ) A ( ) ( ) Re co A ( ) ( ) ( ) ( ) ( ) A ( ) Re ( ) A ( ) We use Wch mehod o approxmae he egra ha coas coage fuco ad oba co ( ) d K( ) ( ) K( ) ( ) co f - s eve f - s odd (45) The ef-had sde of (43) ca aso be cacuaed drecy by usg ahemaca Defe he marces f ( ( )) f ( ( )) f ( z) ( ) d ( ) d ( ) z ( ) z (48) To crease he accuracy of f (z) we sha use he foowg formua Based o he fac ha d z we ca wre f (z) as f ( ( )) f ( ( )) ( ) d ( ) d ( ) z ( ) z (49) f( z) ( ) ( ) d d ( ) z ( ) z The usg he Nysröm mehod wh he rapezoda rue o dscreze he egras (49) we oba he approxmao f ( ( )) f ( ( )) ( ) ( ) ( ) z ( ) z ( ) ( ) f( z) ( ) z ( ) z (4)

5 53 S Zuaha Aspo e a / Jura Teoog (Sceces & Egeerg) 7: (4) Ths has he advaage ha he deomaor hs formua compesaes for he error he umeraor Nex subsue f() z gve (4) o Equao Fz () (4) ad by ag he rea par of (4) gves (8) e u( z) Re F( z) Fay by usg u (z) we ca compue he Gree s fuco G ( ) z z by he foowg formua (5) e G ( z z) u( z) z z 5 NUERICAL EXAPLES Exampe I Exampe we cosder a auus as show (Fgure ) The boudary of hs rego s parameerzed by he fuco : ( ) e : ( ) pe wh p5 z 75 z e ad he fe seres coverges uformy for z ad a eas e a geomerc seres wh rao We descrbe he error by maxmum error orm G( z z) G ( z z) s he umber of odes ad G( z z ) s he umerca approxmao of G( z z ) We choose some es pos sde he rego The resus are show Tabe z Tabe The error G( z z) G ( z z) The 3D po of he surface of G ( ) z z s show (Fgure 3) z ad z Fgure 3 The 3D po of Gree s fuco for Exampe Exampe I Exampe we cosder a Eprochod as show (Fgure 4) The boudary of hs rego s parameerzed by he fuco Fgure The es rego Ω for Exampe : ( ) e pe : ( ) qe The exac Gree s fuco of hs rego s gve by Log e G( z z) Log Log Log e cos wh p3333 q z 75 z z ad z

6 54 S Zuaha Aspo e a / Jura Teoog (Sceces & Egeerg) 7: (4) u( z) z u( ) z (6) Fgure 4 The es rego Ω for Exampe The 3D po of he surface of G ( ) z z s show (Fgure 5) o ha rego by meas of sovg a egra equao umercay usg Nysröm mehod wh he rapezoda rue Oce we go he souo uz () he Gree s fuco of ca be compued by usg he formua G ( z z) u( z) z z (6) The umerca exampe usraes ha he proposed mehod ca be used o produce approxmaos of hgh accuracy Acowedgeme Ths wor was suppored par by he aaysa sry of Hgher Educao (OHE) hrough he Research aageme Cere (RC) Uvers Teoog aaysa (GUPQJ3564H6) Refereces Fgure 5 The 3D po of Gree s fuco for Exampe 6 CONCLUSION Ths sudy has preseed a mehod for compug he Gree s fuco o douby coeced regos by usg a ew approach based o boudary egra equao wh geerazed Neuma ere The dea for compug he Gree s fuco o s o sove he Drche probem [] QH Q 7 Gree s Fuco ad Boudary Eemes of ufed aeras Esever Scece [] P Herc 986 Apped ad Compuaoa Compex Aayss New Yor: Joh Wey 3: 55 7 [3] D Crowdy J arsha 7 IA J App ah 7: 78 3 [4] R Wegma S Nasser 8 J Compu App ah 4: [5] S Nasser A H urd Isma E A Aeay J App ah ad Comp 7: [6] A Aagee Iegra Equao Approach for Compug Gree s Fuco o Smpy Coeced Regos Sc Dsserao Uvers Teoog aaysa Suda [7] K E Aso 997 The Numerca Souo of Iegra Equaos of he Secod Kd Cambrdge: Cambrdge Uversy Press [8] L V Ahfors 979 Compex Aayss Ieraoa Sude Edo Sgapore: cgraw-h [9] S Nasser 9 SIA J Sc Compu 3: [] S Nasser 9 Compu ehods Fuc Theor 9: 7 43 [] J Hesg R Oaa 8 J Compu Physcs 7: 899 9