Reduced Differential Transform Method for Solving Klein Gordon Equations

Transcription

1 Proceedigs of he World Cogress o Egieerig 0 Vol I WCE 0, July - 8, 0, Lodo, UK Reduced Differeial Trasfor Mehod for Solvig Klei Gordo Equaios Yıldıray Kesi, Sea Servi ad Gali Ouraç Absrac Reduced differeial rasfor ehod (RDTM) is ileeed for solvig he liear ad oliear Klei Gordo equaios The aroiae aalyical soluio of he equaio is calculaed i he for of a series wih easily couable cooes Coarig he ehodology wih soe oher ow echiques shows ha he rese aroach is effecive ad owerful Three es odelig robles fro aheaical hysics are discussed o illusrae he effeciveess ad he erforace of he roosed ehod Ide Ters Reduced differeial rasfor ehod, Variaioal ieraio ehod, Klei Gordo equaios I INTRODUCTION Oe of he os iora of all arial differeial equaios occurrig i alied aheaics is ha associaed wih he ae of Klei Gordo The Klei Gordo equaio lays a iora role i aheaical hysics such as lasa hysics, solid sae hysics, fluid dyaics ad cheical ieics [-3] We cosider he Klei Gordo equaio u u unu(, ) f(, ) () subjec o iiial codiios u (,0) g ( ), u (,0) h ( ) () u is a fucio of ad, Nu(, ) is a oliear fucio, ad f (, ) is a ow aalyic fucio May uerical ehods were develoed for his ye of oliear arial differeial equaios such as he Adoia Decoosiio Mehod (ADM) [4-7], he EXP fucio ehod [8], he Hoooy Perurbaio Mehod (HPM) [9], he Hoooy Aalysis Mehod (HAM) [0] ad Variaioal Ieraio Mehod (VIM) [-4] I his aer, we solve soe Klei Gordo equaios by he reduced differeial rasfor ehod [5-8] which is reseed o overcoe he deeri of cole calculaio of differeial rasfor ehod (DTM) [9] The ai advaage of he ehod is he fac ha i rovides is user wih a aalyical aroiaio, i ay cases a eac soluio, i a raidly coverge sequece wih elegaly coued ers The srucure of his aer is orgaized as follows I secio, we begi wih soe basic defiiios ad he use of he roosed ehod I secio 3, we aly he reduced differeial rasfor ehod o solve hree es eales i order o show is abiliy ad efficiecy Mauscri received March, 0 G Ourac is wih he Selcu Uiversiy, Deare of Maheaics, Koya, 4003 Turey (corresodig auhor o rovide hoe: ; fa: ; e-ail: gourac@ selcuedur) Y Kesi is wih he Selcu Uiversiy, Deare of Maheaics, Koya, 4003 Turey e-ail: yildirayesi@ yahooco ISBN: ISSN: (Pri); ISSN: (Olie) II TRADITIONAL DIFFERENTIAL TRANSFORM METHOD A Oe Diesioal Differeial Trasfor Mehod The differeial rasfor of he fucio defied as follows: d W w! d () 0 w is he origial fucio ad d rasfored fucio Here d wih resec o The differeial iverse rasfor of w is W is he eas he he derivaive w W is defied as W () 0 Cobiig () ad () we obai d w w 0! d (3) 0 Fro above defiiios i is easy o see ha he coce of differeial rasfor is derived fro Taylor series easio Wih he aid of () ad () he basic aheaical oeraios are readily be obaied ad give i Table Table Oe-diesioal differeial rasforaio Fucioal For Trasfored For u v U V cu! cu d u d U! u v UrV r r 0 B Two Diesioal Differeial Trasfor Mehod Siilarly, he wo diesioal differeial rasfor of w, ca be defied as follows: he fucio h W, h w, h (4) h!! (0,0) w, is he origial fucio ad W, h is he rasfored fucio The differeial iverse rasfor of W, h is h W h 0 h0,, w 5) WCE 0

2 Proceedigs of he World Cogress o Egieerig 0 Vol I WCE 0, July - 8, 0, Lodo, UK The cobiig equaio (4) ad (5) we wrie h h w, w, h 0 h0 h!! () (0,0) Therefore we ca obai basic aheaical oeraios of wo-diesioal differeial rasfor as follows i Table Table Two diesioal differeial rasforaio Fucioal For Trasfored For u, v, U, h V, h cu, cu, h u, U, h u, rs u, r s h U, h r hs!! U r, h s! s! u, v, Ur, hsv r, s h r0 s0 Now we ca sae our ai resuls i he e secio III REDUCED DIFFERENTIAL TRANSFORM FOR KLEIN GORDON EQUATIONS The basic defiiios ad oeraios of reduced differeial rasfor ehod [5-7] are iroduced as follows: Defiiio If fucio u, is aalyic ad differeiaed coiuously wih resec o ie ad sace i he doai of ieres, he le U ( ) u, (3)! 0 he -diesioal secru fucio U rasfored fucio I his aer, he lowercase u, is he rerese he origial fucio while he uercase U sad for he rasfored fucio Defiiio The reduced differeial rasfor of a sequece U 0 is defied as follows: (3) 0, u U The cobiig equaio (3) ad (3) we wrie u, u, 0! (33) 0 Soe basic roeries of he reduced differeial rasforaio obaied fro defiiios (3) ad (3) are suarized i Table 3 The roofs of Table 3 ad he basic defiiios of reduced differeial rasfor ehod are available i [8] To illusrae, Cosider he followig Klei Gordo equaios (): Lu (, ) Lu (, ) u (, ) Nu (, ) f(, ) (34) wih iiial codiios u (,0) g ( ), u (,0) h ( ) (35) L, L, Nu(, ) is a oliear er Assue ha we ca wrie,, u U 0 0 The, we ge ers U, U, U,, U, U, U,, U, U, 0,0 0, 0,,0,,,0, The firs grou ca be wrie 0 U0, grou as U, 0 0, he secod, he hird grou as U, ec 0 Table 3 Basic oeraios of RDTM Fucioal For Trasfored For u (, ) u,!, v, u U ( ) V ( ) 0 u, U ( ) ( is a cosa) ( ) u (, ) U ( ) u, v, r u (, ) r u (, ) Nu(, ) r 0 U ( ) V ( ) r r ( r)! Ur( )! U ( ) Male code NF:=Nu(,):#Noliear fucio odr:=3:# Order u[]:=su(u[b]*^b,b=0odr): NF:=subs({Nu(,)=u[]},NF): s:=ead(nf,): d:=ualy(s,): for i fro 0 o odr do [i]:=((d@@i)(d)(0)/i!): ri(n[i],[i]); # Trasfor Fucio od: ISBN: ISSN: (Pri); ISSN: (Olie) WCE 0

3 Proceedigs of he World Cogress o Egieerig 0 Vol I WCE 0, July - 8, 0, Lodo, UK Thus u, U ( ) 0 U ( ) U, so ha he double series urs o 0 a sigle series Le he oliear er Nu(, ),wrie 0 0 Nu(, ) N ( U ( ),, U ( )) N ( ) Calculaio of N ( ) was give i he Table 3 The aroiae soluio usig he arial soluio is give by: u (,) L f(,) L Lu (,) L Nu (,) Lu (,) (3) u(0, ) u (0, ) g( ) h( ) ad have L dd Subsiuig for u (, ), (, ) 0 o Nu we U ( ) g( ) h( ) F ( ) dd U ( ) dd N ( ) dd U ( ) dd We ow carry ou he above iegraios o wrie U( ) g( ) h( ) F( ) ( )( ) 0 0 U ( ) N( ) ( )( ) ( )( ) 0 0 U ( ) 0 ( )( ) Le o he righ side The U ( ) g( ) h( ) F ( ) ( ) 0 ( ) ( ) U ( ) U ( ) N ( ) ( ) (37) Fially, equaio coefficies of lie owers of, we derive he recursio forula for he coefficies (accordig o he RDTM ad Table 3) U0( ) g( ), U( ) h( ) (38) ad ( )! U ( ) F( ) U ( ) ( ) ( ) N U! (39) U ( ), F ( ) ad N ( ) are he rasforaios of he fucios u (, ), f(, ) ad Nu(, ) resecively Subsiuig (38) io (39) ad by a sraigh forward ieraive calculaios, we ge he followig U ( ) values ISBN: ISSN: (Pri); ISSN: (Olie) The he iverse rasforaio of he se of values U ( ) 0 give aroiaio soluio as, u (, ) U ( ) 0 is order of aroiaio soluio Therefore, he eac soluio of roble is give by u (, ) li u (, ) IV APPLICATIONS To show he efficiecy of he ew ehod described i he revious ar, we rese soe eales A Eale We firs cosider he hoogeeous Klei Gordo equaio [] u u u 0 (4) wih iiial codiios: u (,0) si( ), u (,0) 0 (4) u u, is a fucio of he variables ad The, by usig he basic roeries of he reduced differeial rasforaio, we ca fid he rasfored for of equaio (4) as ( )! U ( ) U ( ) ( ) U (43)! Usig he iiial codiios (4), we have U0( ) si( ), U( ) 0 (44) Now, subsiuig (44) io (43), we obai he followig U ( ) values successively U( ), U3( ) 0, U4( ), 4 U5( ) 0, U( ) 70 0, for is odd U ( ),for is eve! Fially he differeial iverse rasfor of U ( ) gives u, U ( ) si( ) 0 0,,4,! Hece he closed for of (35) is u, si( ) cosh( ) (45) which is he eac soluios of (4) (4) B Eale We e cosider he ihoogeeous oliear Klei- Gordo equaio [] u u u cos( ) cos ( ) (4) wih he iiial codiios: u (,0), u (,0) 0 (47) Taig differeial rasfor of (4) ad he iiial codiios (47) resecively, we obai ( )! U ( ) U ( ) ( ) ( ) N F! (48) u, N ad F are rasfored for of ad cos( ) cos ( ) WCE 0

4 Proceedigs of he World Cogress o Egieerig 0 Vol I WCE 0, July - 8, 0, Lodo, UK For he easy o follow of he reader, we ca give he firs few oliear er are F N0 U0( ) 0 F N U0( ) U( ) 0 N U0( ) U( ) U ( ) F N3 U0( ) U3( ) U( ) U( ) F3 0 The rasfored iiial codiios U0( ), U( ) 0 (49) The subsiuig (49) io (48) we have U( ), U3( ) 0, U4( ), 4 U5( ) 0, U( ) 70 ad 0, for is odd U ( ) / ( ),for is eve! Fially he differeial iverse rasfor of U ( ) gives / ( ) u, U ( ) cos( ) 0 0! which is he eac soluio [0] C Eale 3 We ow cosider he oliear Klei-Gordo equaio [] u u u u 0 (40) 4 wih iiial codiios u (,0) sech( ), u (,0) sech( )ah( ) (4) The eac soluio of his roble is u (, ) sech If we wa o solve his equaio by eas of RDTM, usig Table 3, we ca fid he rasfored for of equaio (40) as ( )! 3 U ( ) U ( ) ( ) ( ) U N! 4 (4) 3 N is rasfored for of, u ad he rasfored iiial codiios U0( ) sech( ), U( ) sech( )ah( ) (43) Subsiuig (43) io (4), we obai he followig U ( ) values successively The, he iverse rasforaio of he se of values U ( ) 0 gives si-er aroiaio soluio as Therefore, he eac soluio of roble is give by uy (, ) li u ( y, ) This soluio is coverge o he eac soluio [] ad he sae as aroiae soluio of he variaioal ieraio ehod [] (see Figure ) sih( ) u (, ) U ( ) 0 cosh( ) cosh( ) (cosh ( ) ) (cosh ( ) )sih( ) cosh ( ) 48cosh ( ) 4 (cosh ( ) 0 cosh ( ) 4) 4 (44) 5 384cosh ( ) 4 (cosh ( ) 0 cosh ( ) 0) sih( ) cosh ( ) 4 (cosh ( ) 8 cosh ( ) 840 cosh ( ) 70) cosh ( ) Figure The coariso of he RDTM aroiaio ad he eac soluio Figure shows he coariso of he RDTM aroiaio soluio of order si ad he eac soluio u (, ) sech, he solid lie rereses he soluio by he reduced differeial rasfor ehod, while he circle rereses he eac soluio Fro he figure, i is clearly see ha he RDTM aroiaio ad he eac soluio are i good agreee V CONCLUSIONS Aalyical soluios eable researchers o sudy he effec of differe variables or araeers o he fucio uder sudy easily Is sall size of couaio i coariso wih he couaioal size required i oher uerical ehods, ad is raid covergece show ha he ehod is reliable ad iroduces a sigifica irovee i solvig Klei-Gordo equaios over eisig ehods Fro his sudy cocluded ha, i ca be he reduced differeial rasfor ehod oulied i he revious secio fids quie racical aroiae aalyical resuls wih less couaioal wor ISBN: ISSN: (Pri); ISSN: (Olie) WCE 0

5 Proceedigs of he World Cogress o Egieerig 0 Vol I WCE 0, July - 8, 0, Lodo, UK ACKNOWLEDGMENT This sudy was suored by he Coordiaorshi of Selcu Uiversiy s Scieific Research Projecs (BAP) REFERENCES [] L Debah, Noliear Parial Differeial Equaios for Scieis ad Egieers, Birhauser, Boso, 997 [] PG Drazi, RS Johso, Solios: A Iroducio, Cabridge Uiversiy Press, Cabridge, 983 [3] MJ Ablowiz, PA Clarso, Solios, Noliear Evoluio Equaios ad Iverse Scaerig Trasfor, Cabridge Uiversiy Press, Cabridge, 990 [4] AM Wazwaz, Parial differeial equaios: ehods ad alicaios The Neherlads: Balea Publishers, 00 [5] Saau Saha Ray, A alicaio of he odified decoosiio ehod for he soluio of he couled Klei Gordo Schrödiger equaio, Couicaios i Noliear Sciece ad Nuerical Siulaio, 3(7) 008, 3-37 [] SM El-Sayed, The decoosiio ehod for sudyig he Klei-Gordo equaio, Chaos, Solios & Fracals 8 (003) [7] K C Basa, P C Ray, R K Bera, Soluio of o-liear Klei Gordo equaio wih a quadraic o-liear er by Adoia decoosiio ehod, Couicaios i Noliear Sciece ad Nuerical Siulaio, 4(3) 009, [8] A Ebaid, Eac soluios for he geeralized Klei Gordo equaio via a rasforaio ad E-fucio ehod ad coariso wih Adoia s ehod, Joural of Couaioal ad Alied Maheaics, 3() 009,78-90 [9] Z Odiba, S Moai, A reliable reae of hoooy erurbaio ehod for Klei Gordo equaios, Physics Leers A, 35(5-) 007, [0] Q Su, Solvig he Klei Gordo equaio by eas of he hoooy aalysis ehod, Alied Maheaics ad Couaio,9 () 005, [] E Yusufoglu, The variaioal ieraio ehod for sudyig he Klei Gordo equaio, Alied Maheaics Leers, (7) 008, 9-74 [] N Bildi, A Koural, The use of variaioal ieraio ehod, differeial rasfor ehod ad Adoia decoosiio ehod for solvig differe yes of oliear arial differeial equaios, Ieraioal Joural of Noliear Scieces ad Nuerical Siulaio 7 () (00) 5 70 [3] Q Wag, D Cheg, Nuerical soluio of daed oliear Klei Gordo equaios usig variaioal ehod ad fiie elee aroach, Alied Maheaics ad Couaio, () 005,38-40 [4] J Biazar, H Ghazvii, He s variaioal ieraio ehod for solvig hyerbolic differeial equaios, Ieraioal Joural of Noliear Scieces ad Nuerical Siulaio 8 (3) (007) 3 34 [5] Y Kesi, G Ourac, Reduced Differeial Trasfor Mehod for Parial Differeial Equaios, Ieraioal Joural of Noliear Scieces ad Nuerical Siulaio, 0 () (009) [] Y Kesi, G Ourac, Reduced differeial rasfor ehod for solvig liear ad oliear wave equaios, Iraia Joural of Sciece & Techology, Trasacio A, Vol 34(A) 00, 3- [7] Y Kesi, G Ourac, Reduced Differeial Trasfor Mehod for fracioal arial differeial equaios, Noliear Sciece Leers A () (00) -7 [8] Y Kesi, PhD Thesis, Selcu Uiversiy (o aear) [9] ASV Ravi Kah, K Arua, Differeial rasfor ehod for solvig he liear ad oliear Klei Gordo equaio, Couer Physics Couicaios, 80(5) 009, [0] Saeid Abbasbady, Nuerical soluio ofo-liear Klei Gordo equaios by variaioal ieraio ehod, Ieraioal Joural for Nuerical Mehods i Egieerig, 70(7) 00, [] Wazwaz A The odified decoosiio ehod for aalyic reae of differeial equaios Alied Maheaics ad Couaio 00; 73:5 7 [] X Zhao, H Zhi, Y Yu, H Zhag, A ew Riccai equaio easio ehod wih sybolic couaio o cosruc ew ravellig wave soluio of oliear differeial equaios, Alied Maheaics ad Couaio 7 (00), 4-39 ISBN: ISSN: (Pri); ISSN: (Olie) WCE 0