On The Geometrıc Interpretatıons of The Kleın-Gordon Equatıon And Solution of The Equation by Homotopy Perturbation Method

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1 Available a hp://pvam.ed/aam Appl. Appl. Mah. SSN: Vol. 7, sse (December ), pp Applicaios ad Applied Mahemaics: A eraioal Joral (AAM) O The Geomerıc erpreaıos of The Kleı-Gordo Eqaıo Ad Solio of The Eqaio by Homoopy Perrbaio Mehod Hasa Bl ad H. Mehme Başkoş Deparme of Mahemaics Fira Uiversiy 9 Elazig, Trkey hbl@fira.ed.r, hmbaskos@gmail.com Received: Je 6, ;Acceped: December 5, Absrac This paper is orgaized i he followig ways: he firs par, we obaied he Klei Gordo Eqaio (KGE) i he Galilea space. he secod par, we applied Homoopy Perrbaio Mehod (HPM) o his differeial eqaio. he hird par, we gave wo eamples for he Klei Gordo eqaio. Fially, We compared he merical resls of his differeial eqaio wih heir eac resls. We also showed ha approach sed is easy ad highly accrae. Keywords: The Galilea space, The Homoopy perrbaio mehod, The Klei-Gordo Eqaio, Liear ad oliear parial differeial eqaios MSC No.: 5A, 5A5, 8Q5. rodcio The Klei-Gordo, Sie-Gordo ad Sih-Gordo Eqaios The relaivisic wave eqaio of he moio of a free paricle wih zero spi fod by physiciss O. Klei ad V. Gordo is called Klei-Gordo eqaio (KGE). f he paricle is characerized by oe space coordiae, ad he he eqaio has he form as 69

2 6 H. Bl ad H. M. Başkoş m. (.) where is ime,, is wave fcio, ad m is he mass of he paricle. Noe ha Eqaio (.) for m is a classical oe-dimesioal wave eqaio (he eqaio of a vibraig srig) whose solio, has he form f f, where f is a arbirary fcio. By aalogy wih KGE he eqaios si (.) sih (.) are called Sie-Gordo eqaios (SGE) ad Sih-Gordo eqaio (SHGE), respecively. Eqaio (.) has also a impora physical meaig: sice he lef-had side of his eqaio coicides wih he lef-had side of he eqaio of a vibraig srig of he wave [c ad Ugrl (7)] Eqaio (.), i is also a wave eqaio, b, like Eqaio (.), i is oliear (Liao, He (, 4, 6, 7, Bekas e al. (4)) ad describes physical processes relaed o he oliear [Adomia (986)] waves, i pariclar soliary waves (solios) [Drazi(989)] which preserve heir shape der ieracio. This heory is very impora for he heory of plasm. The -dimesioal Galilea ad psedo-galilea spaces ad ca be defied as he affie space E whose hyperplae a ifiiy is edowed by he geomery of he Eclidea space Rosefeld (969) R or he psedo Eclidea space R [see, Rosefeld (969) pp ]. f a sysem of affie coordiaes i he space E is chose sch ha he basis vecors e, e,, e are direced o he hyperplae a ifiiy of R or R, he disace d bewee X,,, Y y, y,, y is eqal o y. f y, whe wo pois ad d, he hese pois have aoher disace d eqal o he disace bewee he X,,, Y y, y,, y i R or R, respecively. The moios i pois ad ad have he form a, A A a i, j,,,, i i i i i j i where A j is a orhogoal or psedo orhogoal mari, respecively (here he Eisei rle for smmaio is sed). These formlas coicide wih he formlas of rasformaio of orhogoal coordiaes i he -dimesioal isoropic or psedo isoropic spaces or, respecively, is he -dimesioal affie space E whose hyperplae a ifiiy is edowed wih he geomery of he co-eclidea space R * or he copsedo Eclidea space R * correspodig o R ad R i he daliy priciple of he projecive space P. The moios i ad have he from

3 AAM: er. J., Vol. 7, sse (December ) 6 i, i i i j i A a A a, i, j,,,. i j The hyperplae a ifiiy of R ad plae i he hyperplae -plae, which is he iersecio of all hyperspheres i R or R, form he absoles of ad. For ad he absoles cosis of he plae a ifiiy, lie, ad wo imagiary cojgae or real pois o his lie. Depedig o a posiio relaive o he absole of For ad he lies ad plaes i hese spaces are divided io wo classes: lies of geeral posiio which do o mee he lie, ad special lies which mee he lie: plaes of geeral posiio which do o coai he lie (he plaes or ),ad special plaes which coai his lie (he plaes R or R ). R, ha is, he, ad he absole imagiary or real hyperqadric i his A each poi X i or we deermie orhoormal frames which cosis of vecors e, e, e of legh or i sch ha he lie Xe is a lie of geeral posiio, ad he lies Xe, Xe are special lies which divide harmoically he lies joiig X wih wo imagiary cojgae or real pois of he absole whose eqaios will be wrie as g g, where g g for g g for ad. f a poi X is characerized by a posiio vecor, he he derivaio formlas for hese frames are i e, de e, de e, i d e, de, where i, j,,,,, for eqaios gives ad for. Eerior differeiaio of hese d, d i, i v d, d. v, (.4) where v,. Formlas (.4) show ha he liear forms ad are locally eac differeials, herefore d, dv. (.5) Cosider a crve C of geeral posiio i or. f X is a poi o his crve, e is age vecor o his crve a X, e is a special vecor of he oscillaig plae of his crve a X, ad e is he hird vecor of a orhoormal frame. The derivaio eqaios of his crve are

4 6 H. Bl ad H. M. Başkoş d d de de de e, ke, e, e, (.6) d d d where is he aral parameer He (), k ad are he crvare ad he orsio of he crve. Cosider a srface S of geeral posiio i or. Le s sppose ha he iersecios of his srfaces wih Eclidea or psedo-eclidea plaes cos. are o sraigh lies, ha is, his srface has o special reciliear geeraors. We deermie a a poi X of his srface he orhoormal frame, whose vecors e ad e are age vecors o S a X ad e is he ormal vecor o S a X, ha is, his vecor is orhogoal o e (he vecors e ad e of his are i a plae = ). The differeial eqaio of Pfaff of he srface S is. (.7) The eerior differeiaio of he eqaio gives, (.8) hece, by meas of he Cara lemma, we obai a b, b c. (.9) The firs fdameal forms of he srface S for geeral crves are ds, (.) ad for special crves are ds g, (.) ha is, iersecios of S, wih plaes. The secod fdameal form of he srface S is d a, e g g a b c. (.) We call a srface i spacelike if g ad imelike if g. The lie of he absole deermies o he srface S, he Koeigs e cosisig of special crves which are iersecios of S wih he plaes, ad of crves of agecy of coes wih apices o he lie of he

5 AAM: er. J., Vol. 7, sse (December ) 6 absole age o S. The crves of his e are crvare lies of S, sice ormal lies o S alog crves of his e forms developable srface. A he pois of he crvare lies of S of geeral posiio, vecors e of movig frames have cosa direcios, sice hey are direced o he apices of coes, herefore for crvare crves of geeral posiio,. follows from (.5), ha he eqaios of crvare crves of S are cos. v cos. The coordiaes ad v are called caoical coordiaes o he srface S. The pricipal crvares of S, ha is ormal crvares are, respecively, k / for crves ad ac b k c, k g. (.) c Hece, he Gassia crvare K = K e = k k of he srface S is k g ac b K k. (.4) Therefore, he Gassia crvare of a srface S i ad of a imelike srface i is eqal o ac b ad of a spacelike srface i is eqal o b ac. Noe ha he codiio for he srfaces S i ad which have o special reciliear geeraors is he eqaliy c =. The crves o a srface S which are deermied by he eqaio = are asympoic crves. Sice he form is epressed by he formla (.), he codiio for fidig asympoic direcios = / of geeral posiio is a b c. (.5) he case whe vecor e of he movig frame is fied a every poi A o S, he movig frame is caoical ad all oher forms are pricipal, ha is a. (.6) The eerior differeiaio of he forms (.9) ad (.6) ad he sbsiio of epressio (.7), (.6), (.4) ad (.6) io (.4) give - K, (.7) g - a b b ca, (.8) -b c c, (.9)

6 64 H. Bl ad H. M. Başkoş where he idices p ad p mea Pfaffia derivaives deermied by he formla dp p p. Le he vecor e e be age o a crvare crve of geeral posiio, ha is, he coordiae sysem o he srface S is caoical sysem, v, ad le s fid he correspodig differeial forms of he movig frame. Sice he crvare of a special crvare crve is k dv ds, where v is o agle bewee age lies ad s is he legh of special crve he formlas (.) ad (.) imply ha dv cw. Le s deoe he radis of crvare of his crve by k c (.) he we have dv. (.) By he sbsiio (.) io he secod formla (.6) ad by formlas (.5) ad a, we obai b. (.) Therefore, he formlas (.) have he from k c, k g a. (.) By formlas (.5) ad (.) we ca epress he Pfaffia derivaives p ad p hrogh he parial derivaes p ad p v i he form p p, p cpv. Therefore, by he eqaios (.8), (.9) ad (.) we obai c, av, ha is, a a d dv. v his case formla (.7) ca be wrie as a vv v a. (.4) This formla shows ha he codiios of iegrabiliy of he differeial eqaios of a srface S are redced o sigle differeial eqaio for is pricipal crvares k ad k.

7 AAM: er. J., Vol. 7, sse (December ) 65 Le a srface S have a crvare K cos = m where. Whe a srface S is referred o he caoical coordiaes, le s show ha he eqaio (.4) ca be redced o he form (.). his case formlas (.) ca be wrie as k c, k Kk m ad also formla (.4) ca be wrie as g m vv m g. (.5) Le s se for a srface i m m, ha is, if we deoe by, vv. ad for a spacelike srface i ; he we obai v / m by, he fcio by for m =, we obai a eqaio. A Aalysis of he HPM A lo of mehods have bee sed o obai solios of parial differeial eqaios i he lierare [He ad Elaga ()]. We cosider HPM which is oe of he mos sed mehods. The firs of all, we ms obai form HMP [Abbasbady (6), He (999)] for HPM. To illsrae he basic ideas of his mehod, we cosider he followig eqaio: f r, r, A (.) wih bodary codiio B,, r, (.) where A is a geeral differeial operaor, B a bodary operaor, f (r) a kow aalyical fcio ad Γ is he bodary of he domai Ω. A ca be divided io wo pars which are L ad N, where L is liear ad N is oliear. Eqaio (.) ca herefore be rewrie as follows; LN f r, r. (.) Homoopy perrbaio srcre is show as followig;

8 66 H. Bl ad H. M. Başkoş where v p p Lv L p Av f r,, r, p:, (.4) v. (.5) Eqaio (.4), p [, ] is a embeddig parameer ad is he firs approimaio ha saisfies he bodary codiio. We ca assme ha he solio of Eqaio (.4) ca be wrie as a power series i p, as followig; v v pv p v p v (.6), ad he bes approimaio for solio is lim vv v v v, (.7) p The covergece of series Eqaio (.7) has bee proved by He (). This echiqe ca have fll advaage of he radiioal perrbaio echiqes. The series Eqaio (.7) is coverge rae depeds o he o-liear operaor Av ( ) (he followig opiios are sggesed by He (): () The secod derivaive of Nv ( ) wih respec o v ms be small becase he parameer may be relaively largee, i.e., p. () The orm of L ( N / v) ms be smaller ha oe so ha he series coverges.. Applicaios of The HPM Eample. We cosider a liear parial differeial eqaio i order o illsrae he echiqe discssed above. The problem of he form is, (.) where eac solio of he differeial Eqaio (.) is give by; (, ) ( )si (.) ad iiial codiios

9 AAM: er. J., Vol. 7, sse (December ) 67 (,). Srcre of HPM He (4,,, 9) for Eqaio (.) is p [ Y] p YY Y, '' '', Y py p py py py '', (.) Y p py py (.4) y where Y, Y form as followig; " y ad p,. We sppose ha he solio of Eqaio (.) has he Y Y py p Y p Y p Y, (.5) Y Y py p Y p Y, Y Y py p Y p Y. " " " " " The, sbsiig Eqaio (.5) io Eqaio (.4), ad rearragig based o powers of p - erms, we obai: Y py py py p py py py py py py py py " " " ", " " ", Y py p Y p Y p py p Y p Y py p Y p Y p Y : (.6) " p : Y Y Y (.7) " p : Y Y Y (.8) " p : Y Y Y (.9) wih solvig Eqaio (.6-.9); p Y Y : (,), (.)

10 68 H. Bl ad H. M. Başkoş " " : p Y Y Y Y Y Y " Y Y Y dd Y, 6 " " : p Y Y Y Y Y Y " Y Y Y dd (.) 5 Y, 5! " " : " p Y Y Y Y Y Y Y Y Y dd (.) 7 Y ( ), 7! he erms of Eqaio (.5) cold easily calclaed. Whe we cosider he series Eqaio (.5) wih he erms Eqaios (.6-.9) ad sppose p, we obai approimaio solio of Eqaio (.) as followig;, Y Y Y Y. (.) As a resl, he compoes Y, Y, Y, Y, are ideified. We obai aalyic solio of Eqaio (.) as followig; 5 7, ( ) ,! 5! 7!

11 AAM: er. J., Vol. 7, sse (December ) a. Eac Solio b. Approimaio Solio HPM Figre. The D srfaces for Y i compariso wih he aalyic solio (, ) whe =.5 wih iiial codiio of Eqaio (.) by meas of HPM Eac. Sol App. Hpm a. Eac Solio b. Approimaio Solio HPM Figre. The D plos of he merical resls for Y i compariso wih he aalyic solio (, ) whe =.5 wih iiial codiio of Eqaio (.) by meas of HPM Eample. We cosider a oliear parial differeial eqaio give as,,. (.4) The bodary codiios ad iiial codiio are (,) (,) e,, (, ) e, ( ). (.5) Srcre of HPM He (4,,, 9) for Eqaio (.) is

12 6 H. Bl ad H. M. Başkoş [ ], '' ' p Y p Y Y Y YY '' ', Y py p py py py pyy '' ', (.6) Y p py py pyy (.7) y " y ' y where Y, Y, Y, (.4) has he form as followig; ad p,. We sppose ha he solio of Eqaio Y Y py p Y p Y p Y,, (.8), Y Y py p Y p Y " " " " ", Y Y py p Y p Y (.9) Y Y py p Y p Y ' ' ' ' '. The, sbsiig Eqaio (.9) io Eqaio (.7), ad rearragig based o powers of p - erms, we obai; p Y :, (.) " ' p : Y Y YY Y, (.) " ' ' p : Y Y YY YY Y Y, (.) " ' ' ' p : Y Y YY YY Y YY Y Y, (.) wih solvig Eqaio (.-.); p Y Y Y e : (,) ( ), (.4) " ' " ' p : Y Y YY Y Y Y YY Y, " ' Y Y YY Y d d, " ' Y Y YY Y dd, Y e,!!

13 AAM: er. J., Vol. 7, sse (December ) 6 " ' ' p : Y Y YY YY Y Y, " ' ' Y Y YY YY Y Y, " ' ' Y Y YY YY Y Y dd, 4 5 Y e, 4! 5! " ' ' ' p : Y Y YY YY Y YY Y Y, " ' ' ' Y Y YY YY Y YY Y Y, " ' ' ' Y Y YY YY Y YY Y Y dd, 6 7 Y e, 6! 7! (.5) (.6) he erms of Eqaio (.8) cold calclaed. Whe we cosider he series Eqaio (.8) wih he erms Eqaio (.4)- Eqaio (.6) ad sppose p, we obai approimaio solio of Eqaio (.4) as followig;, Y Y Y Y (.7) As a resl, he compoes Y, Y, Y, Y, are ideified. We obai aalyic solio of Eqaio (.4) , e.!! 4! 5! 6! 7! a: Eac Solio b: Approimaio Solio HPM Figre. The D srfaces for Y i compariso wih he aalyic solio (, ) whe =.5 wih iiial codiio of Eqaio (.4) by meas of HPM

14 6 H. Bl ad H. M. Başkoş Table. Absole errors obai for Eample The merical resls for Y i compariso wih he aalyic solio (, ) whe. wih iiial codiio of Eqaio (.) by meas of HPM, ADM, VM. Eac Hpm Eac Vim Eac Adm.,8778E 7 -,96,8778E 7. -5,55E 7 -,6577-5,55E 7. -,98.4,5E 4 -,966,5E 4.5,58E 4 -,6445,58E Eac Sol Appr. Hpm a. Eac Solio b. Approimaio Solio HPM Figre 4. The D plos for Y i compariso wih he aalyic solio (, ) whe =.5 wih iiial codiio of Eqaio (.4) by meas of HPM Table. Absole errors obai for Eample.. The merical resls for Y i compariso wih he aalyic solio (, ) whe.wih iiial codiio of Eq.(.4) by meas of HPM, ADM, VM. Eac Vim Eac Hpm.,77E 9,6547E 8,77E. 7,768E,484E 6 7,768E.,866E 9,5485E 5,866E 9.4,87949E 8,97E 4,87949E 8.5,E 7,65E 4,E 7 Eac Adm

15 AAM: er. J., Vol. 7, sse (December ) 6 4. Nmerical Compariso Tables ad show he differece of aalyical solio ad merical solio of he absole error. We also demosrae he merical solio of Eqaio (.) i Figre (a), he correspodig approimae merical solio i Figre (b) ad Eqaio (.4) i Figre (a), he correspodig approimae merical solio i Figre (b). We oe ha oly erms were sed i evalaig he approimae solio. We achieved a very good approimaio wih he acal solio of he eqaios by sig erms oly of he Homoopy perrbaio mehod above. is evide ha he overall errors ca be made smaller by addig ew erms o he perrbaio. Nmerical approimaios shows a high degree of accracy ad i mos cases, he -erm approimaio, is accrae for qie low vales of. The solio is very rapidly coverge by ilizig he homoopy perrbaio mehod. We obaied ha he re of his mehodology jsify from his merical resls, eve i he few erms approimaio is accrae (a) Eac Solio (b) Nmerical Solio (HPM) (a) Eac Solio (b) Nmerical Solio (ADM) (a) Eac Solio Figres A b) Nmerical Solio (VM).8.6.4

16 64 H. Bl ad H. M. Başkoş (a) Eac Solio (b) Nmerical Solio (HPM) (a) Eac Solio (b) Nmerical Solio (ADM) (a) Eac Solio (b) Nmerical Solio (VM) Figres B 5. Coclsio The mehod have bee sed for solvig a lo of differeial eqaios sch as ordiary, parial, liear, oliear, homogeeos, ohomogeeos by may researchers. his research, we sed for solvig liear ad oliear Klei-Gordo eqaios wih iiial codiios. Accordig o hese daas sch as D, D graphics ad Tables, oe realize ha oe of he advaages of HPM displays a fas covergece of he solios.

17 AAM: er. J., Vol. 7, sse (December ) 65 REFERENCES Adomia G. (986). Noliear Sochasic Operaor Eqaios, Academic Press, Sa Diego. Abbasbady S. (6). Applicaio of He s homoopy perrbaio mehod for Laplace rasform, Chaos Solios Fracals,, 6. Bekas, M., Bl, H. ad Erg, M. (4). Geomeric erpreaios of The Klei-Gordo Eqaio ad Solio of The Eqaio By The Adomia Decomposiio Mehod, Fira Uiversiesi Fe ve Mühedislik Bilimler Dergisi, 6(), 4 4. Bl, H. ad Baskos, H.M. (9). A Comparsio Amog Homoopy Perrbaio Mehod Ad The Decomposiio Mehod Wih The Variaioal eraio Mehod For Dispersive Eqaio eraioal Joral of Basic & Applied Scieces JBAS, 9(), 4. Drazi, P. G. ad Johso R.S. (989). Solios: a rodcio, Cambridge Uiversiy Press, New York, USA. He, J.H. (999). Homoopy perrbaio echiqe, Comper Mehods i Applied Mechaics ad Egieerig, 78, He, J.H. (6). Some asympoic mehods for srogly oliear eqaios. eraioal Joral of Moder Physics. B, (), He, J.H. (4). The homoopy perrbaio mehod for oliear oscillaors wih discoiiies, Applied Mahemaics ad Compaio, 5, He, J.H. (). Deermiaio of limi cycles for srogly oliear oscillaors. Physical Review Leer, 9(7), 74-. He, J.H. (). Bookkeepig parameer i perrbaio mehods, eraioal Joral No- Liear Sciece Nmerical Simlaio. (4), 7. He, J.H.(). A coplig mehod a homoopy echiqe ad a perrbaio echiqe for oliear problems, eraioal Joral No-Liear Mechaic, 5, 7 4. He, J.H. ad Elaga S.K. (). A firs corse i fcioal aalysis, Asia Academic Pblisher Limied, Hog Kog. c, M. ad Ugrl, Y. (7). Nmerical Simlaio of he reglarized log wave eqaio by He's homoopy perrbaio mehod, Phys. Le. A 69, Liao, S.J. (4). O he homoopy aalysis mehod for oliear problems, Applied Mahemaics ad Comper, 47, Rosefeld, B. A. (969). No Eclidea Spaces, Naka, Moscow, RUSSA.