SOLUTION OF THE HYPERBOLIC KEPLER EQUATION BY ADOMIAN S ASYMPTOTIC DECOMPOSITION METHOD

Transcription

1 Romaia Rports i Physics 70, XYZ (08) SOLUTION OF THE HYPERBOLIC KEPLER EQUATION BY ADOMIAN S ASYMPTOTIC DECOMPOSITION METHOD ABDULRAHMAN F ALJOHANI, RANDOLPH RACH, ESSAM EL-ZAHAR,4, ABDUL-MAJID WAZWAZ 5, ABDELHALIM EBAID,* Dpartmt of Mathmatics, Faculty of Scic, Uivrsity of Tabuk, POBox 74, Tabuk 749, Kigdom of Saudi Arabia Th Gorg Adomia Ctr for Applid Mathmatics, 6 South Mapl Strt, Hartford, Michiga , USA Dpartmt of Mathmatics, Faculty of Scics ad Humaitis, Pric Sattam Bi Abdulaziz Uivrsity, Alkharj, 94, Kigdom of Saudi Arabia 4 Dpartmt of Basic Egirig Scic, Faculty of Egirig, Shbi El-Kom, 5, Mofia Uivrsity, Egypt 5 Dpartmt of Mathmatics, Sait Xavir Uivrsity, Chicago, IL 60655, USA * Corrspodig author, abaid@utdusa Rcivd Octobr 6, 07 Abstract Th hyprbolic Kplr quatio is of practical itrst i astroomy It is oft usd to dscrib th cctric aomaly of a comt of xtrasolar origi i its hyprbolic trajctory past th Su Efficit dtrmiatio of th radial distac ad/or th Cartsia coordiats of th comt rquirs accurat calculatio of th cctric aomaly, hc th d for a covit, robust mthod to solv Kplr s quatio of hyprbolic typ I this papr, th Adomia s asymptotic dcompositio mthod is proposd to solv this quatio Our calculatios hav dmostratd a rapid rat of covrgc of th squc of th obtaid approximat solutios, which ar displayd i svral graphs Also, w hav show i this papr that oly a fw trms of th Adomia dcompositio sris ar sufficit to achiv xtrmly accurat umrical rsults v for much highr valus tha thos i th litratur for th ma aomaly ad th cctricity of th orbit Th mai charactristic of th obtaid approximat solutios is that thy ar all odd fuctios i th ma aomaly, which w hav illustratd through graphs I additio, it is foud that th absolut rmaidr rror usig oly thr compots of Adomia s solutio dcrass across a spcifid domai ad approachs zro as th cctric aomaly tds to ifiity Morovr, th absolut rmaidr rror dcrass by icrasig th umbr of compots of th Adomia dcompositio sris Fially, th currt aalysis may b th first to mak a ffctiv applicatio of th Adomia s asymptotic dcompositio mthod i astroomical physics Ky words: Adomia s asymptotic dcompositio mthod; Adomia polyomials; hyprbolic Kplr s quatio; sris solutio INTRODUCTION I clstial mchaics, Kplr s quatios of lliptical ad hyprbolic typs play importat rols Efficit dtrmiatio of th accurat positio of a objct (a plat, comt or a astroid) wh orbitig th Su rquirs solvig ithr of ths

2 Articl o A F Aljohai, R Rach, E El-Zahar, A-M Wazwaz, A Ebaid quatios with a solutio mthod of high accuracy I our solar systm, som comts of xtrasolar origi follow hyprbolic trajctoris past th Su Such comts tr th solar systm comig from th Oort cloud ad itrstllar spac ad may xit th solar systm through hyprbolic trajctoris I astroomy, th origial orbits of such comts may chag from lliptical to hyprbolic, spcially, wh takig ito accout th possibl gravitatioal attractio of th plats of substatial mass, g, Jupitr I this papr, Kplr s quatio of hyprbolic typ is cosidrd i th stadard form [], sih( H ( t) ) H ( t) = M ( t), <, 0 M <, () μ whr H is dfid as th cctric aomaly, M ( t) = ( t τ ) is th ma a aomaly, a is th smi-major axis, is th cctricity of th orbit, μ = GM is th gravitatioal paramtr of th ctral body of mass M, whr G is th uivrsal gravitatioal costat ad τ is th tim of passag through th closst poit of approach to th focus Th polar quatio of a hyprbola with its focus at th origi may b writt as a( ) r = () + cos f This agl f is giv i trms of H through th followig rlatioship [], f + H ta = ta () I additio, at a istat t, th ( x, y) coordiats of a objct i th stadard fram of rfrc (whr th ctral body is at th origi ad th x -axis poits towards th priapsis) ar giv i trms of H by x = a( cosh H ), (4) (5) y = a sih H So, i ordr to calculat th radial distac r ad th tru aomaly f of a comt wh orbitig th Su at a spcifid tim t, th hyprbolic Kplr quatio () is first solvd for H at that tim t ad th Eqs () ad () ar applid As show by Eq (), th hyprbolic Kplr quatio is a trascdtal quatio that has o xact closd-form aalytic solutio Although may authors [ 0] hav dvisd various umrical ad aalytical solutios for Kplr s quatio of lliptical typ, littl ffort has b dvotd to ivstigat th hyprbolic form of this quatio [ 4] Sarchig for a w accurat but simpl aalytical solutio for th hyprbolic Kplr quatio is still of maifst practical itrst I ordr to cotribut to a improvd solutio of this problm, th authors bliv that th

3 Solutio of th hyprbolic Kplr quatio by Adomia s dcompositio mthod Articl o Th Adomia s dcompositio mthod (ADM) ca b ffctivly applid to solv th hyprbolic form of this quatio Th ADM is a systmatic aalytic approximatio mthod for solvig algbraic ad trascdtal quatios, matrix quatios, oliar itgral quatios, ad oliar diffrtial quatios icludig both oliar iitial valu problms ad oliar boudary valu problms v for irrgular boudary cotours It has b widly implmtd to solv a larg umbr of frotir problms i th applid scics ad girig ad ca b also xtdd to covr may scitific modls [5 4] It xprsss th solutio i th form of a ifiit sris Udr physically appropriat coditios, this sris oft rapidly covrgs ad hc a fw trms of Adomia s mthod ar sufficit to obtai accurat umrical rsults for th ivstigatd problm I this cas, th squc of approximat solutios by Adomia s mthod covrgs to a crtai curv or fuctio For xampl, th ADM has b applid by Ebaid [] to solv th Thomas-Frmi quatio that has o xact solutio I that papr, h showd gomtrically that th squc of th approximat solutios covrgs to a crtai curv, which may v b th xact solutio for that problm Th objctiv of this papr is to aalyz th hyprbolic Kplr quatio by usig th ADM W show that th Adomia s dcompositio sris solutio to th currt problm closly coicids with thos i th litratur for all valus of th ma aomaly paramtr M [0, ) ad for all valus of th cctricity of th orbit usig oly a fw of Adomia s solutio compots I additio, it will b show i this papr that th squc of th Adomia s dcompositio approximat solutios covrgs rapidly i a much widr rag tha thos cosidrd i th litratur for th paramtrs M ad APPLICATION OF ADOMIAN S ASYMPTOTIC DECOMPOSITION METHOD Th Adomia s asymptotic dcompositio mthod (AADM) is applid i this Sctio to calculat a squc of approximat aalytic solutios for th hyprbolic Kplr quatio W rwrit Eq () i th caoical form as M ( t) sih( H ( t) ) = + H ( t) (6) Th sris of th Adomia polyomials ad th Adomia dcompositio sris ar ( H t ) A t A t A ( H t H t ) sih () = (), ()= (),, (), =0 =0 Ht ()= H() t 0 (7)

4 Articl o A F Aljohai, R Rach, E El-Zahar, A-M Wazwaz, A Ebaid 4 Upo substitutio of Eq (7) ito Eq (6), w obtai M ( t) A ( t) = + H ( t) (8) =0 =0 Accordigly, th followig algorithm ca b stablishd usig th Adomia rcursio schm, M ( t) A 0( t) =, A + ( t) = H, 0 (9) Th Adomia polyomials for th hyprbolic si oliarity wr dfid by Adomia ad Rach i 98 [] as d = sih m A λ H ( t) m (0)! dλ m= 0 λ=0 Applyig this formula, th first svral Adomia polyomials for th hyprbolic si oliarity ar A 0 = sih( H 0 ), A = H cosh( H 0 ), A = H cosh( H 0 ) + H sih( H0 ),! A H cosh 0 0 +! ( H ) + H H sih( H ) H cosh( ), = H0 () Hc A A A A = H 0 = H = H 4 = H,,,, ()

5 5 Solutio of th hyprbolic Kplr quatio by Adomia s dcompositio mthod Articl o ad so o Combiig () ad (), w obtai M sih( H 0 ) =, H cosh( H0 ) = H0, H cosh( H0 ) + H sih( H0 ) = H,! H cosh H 0 + HH sih H0 + H cosh H 0 = H! Thrfor = M H, 0 sih M sih H =, + M H ( ) ( ) ( ) M M ( + M )sih M + M sih =, ( + M ) M M 6( + M ) sih 9 M + M sih ( M ) sih M H =, 5/ 6( + M ) () (4) Th ADM givs th -trm approximat aalytic solutio Φ (t) for th hyprbolic Kplr quatio as Accordigly, w calculat M Φ ()= t, + sih + M i=0 Φ ( t) = H i ( t) (5)

6 Articl o A F Aljohai, R Rach, E El-Zahar, A-M Wazwaz, A Ebaid 6 M M M Φ M + M ( + M ) ()= t sih, / sih (6) + M M M Φ ()= t / sih / + M ( + M ) ( + M ) M ( M ) M sih 5/ sih 6( + M ) I a subsqut Sctio, w will show that th squc of th approximat solutios i (6) for th hyprbolic Kplr quatio is covrgt i a widr rag tha thos rportd i th litratur for M ad I additio, th accuracy of th prst umrical rsults will b validatd by calculatig th absolut rmaidr rror RE + ( t) dfid by RE + ( t) = sih( Φ + ( t) ) Φ+ ( t) M, 0, (7) by usig th -trm approximat solutio to stimat th cctric aomaly H Morovr, th advatag ad th ffctivss of th prst low-ordr approximat aalytic solutios for th hyprbolic Kplr quatio ovr svral xistig mthods i th litratur will b provd for crtai highr valus of th paramtrs M ad DISCUSSION I th prvious Sctio, Adomia s asymptotic dcompositio mthod has b applid to obtai th approximat solutios of th hyprbolic Kplr quatio Such approximat solutios ar applid i this discussio to obtai svral plots Lt us bgi by graphically dmostratig th covrgc of th prst approximat solutios I Fig, th approximat solutios Φ ( t), Φ 5 ( t), ad Φ7 ( t) ar plottd for = 5 vrsus th ma aomaly M A rapid covrgc is obsrvd i this figur usig oly a fw trms of th Adomia asymptotic solutios

7 7 Solutio of th hyprbolic Kplr quatio by Adomia s dcompositio mthod Articl o Figur : Covrgc of th approximat solutios vrsus M at = 5 Th mai rsult hr is that th rat of covrgc is icrasd for highr valus of M, whr at M 4 th thr-trm approximat solutio Φ ( t) of Adomia s asymptotic dcompositio mthod is sufficit to provid a rmarkably accurat solutio, whil at th lowr valus of M i th domai [0,4) a highrordr approximat solutio such as Φ (t) for 5 is rquird to achiv a similarly high accuracy I additio, ths approximat solutios ar all odd fuctios i th ma aomaly M which has b show i Fig for a slctd highr valu of th cctricity paramtr = 05

8 Articl o A F Aljohai, R Rach, E El-Zahar, A-M Wazwaz, A Ebaid 8 Figur : Th odd proprty of th approximat solutios vrsus M at = 05 I a widr rag of M [0,00], it has b also show from Fig that th approximat solutios Φ ( ), Φ ( ), ad Φ ( ) covrg to a crtai t 5 t curv/fuctio Bsids, th odd proprty is also show i Fig 4 for a furthr widr rag M [ 00,00] Morovr, Fig 5 ad Fig 6 also dmostrat th rapid rat of covrgc of Adomia s squc of aalytic approximat solutios vrsus th cctricity paramtr i th domai [5,55] at th lowst ad th highst valus of M, which was cosidrd by Sharaf t al [], rspctivly It is radily appart from ths figurs that Adomia s solutios also covrg, v at th highst cosidrd valu, M = Th aformtiod discussio corroborats th ffctivss of Adomia s asymptotic dcompositio mthod i quickly ad accuratly solvig th hyprbolic Kplr quatio 7 t

9 9 Solutio of th hyprbolic Kplr quatio by Adomia s dcompositio mthod Articl o Figur : Covrgc of th approximat solutios vrsus M at = 5 i a widr rag Figur 4: Th odd proprty of th approximat solutios vrsus M at = 05 i a widr rag

10 Articl o A F Aljohai, R Rach, E El-Zahar, A-M Wazwaz, A Ebaid 0 Figur 5: Covrgc of th approximat solutios vrsus at M = 58 Figur 6: Covrgc of th approximat solutios vrsus at M = 75005

11 Solutio of th hyprbolic Kplr quatio by Adomia s dcompositio mthod Articl o Th ffctivss ad fficicy of th currt lowr-ordr approximat solutios ar umrically validatd i Figs 7-8 I ths figurs, th corrspodig absolut rmaidr rrors RE, RE5, ad RE7 ar displayd at two slctd valus of th cctricity = 5 ad = 00 for two slctd domais of th ma aomaly M [0,00] i Fig 7 ad M [0,0000] i Fig 8 W coclud from figurs 7-8 that th absolut rmaidr rror RE usig oly thr compots of Adomia s asymptotic dcompositio mthod dcrass ad approachs zro as M tds to ifiity Howvr, th maximum valu of th absolut rmaidr rror RE is about 0 radias i th rgio M (0,) ad it is about 008 radias M (,6) as i Fig 7 ad th it dcrass with icrasig M th absolut rmaidr rror dcrass with icrasig th cctricity For xampl, th maximum valu of th absolut rmaidr rror RE is about radias as i Fig 8 wh = 00 th absolut rmaidr rror dcrass by icrasig th umbr of compots as illustratd by th curvs of RE ad RE i Figs Figur 7: Th absolut rmaidr rror vrsus M at = 5

12 Articl o A F Aljohai, R Rach, E El-Zahar, A-M Wazwaz, A Ebaid Figur 8: Th absolut rmaidr rror vrsus M at = 00 Figur 9: Th absolut rmaidr rror vrsus at M = 500

13 Solutio of th hyprbolic Kplr quatio by Adomia s dcompositio mthod Articl o Figur 0: Th absolut rmaidr rror vrsus at M = For a furthr validatio of th currt umrical rsults, two additioal plots ar displayd i Figs 9-0 for th absolut rmaidr rrors RE, RE5 ad RE7 vrsus th cctricity i th domai [5,00] for M = 500 ad M = 75000, rspctivly Th rsults plottd i ths two figurs rval that th approximat solutio usig oly thr trms of th asymptotic Adomia s sris is also accurat as M at all cosidrd valus of th cctricity paramtr Morovr, th absolut rmaidr rrors RE5 ad RE7 approach zro v at ths highr valus of th ma aomaly ad th cctricity This, of cours, provs th svral rmarkabl advatags of th Adomia s asymptotic dcompositio mthod ovr th xistig mthods i th litratur Fially, th authors of th prst papr rcommd th currt approach as th most ffctiv aalytical tchiqu to solv th hyprbolic Kplr quatio 4 CONCLUSION I this papr, Kplr s quatio for hyprbloic orbits has b aalytically solvd by usig th Adomia s asymptotic dcompositio mthod (AADM) Th odd proprty of th obtaid approximat solutios has b dmostratd through a sris of graphs Also, w hav show that oly a fw trms of th AADM ar sufficit to obtai xtrmly accurat umrical rsults v at highr valus tha

14 Articl o A F Aljohai, R Rach, E El-Zahar, A-M Wazwaz, A Ebaid 4 thos i th litratur for th ma aomaly ad th cctricity paramtrs I compariso to th mthods rportd i th litratur, th prst approach is ot oly ffctiv but also vry simpl i tchiqu Morovr, th advatags of th currt rsults for th cctric aomaly ar valid i ay spcifid domai Bsids, vry small absolut rmaidr rrors hav b achivd usig oly a fw trms of th Adomia s asymptotic dcompositio sris Th ffctivss provd i this papr for Adomia s asymptotic mthod to quickly ad asily solv th hyprbolic Kplr quatio may b advatagously xtdd to othr problms i physics ad girig REFERENCES FR Moulto, A Itroductio to Clstial Mchaics, d rvisd ditio, Dovr, Nw York, 970 NI Ioakimids ad KE Papadakis, Clstial Mchaics ad Dyamical Astroomy 5 (4), 05 6 (985) NM Swrdlow, J for History of Astroomy, 9 4 (000) 4 L Stumpf, Chaotic Clstial Mchaics ad Dyamical Astroomy 74, (999) 5 M Palacios, J Comput Appl Math 8, 5 46 (00) 6 P Colwll, Amrica Mathmatical Mothly 99(), 45 48(99) 7 P Colwll, Solvig Kplr s quatio ovr thr cturis, Willma-Bll Ed, Richmod, 99 8 K Boubakr, Amrica Mathmatical Mothly, Apiro 7 (), (00) 9 Rza Esmalzadh ad Hossi Ghadiri, It J Computr Applicatios 89(7), 8(04) 0 Curila Mirca ad Curila Sori, Aall Uivrsitatii di Orada, Fascicula Protc tia Mdiului, Vol XXII, 9 4 (04) M Avdao, V Marti-Molia, ad J Ortigas-Galido, arxiv:50064v [physicsclass-ph], (0 Fb 05) T Fukushima, Clstial Mchaics ad Dyamical Astroomy 68, 7(997) MA Sharaf, MA Baajh, ad AA Alshaary, J Astrophys Astr 8, 9 6 (007) 4 A Alshary ad A Ebaid, Acta Astroautica 40, 7 (07) 5 G Adomia ad R Rach, J Math Aal Applic 05, 4 66 (985) 6 G Adomia ad R Rach, J Math Aal Applic (), 6 40(985) 7 G Adomia ad R Rach, Kybrts 5 (), 7(986) 8 AM Wazwaz, Rom J Phys 6, (06) 9 AM Wazwaz t al, Rom Rp Phys 69, 0 (07) 0 Y Zhag, D Balau, ad X J Yag, Proc Romaia Acad A 7, 0 6 (06) AH Bhrawy t al, Proc Romaia Acad A 8, 7 4 (07) G Adomia, R Rach, J Math Aal Applic 9 (), 9 46 (98) I A Cristscu, Rom Rp Phys 68, (06) 4 RM Hafz t al, Rom Rp Phys 68, 7 (06) 5 L Bougoffa, Rom J Phys 6, 0 (07) 6 AM Wazwaz, Rom J Phys 60, 56 7 (05) 7 AM Wazwaz t al, Rom Rp Phys 69, 0 (07) 8 K Parad, H Yousfi, ad M Dlkhosh, Rom J Phys 6, 04 (07) 9 A Ebaid, Z Naturforschug A 66, 4 46 (0) 0 JS Dua ad R Rach, Appl Math Comput 8, (0) A Ebaid, J Comput Appl Math 5, (0) AM Wazwaz, R Rach, ad JS Dua, Appl Math Comput 9, (0)

15 5 Solutio of th hyprbolic Kplr quatio by Adomia s dcompositio mthod Articl o EH Ali, A Ebaid, ad R Rach, Comput Math Applic 6, (0) 4 I A Cristscu, Rom Rp Phys 68, (06) 5 C Chu, A Ebaid, M L, ad E Aly, ANZIAM Joural 5, 4 (0) 6 A Ebaid, MD Aljoufi, ad A-M Wazwaz, Appl Math Ltt 46, 7 (05) 7 HO Bakodah t al, Rom Rp Phys 70, XYZ (08) 8 AA Gabr ad A Ebaid, Rom Rp Phys 70, XYZ (08) 9 H Lblod, H Triki, ad D Mihalach, Phys Rv A 85, 0586 (0) 40 D Mihalach, Rom J Phys 59, 95 (04) 4 D Mihalach, Proc Romaia Acad A 6, 6 69 (05) 4 D Mihalach, Rom Rp Phys 69, 40 (07)