A residual power series technique for solving systems of initial value problems

Transcription

1 A residual power series echnique for solving sysems of iniial value problems Omar Abu Arqub, Shaher Momani,3, Ma'mon Abu Hammad, Ahmed Alsaedi 3 Deparmen of Mahemaics, Faculy of Science, Al Balqa Applied Universiy, Sal 97, Jordan Deparmen of Mahemaics, Faculy of Science, The Universiy of Jordan, Amman 94, Jordan 3 Nonlinear Analysis and Applied Mahemaics (NAAM) Research Group, Faculy of Science, King Abdulaziz Universiy, Jeddah 589, Saudi Arabia Absrac. In his aricle, a residual power series echnique for he power series soluion of sysems of iniial value problems is inroduced. The new approach provides he soluion in he form of a rapidly convergen series wih easily compuable componens using symbolic compuaion sofware. The proposed echnique obains Taylor expansion of he soluion of a sysem and reproduces he exac soluion when he soluion is polynomial. Numerical examples are included o demonsrae he efficiency, accuracy, and applicabiliy of he presened echnique. The resuls reveal ha he echnique is very effecive, sraighforward, and simple. Keywords: Sysems of iniial value problems; Residual power series; Taylor expansion AMS Subjec Classificaion: 35F55; 74H0; 34K8. Inroducion In real life siuaions quaniies and heir rae of changes depend on more han one variable. For example, he rabbi populaion, hough i may be represened by a single number, depends on he size of predaor populaions and he availabiliy of food. In order o represen and sudy such complicaed problems we need o use more han one dependen variable and more han one equaion. Sysems of differenial equaions are he ools o use. These inds of equaions can be found in almos all branches of sciences, engineering, and echnology, such as elecromagneic, solid sae physics, plasma physics, elasiciy, fluid dynamics, oscillaion heory, mahemaical biology, chemical ineics, biomechanics, and conrol heory [-6]. In he presen paper, we invesed he residual concep in he power series mehod o obain a simple echnique (we call i residual power series (RPS) [7-5]) o find ou he coefficiens of he series soluions. This echnique helps us o consruc a power series soluion for srongly linear and nonlinear sysems. The RPS echnique is effecive and easy o use for solving linear and nonlinear sysems of iniial value problems (IVPs) wihou linearizaion, perurbaion, or discreizaion. Differen from he classical power series mehod, he RPS echnique does no need o compare he coefficiens of he corresponding erms and recursion relaions are no required. This echnique compues he coefficien of he power series by a chain of linear equaions of n-variable, where n is number of equaions in he given sysem. The RPS echnique is differen from he radiional higher order Taylor series mehod. The Taylor series mehod is compuaionally expensive for large orders. The RPS echnique is an alernaive procedure for obaining analyic Taylor series soluion of sysems of IVPs. By using residual error concep, we ge a series soluion, in pracice a runcaed series soluion. The RPS echnique has he following characerisics [7-5]; firs, he echnique obains Taylor expansion of he soluion; as a resul, he exac soluion is available when he soluion is polynomial. Moreover he soluions and all is derivaives are applicable for each arbirary poin in he given inerval. Second, i does no require any modificaion while swiching from he firs order o he higher order; as a resul he echnique can be applied direcly o he given problem by choosing an appropriae value for he iniial guesses approximaions. Third, he RPS echnique needs small compuaional requiremens wih high precision and less ime. The purpose of his paper is o obain symbolic approximae power series soluions for sysem of IVPs which is as follows: * Corresponding auhor: address: s.momani@ju.edu.jo (Shaher Momani).

2 x () = f (, x (), x (),, x n ()), x () = f (, x (), x (),, x n ()), x n () = f n (, x (), x (),, x n ()), subjec o he iniial condiions x ( 0 ) = x, x ( 0 ) = x,, x n ( 0 ) = x n, () where [ 0, 0 + a], f i : [ 0, 0 + a] R n R are nonlinear coninuous funcions of, x, x,, x n, x i () are unnown funcions of independen variable o be deermined, and 0, a are real finie consans wih a > 0. Throughou his paper, we assume ha f i, x i are analyic funcions on he given inerval. Also, we assume ha f i saisfies all he necessary requiremens for he exisence of a unique soluion. In general, sysems of IVPs do no always have soluions which we can obain using analyical mehods. In fac, many of real physical phenomena encounered, are almos impossible o solve by his echnique. Due o his, some auhors have proposed numerical mehods o approximae he soluions of sysems of IVPs. To menion a few, he homoopy analysis mehod has been applied o solve sysem () and () as described in [6]. In [7] he auhors have developed he homoopy perurbaion mehod. In [8] also, he auhor has provided he differenial ransformaion echnique o furher invesigaion o he above sysem. Furhermore, he reproducing ernel Hilber space mehod is carried ou in [9]. Recenly, a class of collocaion mehods for solving sysem () and () is proposed in [0]. However, none of previous sudies propose a mehodical way o solve sysems of IVPs () and (). Moreover, previous sudies require more effor o achieve he resuls and usually hey are suied for linear form of sysem () and (). On he oher hand, he applicaions of oher versions of series soluions o linear and nonlinear problems can be found in [-6] and references herein. Also, for numerical solvabiliy of differen caegories of differenial equaions one can consul he references [7, 8]. The ouline of he paper is as follows: in he nex secion, we presen he basic idea of he RPS echnique. In secion 3, numerical examples are given o illusrae he capabiliy of proposed approach. This aricle ends in secion 4 wih some concluding remars.. Soluion of sysems of IVPs by RPS echnique In his secion, we employ our echnique of he RPS o find ou series soluion for sysems of IVPs subjec o given iniial condiions. We firs formulae and analyze he RPS echnique for solving such sysems of IVPs. Afer ha, a convergence heorem is presened in order o capure he behavior of he soluion. The RPS echnique consiss in expressing he soluions of sysem of IVPs () and () as a power series expansion abou he iniial poin = 0. To achieve our goal, we suppose ha hese soluions ae he form x i () = x i,m (), i =,,, n, where x i,m are erms of approximaions and are given as x i,m () = c i,m ( 0 ) m. Obviously, when m = 0, since x i,0 () saisfy he iniial condiions (), where x i,0 () are he iniial guesses approximaions of x i (), we have c i,0 = x i,0 ( 0 ) = x i ( 0 ), i =,,, n. If we choose x i,0 () = x i ( 0 ) as iniial guesses approximaions of x i (), hen we can calculae x i,m () for m =,, and approximae he soluions x i () of sysem of IVPs () and () by he h-runcaed series x i () = c i,m ( 0 ) m, i =,,, n. (3) Prior o applying he RPS echnique, we rewrie sysem of IVPs () and () in he form of he following: x i () f i (, x (), x (),, x n ()) = 0, i =,,, n. (4) The subsising of h-runcaed series x i () ino Eq. (4) leads o he following definiion for he h residual funcions: ()

3 Res i () = mc i,m ( 0 ) m m= f i (, c,m ( 0 ) m, c,m ( 0 ) m,, c n,m ( 0 ) m ), i =,,, n, and he following h residual funcions: Res i () = lim Res i (), i =,,, n. I easy o see ha, Res i () = 0 for each [ 0, 0 + a]. This show ha Res i () are infiniely many imes differeniable a = 0. On he oher hand, ds Res d s i ( 0 ) = ds Res d s i ( 0 ) = 0, for each s =,,,. In fac, his relaion is a fundamenal rule in RPS echnique and is applicaions. Now, in order o obain he s-approximae soluions, we pu = and subsiue = 0 ino Eq. (5) and using he fac ha Res i ( 0 ) = Res i ( 0 ) = 0, o conclude c i, = f i ( 0, c.0, c.0,, c n.0 ) = f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 )), i =,,, n. Thus, using s-runcaed series he firs approximaion for sysem of IVPs () and () can be wrien as x i () = x i ( 0 ) + f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 ))( 0 ), i =,,, n. Similarly, o find he nd approximaion, we pu = and x j () = c j,m ( 0 ) m, i =,,, n. On he oher hand, we differeniae boh sides of Eq. (5) wih respec o and subsiue = 0, o ge d d Res i ( 0 ) = c i, f i ( 0, c.0, c.0,, c n.0 ) c j, n j= In fac d d Res i ( 0 ) = d d Res i ( 0 ) = 0. Thus, we can wrie c i, = [ f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 )) + c j, n j= 3 x j x j f i ( 0, c,0, c,0,, c n,0 ), i =,,, n. f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 ))], i =,,, n. Hence, using nd-runcaed series he second approximaion for sysem of IVPs () and () can be wrien as x i () = x i ( 0 ) + f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 ))( 0 ) + [ f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 )) n + c j, j= x j f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 ))] ( 0 ), i =,,, n. This procedure can be repeaed ill he arbirary order coefficiens of RPS soluions for sysem of IVPs () and () are obained. Moreover, higher accuracy can be achieved by evaluaing more componens of he soluion. In oher words, choose large in he runcaion series (3). The nex heorem shows convergence of he RPS echnique. Theorem.. Suppose ha x i (), i =,,, n are he exac soluions for sysem of IVPs () and (). Then, he approximae soluions obained by he RPS echnique are jus he Taylor expansion of x i (), i =,,, n. Proof. Assume ha he approximae soluions for sysem of IVPs () and () are as follows: x i() = c i.0 + c i. ( 0 ) + c i. ( 0 ) +, i =,,, n. (6) In order o prove he heorem, i is enough o show ha he coefficiens c i.m in Eq. (6) ae he form c i.m = x (m) m! i ( 0 ), i =,,, n, (7) for each m = 0,,, where x i () are he exac soluions for sysem of IVPs () and (). Clear ha for m = 0 he iniial condiions () give c i.0 = x i ( 0 ), i =,,, n. (8) Moreover, for m =, subsiue = 0 ino Eq. (), we obain x i ( 0 ) = f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 )), i =,,, n. (5)

4 On he oher hand, from Eqs. (6) and (8), we can wrie x i() = x i ( 0 ) + c i. ( 0 ) + c i. ( 0 ) +, i =,,, n, (9) by subsiuing Eq. (9) ino Eq. () and hen seing = 0, we ge c i. = f i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 )) = x i ( 0 ), i =,,, n. (0) Furher, for m =, differeniaing boh sides of Eq. () wih respec o, we obain n x i () = f i (, x (), x (),, x n ()) + x j () f x i (, x (), x (),, x n ()), i =,,, n, () j j= by subsiuing = 0 in Eq. (), we can conclude ha n x i ( 0 ) = f i( 0, x ( 0 ), x ( 0 ),, x n ( 0 )) + x j ( 0 ) f x i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 )), i =,,, n. () j j= According o Eqs. (9) and (0), we can wrie he approximaion for sysem of IVPs () and () as follows: x i() = x i ( 0 ) + x i ( 0 )( 0 ) + c i. ( 0 ) +,i =,,, n, (3) by subsiuing Eq. (3) ino Eq. () and seing = 0, we obain n c i. = f i( 0, x ( 0 ), x ( 0 ),, x n ( 0 )) + x j ( 0 ) f x i ( 0, x ( 0 ), x ( 0 ),, x n ( 0 )), i =,,, n. (4) j j= Finally, by comparing Eqs. () and (4), we can conclude ha c i. = x i ( 0 ), i =,,, n. By coninuing he above procedure, we can easily prove Eq. (7) for m = 3,4,. So, he proof of he heorem is complee. Corollary.. If some of x i (), i =,,, n is a polynomial, hen he RPS echnique will be obained he exac soluion. I will be convenien o have a noaion for he error in he approximaion x i () x i (). Accordingly, we will le Rem i () denoe he difference beween x i () and is h Taylor polynomial; ha is, Rem i () = x i () x i () = x (m) i ( 0 ) ( m! 0 ) m, i =,,, n. m=+ The funcions Rem i () are called he h remainder for he Taylor series of x i (). In fac, i ofen happens ha he remainders Rem i () become smaller and smaller, approaching zero, as ges large. 3. Numerical resuls and discussion The proposed mehod provides an analyical approximae soluion in erms of an infinie power series. However, here is a pracical need o evaluae his soluion, and o obain numerical values from he infinie power series. The consequen series runcaion and he pracical procedure are conduced o accomplish his as, ransforms he oherwise analyical resuls ino an exac soluion, which is evaluaed o a finie degree of accuracy. In his secion, we consider five examples o demonsrae he performance and efficiency of he presen echnique. Throughou his paper, all he symbolic and numerical compuaions performed by using Maple 3 sofware pacage. To show he accuracy of he presen mehod for our problems, we repor four ypes of error. The firs one is he residual error, Res i (), and defined as Res i () d d x i () f i (, x (), x (),, x n ()), while he exac, Ex, relaive, Rel, and consecuive, Con, errors are defined, respecively, by Ex i (): = x i,exac () x i (), Rel i (): = x i,exac() x i (), x i,exac () Con i (): = x i + () x i (), 4

5 for i =,,, n, where [ 0, 0 + a], x i are he h-order approximaion of x i,exac () obained by he RPS echnique, and x i,exac () are he exac soluion. In mos real life siuaions, he differenial equaion ha models he problem is oo complicaed o solve exacly, and here is a pracical need o approximae he soluion. In he nex wo examples, he exac soluions canno be found analyically. Example 3.. Consider he nonlinear SIR model [9]: S () = βs()i(), I () = βs()i() γi(), R () = γi(), subjec o he iniial condiions S(0) = N S, I(0) = N I, R(0) = N R, where β, γ and N S, N I, N R are posiive real numbers. (5) (6) The SIR model is one common epidemiological model for he spread of disease, which consiss of a sysem of hree differenial equaions ha describe he changes in he number of suscepible, infeced, and recovered individuals in a given populaion. This was inroduced as far bac as 97 by Kermac and McKendric [30], and despie of is simpliciy, i is a good model for many infecious diseases. The reader is ased o refer o [9-37] in order o now more deails abou mahemaical epidemiology, including is hisory and inds, basics of SIR epidemic models, mehod of soluions, and so forh. As we menioned earlier, if we selec he iniial guesses approximaions as S 0 () = N S, I 0 () = N I, and R 0 () = N R hen he Taylor series expansions of soluions for Eqs. (5) and (6) are as follows: S() = c,m m = N S + c, + c, + c,3 3 +, I() = c,m m = N I + c, + c, + c,3 3 +, R() = c 3,m m = N R + c 3, + c 3, + c 3, According o h residual funcions in Eq. (5), we can wrie Res S () = mc,m m m= Res I () = mc,m m m= [ β ( c,m m [β ( c,m m Res R () = mc 3,m m γ [ c,m m ]. m= ) ( c,m m )], ) ( c,m m ) γ c,m m ], In order o find he s-approximae soluions, we pu = hrough Eq. (7) and using he fac ha Res S (0) = Res I (0) = Res R (0) = 0, o conclude c, [ βn S N I ] = 0, c, [βn S N I γn I ] = 0, c 3, [ γn I ] = 0. Based on he above equaions, we can wrie he firs approximaions of he RPS soluion for Eqs. (5) and (6) as S () = N S βn S N I, I () = N I + (βn S N I γn I ), R () = N R + γn I. (7) 5

6 By coninuing wih he similar fashion, he second approximaions of he RPS soluion for Eqs. (5) and (6) ae he form S () = N S βn S N I + c,, I () = N I + (βn S N I γn I ) + c,, (8) R () = N R γn I + c 3,. In order o find he values of he coefficiens c,, c,, and c 3, in Eq. (8), we pu = hrough Eq. (7) and using he fac ha d d Res S (0) = d d Res I (0) = d d Res R (0) = 0, o obain he following resuls: c, [ β(n S )(βn S N I γn I ) β( βn S N I )(N I )] = 0, c, [β(n S )(βn S N I γn I ) + β( βn S N I )(N I ) γ( γn I + βn S N I )] = 0, c 3, [γ( γn I + βn S N I )] = 0. Based on he above equaions, we can wrie he second approximaions of he RPS soluion for Eqs. (5) and (6) as follows: S () = N S βn S N I + (β(n S)(γN I βn S N I ) + β N S N I ), I () = N I + (βn S N I γn I ) + (βn S(βN S N I γn I ) β N S N I + γ(γn I βn S N I )), R () = N R + γn I + γ( γn I + βn S N I ). For numerical resuls, he following values, for parameers, are considered [38]: N S = 499, N I =, N R =, and β = 0.00, γ = 0.. By coninuing wih he similar fashion, he 0h-order approximaions of he RPS soluion for S(), I(), and R() lead o he following resuls: S 0 () = , I 0 () = , R 0 () = These resuls are ploed in Figure for he hree componens S(), I(), R(), and he summaion S() + I() + R(), respecively. Figure.a illusraes he case when we inroduce a small number of infecives I(0) = ino a suscepible populaion. An epidemic will occur and he number of infecives increases; he maximum infecive populaion I max = 4.8 will occur where S has decreased o he value As ime goes on you ravel along he curve o he righ, evenually approaching S = 0 and he disease died ou. The epidemic will end as S approaching o 0 wih I and R approaching some posiive value I = and R = Meanwhile, he number of immune populaion increases, bu he size of he populaion over he period of he epidemic is consan and equal o 500 as shown in Figure.b. 6

7 S() R() S() + I() + R() I() a) b) Figure. Plos of 50h erms RPS approximaions for SIR model (5) and (6): a) S(), I(), and R() versus ime; b) S() + I() + R() versus ime. We menion here ha, he RPS soluion is he same as he Adomian decomposiion soluion obained in [34], he homoopy perurbaion soluion obained in [35], variaional ieraion soluion obained in [36], and he homoopy analysis soluion obained in [37] when ħ i = and μ i =, i =,,3. Example 3.. Consider he nonlinear Genesio sysem [39]: x () = y(), y () = z(), z () = cx() by() az() + x (), subjec o he iniial condiions x(0) = G x, y(0) = G y, x(0) = G z, where a, b, and c are posiive real numbers, saisfying ab < c. (9) (0) The Genesio sysem, proposed by Genesio and Tesi [3], is one of paradigms of chaos since i capures many feaures of chaoic sysems. I includes a simple square par and hree simple ordinary differenial equaions ha depend on hree posiive real parameers. The reader is indly requesed o go hrough [39-44] in order o now more deails abou Genesio sysem, including is hisory and inds, mehod of soluions, is applicaions, and so forh. According o RPS echnique, he iniial guesses approximaions of Eqs. (9) and (0) are x 0 () = G x, y 0 () = G y, and z 0 () = G z. Thus, he firs few approximaions of he RPS soluion for Eqs. (9) and (0) are x () = G x + G y, y () = G y + G z, z () = G z (cg x (G x ) + ag z + bg y ), x () = G x + G y + G z, y () = G y + G z (cg x (G x ) + ag z + bg y ), z () = G z (cg x (G x ) + ag z + bg y ) (a(cg x (G x ) + ag z + bg y ) + G x G y + ag z bg z cg y ). For numerical resuls, he following values, for parameers, are considered [6]: G x = 0., G y = 0.3, G z = 0., and a =., b =.9, c = 6. If we collec he above resuls, hen he 0h-order approximaions of he RPS soluion for x(), y(), and z() are as follows: x 0 () = , y 0 () = , 7

8 z 0 () = While one canno now he error wihou nowing he soluion, in mos cases he consecuive error can be used as a reliable indicaor in he ieraion progresses. In Tables, he value of consecuive error funcions Con x (), Con y (), and Con z () for he wo consecuive approximae consecuive soluions has been calculaed for various in [0,] wih sep size 0. o measure he difference beween consecuive soluions obained from he 0h-order RPS soluions for Eqs. (9) and (0). However, he compuaional resuls below provide a numerical esimae for he convergence of he RPS echnique. Also, i is clear ha he accuracy obained using presen mehod is in advanced by using only few erms approximaions. In addiion, we can conclude ha higher accuracy can be achieved by evaluaing more componens of he soluion. On he oher hand, based on his heurisic, we erminae he ieraion in our mehod. Table : The values of consecuive error funcion Con () when = 0 for differen values of. i Con x 0 () Con y 0 () Con z 0 () From he able, i can be seen ha he RPS echnique provides us wih he accurae approximae soluion for Eqs. (9) and (0). Also, we can noe ha he approximae soluion more accurae a he beginning values of he independen inerval [0,]. Numerical comparisons are sudied nex. Figure, shows a comparison beween he numerical soluion of Genesio sysem for 0h-order RPS approximaion ogeher wih Runge-Kua mehod (RKM) of order four and Predicor-Correcor mehod (PCM) of order four. Throughou his figure, he sep size for he RKM and PCM is fixed a 0.0. The saring values of he PCM obained from he classical fourh-order RKM. I is demonsraed ha he RPS soluions agree very well wih he soluions obained by he RKM and PCM x() z() y() a) b) -0.4 Figure. Plos of RPS soluion vs. RKM and PCM soluions for Genesio sysem (9) and (0) versus ime: a) solid line: 0h erms RPS approximaions, dashed-do-doed line: RKM soluion; b) solid line: 0h erms RPS approximaions, dashed line: PCM soluion x() z() y() 8

9 Example 3.3. Consider he nonlinear sysem of second-order IVP [45]: x () = 4 x () x () x () + x (), () x () = 4 x () x () + x () + x (), subjec o he iniial condiions x (0) =, x (0) = 0, x (0) = 0, x (0) = 0. () As we menioned earlier, if we selec he iniial guesses approximaions as x,0 () =, x, () = 0, x,0 () = 0, and x, () = 0, hen he firs few erms approximaions of he RPS soluion for Eqs. () and () are x, () = 0, x,3 () = 0, x,4 () = 4, x,5 () = 0,, x, () =, x,3 () = 0, x,4 () = 0, x,5 () = 0,. If we collec he above resuls, hen he 0h-runcaed series of he RPS soluion for x () and x () are as follows: x 0 () = x 0 () = = ( ) j ( ) j, (j)! 4 5 j= = ( ) j ( ) +j. ( + j)! Thus, he exac soluions of Eqs. () and () have he general form which are coinciding wih he exac soluions x () = ( ) j ( ) j = cos (j)!, j=0 x () = ( ) j ( ) +j = sin. ( + j)! j=0 Le us now carry ou he error analysis of he RPS echnique for his example. Figure 3 shows he exac soluion x,exac (), x,exac () and he four ieraes approximaions x (), x () for = 5,0,5,0. These graphs exhibi he convergence of he approximae soluions o he exac soluions wih respec o he order of he soluions. j= a) - b) Figure 3. Plos of RPS soluion for Eqs. () and () blue, brown, green, and red solid lines, denoe four ieraes approximaions when = 5,0,5,0, respecively, and blac dashed-do-doed line, denoe exac soluion: a) x () and x,exac (), b) x () and x,exac (). In Figure 4, we plo he error funcions Ex () and Ex () for = 5,0,5,0 which are approaching he axis y = 0 as he number of ieraions increase. These graphs show ha he exac errors are geing smaller as he order of he soluions is increasing, in oher words, as we progress hrough more ieraions. On he oher hand, 9

10 Figure 5 shows he residual error funcions Res () and Res () for = 5,0,5,0 for he wo consecuive soluions. These error indicaors confirm he convergence of he mehod wih respec o he order of he soluions. E-4 9E-5 Ex 5 () Ex 0 () Ex 5 () 8E-5 7E-5 6E-5 5E-5 4E-5 3E-5 E-5 Ex 0 () E-5 a) 0E+0 b) E-4 9E-5 8E-5 7E-5 6E-5 5E-5 4E-5 3E-5 E-5 E-5 0E+0 Ex 5 () Ex 0 () Ex 5 () Ex 0 () Figure 4. Plos of exac error funcions for Eqs. () and () when = 5,0,5,0: a) Ex (), b) Ex (). E- 9E-3 Res 5 () Res 0 () Res 5 () 8E-3 7E-3 6E-3 5E-3 4E-3 3E-3 E-3 Res 0 () a) E-3 0E+0 b) E- 9E-3 8E-3 7E-3 6E-3 5E-3 4E-3 3E-3 E-3 E-3 0E+0 Res 5 () Res 0 () Res 5 () Res 0 () Figure 5. Plos of residual error funcions for Eqs. () and () when = 5,0,5,0: a) Res (), b) Res (). Example 3.4. Consider he nonlinear sysem of second-order IVP [46]: x () = cos + sin x () + cos x (), x () = 4 + x () 5 5 sin, (3) subjec o he iniial condiions x (0) =, x (0) = 0, x (0) = 0, x (0) = π. (4) Assuming ha he iniial guesses approximaions have he form x,0 () = + and x,0 () = π. Then, he 0h-runcaed series of he RPS soluions of x () and x () for Eqs. (3) and (4) are as follows: x 0 () = ()j = ( )j (j)! j=0 x 0 () = π. I easy o see ha, he 0h-runcaed series of he RPS soluions for x () and x () above agree well wih he general form ()j x () = ( ) j = cos(), (j)! j=0 x () = π. So, he exac soluions of Eqs. (3) and (4) will be x () = cos() and x () = π. Our nex goal is o show how he value of in he runcaion series (3) affecs he RPS approximae soluions. To deermine his effec an error analysis is performed. We calculae he approximaions x () and x () for various and obain he exac error funcions. The maximum and average errors when = 5,0,0 for Eqs. (3) and (4) have been lised in Table for i = i, i = 0,,,,0. 0 0

11 Table : The maximum error funcions of x () and x () when = 5,0,5,0. Descripion = 5 = 0 = 5 = 0 max{ex ( i )} max{ex ( i )} max{res ( i )} max{res ( i )} max{rel ( i )} max{rel ( i )} Ex ( i ) Ex ( i ) Res ( i ) Res ( i ) Rel ( i ) Rel ( i ) Conclusion The main concern of his wor has been o propose an efficien algorihm for he soluions of sysem of IVPs. The main goal has been achieved by inroducing he RPS echnique o solve his class of differenial equaions. We can conclude ha he RPS echnique is powerful and efficien echnique in finding approximae soluion for linear and nonlinear IVPs. The proposed algorihm produced a rapidly convergen series wih easily compuable componens using symbolic compuaion sofware. There is an imporan poin o mae here, he resuls obained by he RPS echnique are very effecive and convenien in linear and nonlinear cases wih less compuaional wor and ime. This confirms our belief ha he efficiency of our echnique gives i much wider applicabiliy for general classes of linear and nonlinear problems. References [] P.G. Drazin, R.S. Jonson, Solion: An Inroducion, Cambridge, New Yor, 993. [] G.B. Whiham, Linear and Nonlinear Waves, Wiley, New Yor, 974. [3] L. Debnah, Nonlinear Waer Waves, Academic Press, Boson, 994. [4] L. Collaz, Differenial Equaions: An Inroducion wih Applicaions, John Wiley & Sons Ld, 986. [5] M.W. Hirsch, S. Smale, Differenial Equaions, Dynamical Sysems, and Linear Algebra, Academic Press, 974. [6] I.I. Vrabie, Differenial Equaions: An Inroducion o Basic Conceps, Resuls and Applicaions, World Scienific Pub Co Inc, 004. [7] O. Abu Arqub, A El-Ajou, A. Baaineh, I. Hashim, A represenaion of he exac soluion of generalized Lane-Emden equaions using a new analyical mehod, Absrac and Applied Analysis, Absrac and Applied Analysis 03, Aricle ID , 0 pages, 03. doi:0.55/03/ [8] O. Abu Arqub, Series soluion of fuzzy differenial equaions under srongly generalized differeniabiliy, Journal of Advanced Research in Applied Mahemaics 5 (03) 3-5. [9] O. Abu Arqub, Z. Abo-Hammour, R. Al-Badarneh, S. Momani, A reliable analyical mehod for solving higherorder iniial value problems, Discree Dynamics in Naure and Sociey, Volume 03, Aricle ID 67389, pages, 03. doi.0.55/03/ [0] A. El-Ajou, O. Abu Arqub, Z. Al Zhour, S. Momani, New resuls on fracional power series: heories and applicaions, Enropy 5 (03) [] O. Abu Arqub, A. El-Ajou, Z. Al Zhour, S. Momani, Muliple soluions of nonlinear boundary value problems of fracional order: a new analyic ieraive echnique, Enropy 6 (04)

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