Approximate Series Solution of Nonlinear, Fractional Klein- Gordon Equations Using Fractional Reduced Differential Transform Method

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1 Approxiae Series Soluion of Nonlinear, Fracional Klein- Gordon Equaions sing Fracional Reduced Differenial Transfor Mehod Ean Abueen, Asad Freiha, Mohaed Al-Sadi, Haad Khalil and Raha Ali Khan Deparen of Applied Science, Faculy of Engineering Technology, Al-Balqa Applied niversiy, Aan 94, Jordan Deparen of Applied Science, Ajloun College, Al-Balqa Applied niversiy, Ajloun 686, Jordan Deparen Maheaics, niversiy of Malaand, Chaadara Dir (L), Khyber Pahunhwa, Paisan Absrac: This analysis proposes an analyical-nuerical approach for providing soluions of a class of nonlinear fracional Klein-Gordon equaion subjeced o appropriae iniial condiions in Capuo sense by using he Fracional Reduced Differenial Transfor Mehod (FRDTM). This echnique provides he soluions very accuraely and efficienly in convergen series forula wih easily copuable coefficiens. The behavior of he approxiae series soluion for differen values of fracional-order α is shown graphically. A coparaive sudy is presened beween he FRDTM and Iplici Runge-Kua approach o illusrae he efficiency and reliabiliy of he proposed echnique. Our nuerical invesigaions indicae ha he FRDTM is siple, powerful aheaical ool and fully copaible wih he coplexiy of such probles. Keywords: Nonlinear Parial Differenial Equaion, Fracional Calculus, Series Soluion, Fracional Reduced Differenial Transfor Mehod, Capuo Tie-Fracional Derivaive Inroducion Fracional Parial Differenial Equaions (FPDEs) are widely used in inerpreaion and odeling of any of realis aers appear in applied aheaics and physics including fluid echanics, elecrical circuis, diffusion, daping laws, relaxaion processes, aheaical biology (Klie, 005; Kilbas e al., 00; Baleanu e al., 009; Juarie, 009; Origueira, 00; Mainardi, 00). Fracional derivaives provide ore accurae odels of real-world probles han ineger-order derivaives; hey are acually found o be a powerful ool o describe cerain physical probles. The opic of fracional calculus is a significanly iporan, useful branch of aheaics, plays a criical and serious role o describe a coplex dynaical behavior in reendous scope of applicaion fields, helps o undersand he naure of aer as well as siplified he conrolling design wihou any loss of herediary behaviors and explain even ore coplex srucures. Consider he following nonlinear Klein-Gordon equaions of one-diensional ie fracional odel:, ux, u x au x, x bg u x, f x,, 0 wih iniial condiions () ux,0 g0x ux,0 g x where, a and b are real consans, f(x,), g 0(x) and g (x) are nown analyical funcions, G (u) is a nonlinear funcion, u is an unnown funcion of x and o be deerined. This odel is derived fro well-nown Klein-Gordon equaions (KGEs) by replacing he ie order derivaive wih fracional derivaive of order α. The KGEs are fundaenal class of nonlinear evoluion equaions arising in classical relaivisic and quanu echanics. I go a lo of aenion for sudying solions and condensed aer physics (Yusufoglu, 008; Sweila e al., 0). On he oher hand, analyicalnuerical sudies of he soluion for he FKGEs wih he Capuo or Rieann-Liouville fracional derivaive were used o handle hese probles (Golanhaneh and Baleanu, 0; Gepreel and Mohaed, 0; Yang e al., 04; Marasi and Karii, 04; Khader e al., 04). As we now, here is no classical ehod o handle he nonlinear FPDEs and provide is explici soluion due o he coplexiies of fracional calculus involving hese equaions. For his reason, we need a reliable nuerical approach o find he coefficiens of he fracional series soluions of such equaions. During he pas few decades, any nuerical-analyical ehods were developed for handling he FPDEs and heir syse. For exaples of hese ehods, we refer o he wor in (Hesaeddini and Foros, 0; Moaddy e al., 0; Abdulaziz e al., 008; Hashi e al., 009; Odiba and Moani, 006; Khalil e al., 05a; 05b; 05c; El-Ajou e al., 05; Corresponding Auhor: Mohaed Al-Sadi, Deparen of Applied Science, Ajloun College, Al-Balqa Applied niversiy, Ajloun 686, Jordan. Eail: h.sadi@bau.edu.jo

2 Abu-Gdairi e al., 05; Freiha and Al-Sadi, 0; Moani e al., 04; Al-Sadi e al., 0; 05; 06). On he oher hand, any applicaions for differen probles by using oher nuerical algorihs can be found in (Abu Arqub e al., 0; 0; 04; 05; Moaddy e al., 05; Koashynsa e al., 04; 06). In his analysis, we inend he applicaion of FRDTM o provide nuerical analyical soluions for a class of nonlinear parial differenial equaions included soe well-nown fracional Klein-Gordon equaions. The FRDTM has several advanages for dealing direcly wih suggesed equaions; i needs a few ieraions o ge high accuracy, i is very siple for obaining analyicalapproxiae soluions in rapidly convergen forulas, i allows beer significanly inforaion in providing coninuous represenaion of hese approxiaions, and i has he abiliy for solving oher probles appearing in several scienific fields. This aricle is organized as follows: in he nex secion, necessary deails and preliinaries abou he fracional calculus heory are briefly provided. In secion, he procedure of he RDTM is presened o consruc and predic he series soluion for fracional PDEs (). In secion 4, nueric resuls for cerain ypes of FKGEs are given o verify he validiy and perforance of he presen ehod. Finally, his aricle ends wih soe concluding rears. Maheaical Preliinaries The basic preliinaries, conceps and noaions of fracional inegrals and derivaives in Capuo definiion (Capuo, 967) are inroduced as follows. Here, we adop he Capuo fracional derivaive, which is a odificaion of Rieann-Liouville, because he iniial condiions ha defined during he forulaion of he syse are siilar o hose convenional condiions of ineger order. 0, u x J u x u x, d, 0,0,, 0 Consequenly, he operaor J has he following properies: For u (x,) C,, α,β 0, cr and γ >, one can ge: J J ux, J u x, J J u x, c J c J () Now, we inroduce a odified fracional differenial operaor D proposed by Capuo as follows: x D f x J x x f d, 0 For -<α, N, x and f x C. Definiion For o be he salles ineger ha exceeds α, he Capuo ie-fracional derivaive operaor of order α>0 is defined as: D u x, u x,, u x, u x, J, 0 () Definiion A real funcion u (x,), x R, >0 is said o be in he space C, R, if here exiss a real nuber q>μ such ha u (x,) = q u (x,), where u (x,) C (R [0, )) and i is said o be in he space C if u x, C,. Definiion The Rieann-Liouville inegral operaor of order α 0 of a funcion u (x,) Cμ, μ is defined as: Theore If <α, N, u (x,) C and γ, hen D J u x, u x, and J D ux, u x,,0 u x, where >0. 0! For ore deails abou FDEs, see (Millar and Ross, 99; Podlubny, 999; Sao e al., 99).

3 Descripion of he ehod Le u(x,) be a funcion of wo variables such ha u(x,) = f(x)g(), hen fro he properies of he onediensional Differenial Transfor (DT) ehod, we have: i j i j,, (4) u x f i x g j i j x i0 j0 i0 j 0 where, (i,j) = f(i)g(j) is called he specru of u(x,). Nex, we assue ha u(x,) is coninuously differeniable funcion wih respec o space variable x and ie. Lea (Srivasava e al., 0) Le u(x,) be an analyic funcion, hen he FRDT of u is given by: u x, x 0 (5) where, α is a paraeer describing he order of iefracional derivaive in Capuo sense. The inverse ransfored of is defined by:, 0 (6) 0 u x x Fro Equaions 5 and 6, we have ha u x, 0 0 (7) u x, 0 In paricular, for = 0, Equaion 7 reduces o: u x, 0 (8) u x, Moreover, if =, hen he FRDT of Equaion 7 reduces o he classical RDT ehod. Fro he above lea, he fundaenal operaions of he FRDTM are given by he following heores (Srivasava e al., 0): Theore Le u(x,), v(x,) and w(x,) be any analyic funcions such ha u(x,) = R D x, vx, = R V D x and w(x,) = R D W x, hen: 0 If u(x,) = v(x,) ± w(x,), hen (x) = V (x) ± W (x) If u(x,) = av(x,), hen ( x) = av (x), a is an arbirary consan If u(x,) = x n v(x,), hen (x) = V -n(x) If u(x,) = x n, hen ( x) = x δ(-n), δ() =, If u(x,) = v(x,) w(x,), hen (x) = Vr xw r x W r 0 r x V r x r 0 Theore Le u(x,) and v(x,) be any wo analyic funcions such ha u(x,) = R D x and v(x,) = R D V x, hen: r v x r x r x If u(x,) =,, hen (x) = x r V r If u(x,) = v x,, hen (x) = r V Corollary r x r If u(x,) = e λ+x x, hen x e.! Corollary If u(x,) = sin(x+), v(x,) = cos(x+), hen (x) = sinx and V(x) = cosx!!. The reader is referred o (Kesin and Ouranc, 009; Abazari and Abazari, 0; Secer, 0; Sohail and Mohyud-Din, 0; Al-Ar, 04) and he references herein o now ore deails abou he reduced differenial ransfor echnique, including heir applicaions in various inds of differenial equaions. Now, by applying he FRDTM o Equaion, we obain he following recurrence relaion forula: x x x (9) a x bg x F x where, F (x) and G[ (x)] are he reduced ransforaion of he funcions f(x,) and G(u(x,)), respecively. sing he iniial condiions, we have:, (0) x g x x g x 0 0

4 Subsiuing Equaion 0 ino Equaion 9 and by sraighforward ieraive calculaion, we obain he values (x), for =,,,. Thus, inverse RDT of n x yields ha n n, 0 () 0 u x x Therefore, he closed for soluion given by:, li, () u x u x x n Nuerical Exaples In his secion, soe nuerical exaples are given o verify he sipliciy and applicabiliy of he presen echnique in finding approxiae series soluion for fracional KGEs. The siulaion resuls indicae ha he FRDT ehod is highly accurae and fully copaible wih he coplexiy of he PDEs of fracional-order. Exaple 4. We consider he following one-diensional linear fracional Klein-Gordon equaion: Subsiuing he condiion (5) ino Equaion 4, we ge he values successively (x), =,,,, as follows: x, x, x,..., x Thus, he approxiae soluion can be obained by: u x, sin x T T T The inverse RDTM is given by: u x, sin x sin x , ux, u x u x, 0,0 x wih iniial condiion: () Consequenly, he reduced inverse ransfored of (x) follows he closed for soluion. Seing =, he exac soluion is:, sin u x x e,0 sin x u x where, x, 0. Applying he ransforaion (Li and He, 00), we have: u x, ux, u x, T x By using he FRDTM of Equaion, we have he recurrence relaion forula: x x x x wih ransfored iniial daa: (4) (5) 0 x sin x To deonsrae he efficiency of he presen ehod, we copare he FRDT approxiaion wih he Iplici Runge-Kua (IRK) ehod for =. Figure shows he phase porrai of soluions for Exaple 4. using he FRDTM and IRKM for [0,0.8] and x [0,4]. The nuerical resuls for differen ie levels of α are presened in Fig.. Here, we noe ha he approxiae FRDT are efficiency for ie-fracional KGE and very closed o he IRK soluions. Exaple 4. We consider he nonlinear Klein-Gordon fracional odel in he for:, ux, u x u x x wih iniial condiion:, 0,0 (6),0 sin x u x

5 Applying he ransforaion (Li and He, 00), we have:, ux, u x T x u x, By using he FRDTM, we have he recurrence relaion forula: x x x r x r x r 0 Wih RDTM of iniial condiion: 0 x sin x (7) Following recurrence relaion (7), he sequences coponens (x), =,,,, were copued using he Maheaica pacage, can be successively given by: x x sin sin, cosx 5sin x, sinx cosx 6sin x cosx cos4x sin x sin x x x x and soon The approxiae for soluion is given by: u x, sin x x T x T x T..., Fig.. Coparison of phase plo for u(x,) of Exaple 4. a =, x [0,4] and 0,0.8: he FRDTM; iplici Runge-Kua ehod Fig.. Phase plo of he soluion u(x,) of Exaple 4. using he FRDTM for x [0,4] and [0,0.8]: a = 0.7; a = 0.

6 Accordingly, he inverse RDT is given by:, u x x 0 sin sin sin x x x cosx 5sin x sin x x x cos 6sin cosx cos4x 4 sin x sin x. The series soluion of u follows closed for soluion. The geoeric behaviors of he soluion for Exaple 4. are sudied by drawing he -diensional space figures of he FRDT approxiae soluion ogeher wih is corresponding IRK soluion. Figure shows he coparison beween he FRDT approxiae soluion and IRK soluion a = for x [,] and [0,0.0]. Whileas, Fig. 4 shows he soluion behavior of he nonlinear FKGE for differen specific cases of α in he doain x [,] and [0,0.0]. The perforance errors analysis are obained by he FRDTM a x = and suarized in Table. Nuerically, i is showed ha he RDT ehod is effecive and accurae. Exaple 4. We consider he nonlinear Klein-Gordon fracional odel in he for:, ux, u x u x, u x, 0,0 x wih iniial condiion:,0 sech x u x (8) Fig.. Coparison of phase plo for u(x,) of Exaple 4. a =, x [,] and 0,0.0 : he FRDTM; iplici RKM

7 Fig. 4. Phase plo of he soluion u(x,) of Exaple 4. using he FRDTM for x [,] and [0,0.0]: a = 0.4; a = 0. Table. The error analysis for Exaple 4. when = wih x =. RDTM IRKM Absolue error Relaive error Applying he ransforaion (Li and He, 00), we have: u x, ux, u x, u x, T x By using he FRDTM, we have he recurrence relaion forula: x x x x i x r x r0 i0 r Wih RDT of iniial condiion: 0 x sec h x (9) Following recurrence relaion (9), he sequences coponens (x), =,,,, were copued using he Maheaica 9.0 pacage, can be successively given by: x h x sec, 5 x sech x 4cosh x 5, cosh x 7 8cosh 4 x sec h x x, 4 and soon n By aing he inverse RDT of x 0, we have, u x x 0 sech x sech x 4 sech x 4cosh x 5 5 cosh x 7 sec h x 8cosh 4x. Figure 5 and 6 show he soluion behavior for he nonlinear ie-fracional KGE in Exaple 4. by using he RDTM and IRKM a x [,] and 0,0.0 for differen paricular values of α. Exaple 4.4 We consider he nonlinear Klein-Gordon fracional odel in he fro:, ux, u x u x x wih iniial condiions: ux,0 sin x, ux,0 0, 0, (0) () sing he ransforaion (Li and He, 00), we have:, ux, u x T x u x, By using he FRDTM, we obain he recurrence relaions forula as:

8 x x x r x r x r 0 Wih RDT of iniial condiion: x sin x, x 0 0 () x x sin x sin x, x 0, 4 x cos x 5sin x sin x, 5 x 0, 4 6 x 8 5 cosx 7 0 cosx 6sin x, 4 5sin xsinx Following recurrence relaion () and by sraighforward ieraive seps, yields: and soon The approxiae for soluion is obained by:, sin u x x x T x T x T x T 4 4 Fig. 5. Coparison of phase plo for u(x,) of Exaple 4. a =, x [,] and [0,0.0]: he FRDTM; iplici Runge-Kua ehod Fig. 6. Phase plo of he soluion u(x,) of Exaple 4. using he FRDTM for x [,] and [0,0.0]: a = 0.4; a = 0.

9 Table. Nuerical resuls of u when x = for Exaple 4.4 =.5 =.5 =.75 = Table. The Nuerical resuls for Exaple 4.4 when = wih x =. FRDTM IRKM Absolue error Relaive error Consequenly, he reduced inverse DT of (x) will be given by:, u x x 0 sin sin sin x x x sin x sin x cos x 5sin x sin x cos x 6sin x cos x 4 5sin x sin x 4 In Table, we suarized nuerical values for he soluion of Equaion 0 when =.5,.5,.75 and = wih x =. Also nuerical resuls using our approxiaion and he iplici Runge-Kua ehod for = are suarized in Table. Fro hese ables, i is easier o observe ha he nuerical approxiaions are in agreeen wih each oher s and wih he IRK ehod. Also, hey have sae behavior as hose obained using he iplici Runge- Kua ehod. Concluding rears In his sudy, we presen nuerical algorih for finding approxiae for soluions of a class of Klein-Gordon fracional odel based upon FRDTM. This ehod was used direcly wihou eploying linearizaion and perurbaion. The efficiency and capabiliy of he presen algorih have been checed via several illusraed exaples. The resuls reveal he coplee reliabiliy of his ehod wih a grea poenial in scienific applicaions. Finally, we conclude ha he FRDTM is very powerful, sraighforward and effecive o obain analyical nuerical soluions of a wide variey probles relaed o fracional PDEs applied in aheaics, physics and engineering. Copuaions of his paper have been carried ou by using he copuer pacage of Maheaica 9. References [] Al-Ar, M.O., 04. New applicaions of reduced differenial ransfor ehod. Alex. Eng. J., 5: DOI: 0.06/j.aej

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