Solving Nonlinear Fractional Partial Differential Equations Using the Homotopy Analysis Method

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1 Solving Nonlinear Fracional Parial Differenial Equaions Using he Homoopy Analysis Mehod Mehdi Dehghan, 1 Jalil Manafian, 1 Abbas Saadamandi 1 Deparmen of Applied Mahemaics, Faculy of Mahemaics and Compuer Science, Amirkabir Universiy of Technology, Tehran 15914, Iran Deparmen of Mahemaics, Faculy of Science, Universiy of Kashan, Kashan, Iran Received 0 Sepember 008; acceped 0 January 009 Published online July 009 in Wiley InerScience DOI /num.0460 In his aricle, he homoopy analysis mehod is applied o solve nonlinear fracional parial differenial equaions. On he basis of he homoopy analysis mehod, a scheme is developed o obain he approximae soluion of he fracional KdV, K,, Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like Bm, n equaions wih iniial condiions, which are inroduced by replacing some ineger-order ime derivaives by fracional derivaives. The homoopy analysis mehod for parial differenial equaions of ineger-order is direcly exended o derive explici and numerical soluions of he fracional parial differenial equaions. The soluions of he sudied models are calculaed in he form of convergen series wih easily compuable componens. The resuls of applying his procedure o he sudied cases show he high accuracy and efficiency of he new echnique. 009 Wiley Periodicals, Inc. Numer Mehods Parial Differenial Eq 6: , 010 Keywords: analyical soluion; coupled KdV and Boussinesq-like Bm, n equaions; fracional KdV, K,, Burgers, BBM-Burgers, cubic Boussinesq; fracional parial differenial equaions FPDEs; homoopy analysis mehod HAM I. INTRODUCTION As menioned by several researchers in fracional calculus, derivaives of nonineger order are very effecive for he descripion of many physical phenomena such as rheology, damping laws, and diffusion process 1 5]. Some fundamenal works on various aspecs of he fracional calculus are given by Abbasbandy 6], Capuo 7], Debanh 8], Diehelm e al. 9], Haya e al. 10], Jafari and Seifi 11, 1], Kemple and Beyer 1], Kilbas and Trujillo 14], Kiryakova 15], Miller and Ross ], Momani and Shawagfeh 16], Oldham and Spanier 17], Podlubny ], ec. Several mehods have been used o solve fracional parial differenial equaions, such as Laplace ransform mehod ], Fourier ransform mehod 1], Adomian s decomposiion mehod ADM 16, 18] and so on. We refer he ineresed reader o 19 ] o sudy he main idea behind Correspondence o: Mehdi Dehghan, Deparmen of Applied Mahemaics, Faculy of Mahemaics and Compuer Science, Amirkabir Universiy of Technology, Tehran 15914, Iran mdehghan@au.ac.ir 009 Wiley Periodicals, Inc.

2 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 449 Adomian s decomposiion approach. A subsanial amoun of research work has been direced for he sudy of he nonlinear fracional KdV, K,, Burgers, cubic Boussinesq and BBM-Burgers, coupled KdV and Boussinesq-like Bm, n equaions given by and and D α u u x + u xxx = 0, 1.1 D α u + u x + u xxx = 0, 1. D α u + 1 u x u xx = 0, 1. D α u u xx + u xx u xxxx = 0, 1.4 D α u + u x + uu x au xx bu xx = 0, 1.5 D α u = au xxx + 6uu x ]+bvv x, D α v = v xxx uv x, 1.6 D α u u n xx u m xxxx = 0, m, n>1, 1.7 respecively. In he case of α = 1, Equaion 1.1 is he pioneering equaion ha gives rise o soliary wave soluions. Solions waves wih infinie suppor are generaed as a resul of he balance beween he nonlinear convecion u n x and he linear dispersion u xxx in hese equaions ]. Solions are localized waves 4] ha propagae wihou change of heir shape and velociy properies and are sable agains muual collisions ]. The fracional Kn, n equaion D α u + un x + u n xxx = 0, 1.8 in he case of α = 1, as poined by ], developed in 5], is he pioneering equaion for compacons. In soliary waves heory, compacons are defined as solions wih finie wavelenghs or solions free of exponenial ails 5]. The Burgers equaion appears in fluid mechanics. This equaion incorporaes boh convecion and diffusion in fluid dynamics, and is used o describe he srucure of shock waves ]. The cubic Boussinesq equaion, gives rise o solions and appeared in he works of Priesly and Clarkson 6]. Kaya 7] examined his equaion by using Adomian decomposiion mehod developed in 18], and used horoughly in 19, 8 ]. This equaion has been invesigaed for soliary 0] waves and for raional 1] soluions as well. The fracional Benjamin-Bona-Mahony-Burgers BBM-Burgers equaion sudied by Song and Zhang ] by using he homoopy analysis mehod HAM. Zhu and Xuan 4] invesigaed BBM-Burgers equaion wih dissipaive erm. The fracional KdV ype of equaions have been an imporan class of nonlinear evoluion equaions wih numerous applicaions in physical sciences and engineering fields. In plasma physics hese equaions give rise o he ion acousic soluions 5]. In geophysical fluid dynamics, hey describe a long wave in shallow seas and deep oceans 6]. Boussinesq-like Bm, n equaion appears in nonlinear dispersion in he formaion of paerns in liquid drops, and gives many similariy reducions and a compacion soluion. Auhor of 7] exended he decomposiion mehod o seek more compacon soluions of Eq. 1.7 when m = n Numerical Mehods for Parial Differenial Equaions DOI /num

3 450 DEHGHAN, MANAFIAN, AND SAADATMANDI and α = 1. When m = 1, n = and α = 1, Eq. 1.7 becomes he well-known Boussinesq equaion. When m = 1, n = and α = 1, Eq. 1.7 reduces o he modified Boussinesq equaion which arises from he famous Fermi-Pasa-Ulam problem 8]. Alhough m = 1 and n 4, Eq. 1.7 is he higher-order modified Boussinesq equaion. Auhors of 9] invesigaed exac soliary soluions wih compac suppor for he nonlinear dispersive Boussinesq-like Bm, n equaions. Wazwaz ], examined he KdV, he K,, he Burgers, and he cubic Boussinesq equaions by using variaional ieraion mehod VIM 40 51]. In he case of α = 1, he above equaions reduce o he classical nonlinear parial differenial equaions. Alhough here are a lo of sudies for he classical form of he above equaions, i seems ha deailed sudies of he nonlinear fracional differenial equaion are only beginning. In his work, he homoopy analysis mehod HAM developed by Liao in 5 61] will be used o conduc an analyic sudy on he fracional KdV, K,, Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV and Boussinesq-like Bm, n equaions. I is worh o poin ou ha HAM successfully applied o parial differenial equaions and exended by auhors 11, 1,, 6 64] o solve differen ypes of nonlinear parial differenial equaions 65]. The mehod gives rapidly convergen successive approximaions of he exac soluion if such a soluion exiss, oherwise approximaions can be used for numerical purposes. The homoopy analysis mehod, a new analyic echnique is proposed o solve nonlinear parial differenial equaions wih fracional order. The homoopy analysis mehod is useful o obain exac and approximae soluions of linear and nonlinear parial differenial equaions. In his aricle, we illusrae he validiy of he HAM 57 61] for he nonlinear fracional parial differenial equaions. The curren paper is organized as follows: In Secion II, we describe basic definiions. In Secion III, he homoopy analysis mehod will be inroduced briefly and his echnique will be applied o solve fracional parial differenial equaions. Secion IV conains some es problems o show he efficiency and accuracy of he new mehod. Also a conclusion is given in Secion V. Finally some references are given a he end of his paper. II. BASIC DEFINITIONS In his secion, we give some definiions and properies of he fracional calculus ]. Definiion 1. A real funcion f, >0, is said o be in he space C µ, µ R, if here exiss a real number p>µ, such ha f= p f 1, where f 1 C0,, and i is said o be in he space Cµ n, if and only if f n C µ, n N ]. Definiion. The Riemann-Liouville fracional inegral operaor J α of order α 0, ofa funcion f C λ, λ 1, is defined as ] J α f= D α f= 1 Ɣα 0 τ α 1 fτdτ, α > 0,.1 J 0 f= f,. where Ɣz is he well-known Gamma funcion. Some of he properies of he operaor J α, which we will need here, are given in he following: For f C λ, λ 1, α, β 0 and γ 1: 1 J α J β f= J α+β f, Numerical Mehods for Parial Differenial Equaions DOI /num

4 J α J β f= J β J α f, J α γ = Ɣγ+1 Ɣα+γ +1 α+γ. NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 451 Definiion. The fracional derivaive D α of fin he Capuo s sense is defined as ] D α 1 f= τ n α 1 f n τdτ, α > 0,. Ɣn α 0 for n 1 <α n, n N, > 0, f C 1 n. The following are wo basic properies of he Capuo s fracional derivaive 7]: 1 Le f C 1 n, n N. Then Dα f, 0 α n is well defined and D α f C 1. Le n 1 <α n, n N and f Cλ n, λ 1. Then n 1 J α D α f = f f k 0 + k, >0..4 k! In his aricle only real and posiive α will be considered. Similar o ineger-order differeniaion, Capuo s fracional differeniaion is a linear operaion, 64] k=0 D α λf + µg = λd α f+ µd α g,.5 where λ, µ are consans, and saisfy he so-called Leibniz rule D α f g = k=0 α k g k D α k f,.6 if fτis coninuous in 0, ] and gτ has n + 1 coninuous derivaives in 0, ]. Definiion 4. For n o be he smalles ineger ha exceeds α, he Capuo ime-fracional derivaive operaor of order α>0, is defined as ] 1 D α ux, = α ux, τ n α 1 n ux, τ dτ, if n 1 <α<n, Ɣn α 0 τ = n.7 α n ux,, if α = n N. n For more informaion on he mahemaical properies of fracional derivaives and inegrals one can consul, 7]. For more deails of his secion, he ineresed reader can also see 5, 8, 9, 11, 14, 17]. III. THE HOMOTOPY ANALYSIS METHOD In his aricle, we use he homoopy analysis mehod o solve he problems described in Secion I. This mehod proposed by a Chinese mahemaician Liao 5]. We apply Liao s basic ideas o he nonlinear fracional parial differenial equaions. Le us consider he nonlinear fracional parial differenial equaion NFDux, = 0,.1 Numerical Mehods for Parial Differenial Equaions DOI /num

5 45 DEHGHAN, MANAFIAN, AND SAADATMANDI where NFD is a nonlinear fracional parial differenial operaor, x and denoe independen variables and ux, is an unknown funcion. For simpliciy, we ignore all boundary or iniial condiions, which can be reaed in he same way. On he basis of he consruced zero-order deformaion equaion by Liao 57], we give he following zero-order deformaion equaion in he similar way 1 qlvx, ; q u 0 x, ]=qhnfdvx, ; q],. where q 0, 1] is he embedding parameer, h is a nonzero auxiliary parameer, L is an auxiliary linear nonineger order operaor and i possesses he propery LC = 0, u 0 x, is an iniial guess of ux,, vx, ; q is an unknown funcion on independen variables x,, q. I is imporan o ha one has grea freedom o choose auxiliary parameer h in HAM. The q = 0 and q = 1, give respecively 5 56] vx, ;0 = u 0 x,, vx, ;1 = ux,.. Thus as q increases from 0 o 1, he soluion vx, ; q varies from he iniial guess u 0 x, o he soluion ux,. Expanding vx, ; q in Taylor series wih respec o q, one has vx, ; q = u 0 x, + u m x, q m,.4 m=1 where u m x, = 1 m! m vx, ; q q m..5 q=0 If he auxiliary linear nonineger order operaor, he iniial guess, and he auxiliary parameer h are so properly chosen, he series Eq..4, converges a q = 1. Hence we have 58 61] ux, = u 0 x, + u m x,,.6 which mus be one of he soluion of he original nonlinear equaion, as proved by 57]. As h = 1, Eq.. becomes m=1 1 qlvx, ; q u 0 x, ]+qnfdvx, ; q = 0,.7 which is used mosly in he homoopy perurbaion mehod HPM 66 70]. Thus, HPM is a special case of HAM. The comparison beween HAM and HPM can be found in 71,7]. According o Eq..4, he governing equaion can be deduced from he zero-order deformaion Eq... Define he vecor 57] u n x, ={u 0 x,, u 1 x,,..., u n x, }..8 Differeniaing Eq.., m imes wih respec o he embedding parameer q and hen seing q = 0 and finally dividing hem by m!, we have he so-called mh-order deformaion equaion 57] Lu m x, χ m u m 1 x, ]=hnfr u m 1 x,,.9 Numerical Mehods for Parial Differenial Equaions DOI /num

6 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 45 where NFR u m 1 x, = 1 m 1! m 1 NFDvx, ; q q m 1,.10 q=0 and χ m = { 0, m 1, 1, m>1..11 The mh-order deformaion Eq..9, is linear and hus can be easily solved, especially by means of a symbolic compuaion sofware such as Mahemaica, Maple, Malab, Maxima and so on. We would like o refer he ineresed reader o 7 76] for more deails. IV. TEST PROBLEMS In his secion, we presen several examples, 64] o illusrae he applicabiliy of HAM o solve nonlinear fracional parial differenial equaions inroduced in Secion I. A. The Fracional KdV Equaion We firs consider he fracional KdV equaion, 76] D α ux, + au x x, + bu xxx x, = gx,, 4.1 where a, b are consan, 0 <α 1 and gx, is a funcion of x and. We solve he general nonhomogeneous nonlinear equaion wih using he HAM mehod. In he following, we consider Eq. 4.1 wih he iniial condiion We choose he linear nonineger order operaor ux,0 = fx. 4. Lvx, ; q]=d α vx, ; q. 4. Furhermore, Eq. 4.1, suggess o define he nonlinear fracional parial differenial operaor NFDvx, ; q]=d α vx, ; q + av x x, ; q + bv xxx x, ; q gx,. 4.4 Using he above definiion, we consruc he zeroh-order deformaion equaion 1 qlvx, ; q u 0 x, ]=qhnfdvx, ; q. 4.5 Obviously, when q = 0 and q = 1 respecively, we have vx, ;0 = u 0 x, = ux,0, vx, ;1 = ux,. 4.6 According o Eqs..9.11, we gain he mh-order deformaion equaion Lu m x, χ m u m 1 x, ]=hnfr u m 1 x,, 4.7 Numerical Mehods for Parial Differenial Equaions DOI /num

7 454 DEHGHAN, MANAFIAN, AND SAADATMANDI where NFR u m 1 x, = D α u m 1x, m 1 + a u i u m 1 i x x, + bu m 1 xxx x, 1 χ m gx,. 4.8 i=0 Now he soluion of Eq. 4.7, for m 1 becomes u m x, = χ m u m 1 x, + hl 1 NFR u m 1 x,. 4.9 From Eqs. 4.1, 4. and 4.9 we now successively obain Define hen, we have where Thus, we have u 0 x, = ux,0 = fx, 4.10 u 1 x, = hd α D α u 0 + au 0 x + bu 0 xxx gx, ] = hd α D α fx+ af x x + bf x xxx gx, ] A 1 = af x x + bf xxx x gx,, u 1 x, = hd α af x x + bf xxx x gx, ] = hd α u x, = hh + 1D α u x, = h + 1u + hd α u x, = hh + 1 D α Also define, A 1, A 1 + hd α hd α afa 1 x + ba 1 xxx ] = hh + 1D α A 1 + h D α A, 4.1 A = af A 1 x + ba 1 xxx. a u 1 + u 0u x + bu xxx ], 4.1 A 1 + h h + 1D α A + af A 1 x + ba 1 xxx ]+h D α a D α A 1 + x h D α af A x + ba xxx ] A = A + af A 1 x + ba 1 xxx, A 4 = af A x + ba xxx, u x, = hh + 1 D α A 1 + h h + 1D α A + h D α a D α A + 1 x h D α A 4, 4.15 u 4 x, = h + 1u + hd α au 0 u + u 1 u x + bu xxx ], Numerical Mehods for Parial Differenial Equaions DOI /num

8 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 455 In he above erms we subsiue h = 1, he dominan erms will be remained and he res erms vanish, because hey include facor of h m h + 1 n, m, n N, u 0 x, = fx, u 1 x, = D α A 1, u x, = D α A, u x, = D α a D α A + ] 1 x D α A 4, 4.17 and wih using Eq..6, we have ux, = fx D α A 1 + D α Now consider he following example: A D α a D α A 1 + x D α A D α u u x + u xxx = 0, ux,0 = 6x, 0 <α< Saring wih iniial condiion, he source erm and he auxiliary operaor ux, 0 = fx = 6x, gx, = 0 and Lux, = D α ux,, respecively and pu h = 1, we have Nowwehave A 1 = 6 x, A = 6 5 x, A = x, A 4 = x. u 1 x, = D α 6 6 x x = Ɣα + 1 α, 4.0 u x, = D α 6 5 x = 65 x Ɣα + 1 α, 4.1 u x, = D α For α = 1, ux, reduces o: where ux, = x + D α D α 6 x x = 4 66 x Ɣα + 1 α 6 7 x α Ɣα + 1 α + 1 α, 4.. ux, = u 0 + u 1 + u + =6x + 6 x x x + = 6x = 6x, 6 < 1, x, is he exac soluion ]. 1 6 ] B. The Fracional K, Equaion We nex consider he following K, equaion ] D α ux, + u x x, + u xxx x, = 0, ux,0 = x, 0 <α 1, 4.4 Numerical Mehods for Parial Differenial Equaions DOI /num

9 456 DEHGHAN, MANAFIAN, AND SAADATMANDI ha was examined for compacions. To solve he general homogeneous nonlinear equaion wih he HAM mehod we consider he linear nonineger order operaor Lvx, ; q]=d α vx, ; q. 4.5 Furhermore, Eq. 4.4 suggess o define he nonlinear fracional parial differenial operaor NFDvx, ; q]=d α vx, ; q + v x x, ; q + v xxx x, ; q. 4.6 Applying he above definiion, we consruc he zeroh-order deformaion equaion 1 qlvx, ; q u 0 x, ]=qhnfdvx, ; q. 4.7 I is worh o noe ha, when q = 0 and q = 1 respecively, we can wrie vx, ;0 = u 0 x, = ux,0, vx, ;1 = ux,. 4.8 According o Eqs..9.11, we gain he mh-order deformaion equaion Lu m x, χ m u m 1 x, ]=hnfr u m 1 x,, 4.9 where NFR u m 1 x, = D α u m 1 m 1 m 1x, + u i u m 1 i x x, + u i u m 1 i xxx x,. 4.0 Now he soluion of Eq. 4.9, for m 1 becomes i=0 i=0 u m x, = χ m u m 1 x, + hl 1 NFR u m 1 x,. 4.1 From Eqs. 4.4, 4.8, and 4.1, we now successively obain u 0 x, = ux,0 = x, 4. u 1 x, = hd α D α u 0 + u 0 ] u x 0 xxx = hd α 0 + x + 0 = xh Ɣα + 1 α, 4. u x, = h + 1u 1 x, + hd α u 0 u 1 x + u 0 u 1 xxx ] = xhh + 1 Ɣα + 1 α + u x, = h + 1u + hd α 8xh Ɣα + 1 α, 4.4 u 1 + u 0u + u x 1 + u xxx] 0u, 4.5 Numerical Mehods for Parial Differenial Equaions DOI /num

10 u x, = NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 457 xh1 + h Ɣα + 1 α + 16xh 1 + h 8xh α + Ɣα + 1 Ɣα xh Ɣα + 1 Ɣα + 1 Ɣα + 1 α, 4.6 u 4 x, = h + 1u + hd α u 1 u + u 0 u x + u 1 u + u 0 u xxx ], 4.7 xh1 + h u 4 x, = α + 16xh 1 + h 8xh α + Ɣα + 1 Ɣα + 1 Ɣα xh h + 1 Ɣα + 1 Ɣα + 1 Ɣα + 1 α 16x + Ɣα h Ɣα + 1 h + 1 Ɣα + 1 Ɣα + 1 α + 8xh 1 + h Ɣα + 1 α 64xh 4 Ɣα Ɣα + 1Ɣα + 1Ɣα + 1 4α 16h 4 + Ɣα h4 Ɣα + 1 Ɣα + 1 Ɣ4α + 1 4α, 4.8. In he above erms we subsiue h = 1, he dominan erms will be remained and he res erms vanish, because hey include facor of h m h + 1 n, m, n N and for each α = 1, we have u 0 x, = x, u 1 x, = x, u x, = 4x, u x, = 8x, Thus, he exac soluion of his es problem is as follows ]: C. The Fracional Burgers Equaion u 4 x, = 16x 4,. 4.9 ux, = x = We consider he modified KdV mkdv equaion ] x D α ux, + 1 u x x, u xx x, = 0, ux,0 = x, 0 <α To solve he above problem wih he HAM mehod we choose he linear nonineger order operaor Lvx, ; q]=d α vx, ; q. 4.4 Eq. 4.41, approaches us o define he nonlinear fracional parial differenial operaor NFDvx, ; q]=d α vx, ; q + 1 v x x, ; q v xx x, ; q. 4.4 Numerical Mehods for Parial Differenial Equaions DOI /num

11 458 DEHGHAN, MANAFIAN, AND SAADATMANDI Using 4.4 we have he zeroh-order deformaion equaion 1 qlvx, ; q u 0 x, ]=qhnfdvx, ; q Also noe ha, when q = 0 and q = 1 respecively, we ge vx, ;0 = u 0 x, = ux,0, vx, ;1 = ux, Eqs yield he mh-order deformaion equaion Lu m x, χ m u m 1 x, ]=hnfr u m 1 x,, 4.46 where NFR u m 1 x, = D α u m 1x, + 1 m 1 u i u m 1 i x x, u m 1 xx x, Thus he soluion of Eq. 4.46, for m 1 becomes ] i=0 u m x, = χ m u m 1 x, + hl 1 NFR u m 1 x, Employing Eqs. 4.41, 4.45, and 4.48, we now successively obain u 0 x, = ux,0 = x, 4.49 u 1 x, = hd α D α u ] u 0 x 0 xx = hd α xh 0 + x 0 = Ɣα + 1 α, 4.50 ] 1 u x, = h + 1u 1 x, + hd α u xhh + 1 0u 1 x u 1 xx = Ɣα + 1 α + u x, = h + 1u + hd α u x, = h + 1u + xh 1 + h Ɣα + 1 α +. xh Ɣα + 1 α, 4.51 ] 1 u 1 + u 0u u x xx, 4.5 xh Ɣα xh Ɣα + 1 Ɣα + 1 Ɣα + 1 α, 4.5 In he above erms we subsiue h = 1, he dominan erms will be remained and he res erms vanish, because hey include facor of h m h + 1 n, m, n N and for α = 1wehave u 0 x, = x, u 1 x, = x, u x, = x, u x, = x, u 4 x, = x 4, Numerical Mehods for Parial Differenial Equaions DOI /num

12 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 459 Thus, we ge he exac soluion as ] ux, = x = x D. The Fracional BBM-Burgers Equaion We consider he fracional BBM-Burgers equaion ], D α ux, + u xx, + uu x x, au xx x, bu xx x, = 0, ux,0 = x, >0, 0 <α To solve he general homogeneous nonlinear equaion wih he HAM mehod we choose he linear nonineger order operaor Lvx, ; q]=d α vx, ; q Eq direcs us o define he nonlinear fracional parial differenial operaor NFDvx, ; q]=d α vx, ; q + v xx, ; q + vv x x, ; q av xx x, ; q bv xx x, ; q The zeroh-order deformaion equaion can be consruced, 64] as 1 qlvx, ; q u 0 x, ]=qhnfdvx, ; q Obviously, q = 0 and q = 1 respecively, yield vx, ;0 = u 0 x, = ux,0, vx, ;1 = ux, Considering Eqs..9.11, we gain he mh-order deformaion equaion where Therefore Lu m x, χ m u m 1 x, ]=hnfr u m 1 x,, 4.61 NFR u m 1 x, = D α u m 1 m 1x, + u m 1x x, + u i u m 1 i x x, au m 1 xx x, bu m 1 xx x,. 4.6 u m x, = χ m u m 1 x, + hl 1 NFR u m 1 x,. 4.6 Using Eqs. 4.56, 4.60, and 4.6, we now successively obain u 0 x, = ux,0 = x, 4.64 u 1 x, = hd α D α u ] 0 + u 0 x + u 0 u 0 x au 0 xx bu 0 xx = hd α x + x a = x + x ah α Ɣα + 1 Numerical Mehods for Parial Differenial Equaions DOI /num i=0

13 460 DEHGHAN, MANAFIAN, AND SAADATMANDI Define, A 1 = x + x a, u x, = h + 1u 1 x, + hd α u 1 x + u 0 u 1 x + u 1 u 0 x au 1 xx bu 1 xx ] 4.66 u x, = h + 1u 1 x, + h 10x 4 + 1x 16ax + α 1xh b Ɣα + 1 Ɣα α Again define, A = 10x 4 + 1x 16ax +, u x, = h + 1u + hd α u x + u 0 u x + u 1 u 1 x + u u 0 x au xx bu xx ], u x, = h + 1u + hh + 1D α α 1 hh + 1D α + h D α D α A 1 D α u 1 x + h D α A x h D α 1x + x + 1b Ɣα u 1 xx h D α A xx + h h + 1D α xa 1 ] A 1 x + h h + 1D α x A 1 x h D α x A x + xa ], 4.69 ] u 4 x, = 1 + hu + hd α u x + u i u i x au xx bu xx, i=0 In he above erms, we subsiue h = 1, a = b = 1, only will be remained dominan erms and he res erms vanish, because hey include facor of h m h + 1 n, m, n N and for each α = 1, we have u 0 x, = x, u 1 x, = x + x, u x, = 5x 4 + 6x 8x + 1 1x, u x, = x5 6x + 19x 8x + 104, Thus, he soluion of he given problem is as follows ] ux, = x x + 14x + 5x 4 + 6x 8x x5 6x + 19x 8x Numerical Mehods for Parial Differenial Equaions DOI /num

14 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 461 E. The Fracional Cubic Boussinesq Equaion We consider he fracional cubic Boussinesq equaion ] D α ux, u xx x, + u xx x, u xxxx x, = 0, ux,0 = 1 x, u x,0 = 1 x. 4.7 The given iniial values admi he use of As before, we choose he linear nonineger order operaor u 0 x, = 1 x x Lvx, ; q]=d α vx, ; q Furhermore Eq. 4.7, suggess o define he nonlinear fracional parial differenial operaor NFDvx, ; q]=d α vx, ; q v xx x, ; q + v xx x, ; q v xxxx x, ; q Using he above definiion, we consruc he zeroh-order deformaion equaion 1 qlvx, ; q u 0 x, ]=qhnfdvx, ; q Obviously, when q = 0 and q = 1 respecively, we can wrie vx, ;0 = u 0 x, = ux,0, vx, ;1 = ux, According o Eqs..9.11, we gain he mh-order deformaion equaion Lu m x, χ m u m 1 x, ]=hnfr u m 1 x,, 4.79 where NFR u m 1 x, = D α u m 1 x, u m 1xx x, m 1 j + u i u j i x, u m 1 xxxx x, Hence we have u m 1 j j=0 i=0 xx u m x, = χ m u m 1 x, + hl 1 NFR u m 1 x, Numerical Mehods for Parial Differenial Equaions DOI /num

15 46 DEHGHAN, MANAFIAN, AND SAADATMANDI From Eqs. 4.7, 4.78 and 4.81, we now successively obain u 0 x, = ux,0 = 1 x x, 4.8 u 1 x, = hd α D α u 0 u 0 xx + u 0 xx u ] 0 xxxx = hd α = h! x! +! x x x 7 x 8 α Ɣα + 1 +! ] α+1 x 4 Ɣα α+ x 7 Ɣα + 84 ] α+. x 8 Ɣα u x, = h + 1u 1 x, + hd α u1 xx + 6 u 0 u 1 u ] xx 1 xxxx, 4.84 u x, = h + 1u + hd α u xx + 6 u u 0 + u 0u1 u ] xx xxxx, In he above erms we subsiue h = 1, a = b = 1, he dominan erms will be remained and he res erms vanish, because hey include facor of h m h + 1 n, m, n N and for α = 1, we have u 0 x, = 1 x x, 4.86 u 1 x, = x x !x 7 5!x, 8 u x, = 4 x 5 + small erms, 5 x6 u x, = 6 x 7 + small erms, 7 x8 Thus we ge ux, = 1 x x + x x x 5 5 x x 7 7 x 8 + = 1 x which is he exac soluion ]. F. The Fracional Coupled KdV Equaions We consider he fracional coupled KdV equaions 74, 76]. n 1 n = 1 x x +, n= D α w = aw xxx + 6ww x ]+bvv x, D α v = v xxx wv x, 4.88 Numerical Mehods for Parial Differenial Equaions DOI /num

16 wih iniial condiions NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 46 wx,0 = 1 + a + 6a k + 4k expkx expkx, vx,0 = M 1 + expkx 1 + expkx, 4.89 where, a = 1, ab < 0, M = 4 a b 1 k, and k is consan. To solve he general homogeneous sysem of nonlinear equaions wih he HAM mehod we choose he linear nonineger order operaors L 1 x, ; q]=d α x, ; q, L x, ; q]=d α x, ; q Furhermore, Eq. 4.88, suggess o define he nonlinear fracional parial differenial operaors N 1 FD x, ; q]=d α x, ; q a xxxx, ; q + 6 x x, ; q] b x x, ; q, N FD x, ; q]=d α x, ; q + xxxx, ; q + x x, ; q Using he above definiion, we consruc he zeroh-order deformaion equaion 1 ql 1 x, ; q w 0 x, ]=qhn 1 FD x, ; q, 1 ql x, ; q v 0 x, ]=qhn FD x, ; q. 4.9 Obviously, when q = 0 and q = 1wehave x, ;0 = w 0 x, = wx,0, x, ;1 = wx,, x, ;0 = v 0 x, = vx,0, x, ;1 = vx,, 4.9 respecively. According o Eqs..9.11, we gain he mh-order deformaion equaions L 1 w m x, χ m w m 1 x, ]=hn 1 FR 1 w m 1 x,, L v m x, χ m v m 1 x, ]=hn FR v m 1 x,, 4.94 where ] N 1 FR 1 w m 1 x, = D α w m 1 m 1x, a w m 1xxx x, + 6 w j w m 1 jx x, m 1 b v j v m 1 jx x,, 4.95 j=0 N FR v m 1 x, = D α v m 1 m 1x, + v m 1xxx x, + w j v m 1 jx x,. Now he soluions of Eqs and 4.95 for m 1, become j=0 j=0 w m x, = χ m w m 1 x, + hl 1 N 1 FR 1 w m 1 x,, v m x, = χ m v m 1 x, + hl 1 N FR v m 1 x, Numerical Mehods for Parial Differenial Equaions DOI /num

17 464 DEHGHAN, MANAFIAN, AND SAADATMANDI From Eqs. 4.88, 4.9, and 4.96, we now successively obain w 0 x, = wx,0 = fx= 1 + a + 6a k + 4k expkx 1 + expkx, 4.97 v 0 x, = vx,0 = gx = M expkx 1 + expkx, w 1 x, = hd α D α w ] 0 aw 0xxx 6aNw 0 bmv 0, 4.98 where we define m 1 m 1 Nw m 1 = w i w m 1 i x, Mv m 1 = v i v m 1 i x, hus i=0 Nw 0 = w 0 w 0 x = ff x, Mv 0 = v 0 v 0 x = gg x, w 1 x, = hd α h af xxx 6aff x bgg x = Ɣα + 1 α A 1, A 1 = af xxx 6aff x bgg x, v 1 x, = hd α h g xxx + fg x = Ɣα + 1 α B 1, 4.99 B 1 = g xxx + fg x, w x, = h + 1w 1 x, + hd α aw 1xxx 6w 0 w 1x + w 1 w 0x bv 0 v 1x + v 1 v 0x ] w x, = h + 1w 1 x, + hd α A, A = aw 1xxx 6f w 1x + w 1 f x bgv 1x + g x v 1, and v x, = h + 1v 1 x, + hd α B = v 1xxx + fv 1x + g x w 1, i=0 v 1xxx + fv 1x + g x w 1 ]=h + 1v 1 x, + hd α B 4.10 A 1 = 4ak5 expkx 1 + expkx 1 + a1 + expkx, B 1 = Mak5 expkx1 expkx 1 + a1 + expkx, A = 4a k 8 expkx1 4expkx + expkx h α 1 + a 1 + expkx 4 Ɣα + 1, B = Ma k 8 expkx1 4expkx + expkx h α 1 + a 1 + expkx 4 Ɣα + 1,. Numerical Mehods for Parial Differenial Equaions DOI /num

18 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 465 Thus we have w 1 x, = h 4ak5 expkx 1 + expkx α 1 + a1 + expkx Ɣα + 1, 4.10 v 1 x, = h Mak5 expkx 1 + expkx α 1 + a1 + expkx Ɣα + 1, w x, = 1 + hw 1 x, + h 4a k 8 expkx 1 + expkx 1 + a 1 + expkx 4 α Ɣα + 1Ɣα + 1, v x, = 1 + hv 1 x, + h Ma k 8 expkx 1 + expkx α 1 + a1 + expkx Ɣα + 1Ɣα + 1, and so on, he oher componens can be deermined in a similar way. By repeaing his procedure for h = 1and α = 1 we ge o he soluion as follows wx, = 1 + a + 6a k + 4k expkx 1 + expkx + 4ak5 expkx 1 + expkx 1 + a1 + expkx + a k 8 expkx1 4expkx + expkx 1 + a1 + expkx + a k 11 expkx expkx 11 expkx + expkx 1 + a 1 + expkx 5 +, vx, = 1 + a + 6a k + 4k expkx 1 + expkx + Mak5 expkx 1 + expkx 1 + a1 + expkx + Ma k 8 expkx 1 + expkx 1 + a 1 + expkx 4 + Ma k 11 expkx expkx 11 expkx + expkx +, a 1 + expkx 5 herefore using he Taylor series we obain he closed form soluions 7, 76] wx, = 1 + a + 6a k + 4k expkx + c] 1 + expkx + c], expkx + c] vx, = M 1 + expkx + c], where M, k, a, and c are arbirary consans. G. The Fracional Boussinesq-Like Bm,n Equaion The Boussinesq-like Bm, n equaion wih fully nonlinear dispersion reads 9, 74] D α ux, u n xx x, u m xxxx x, = 0, m, n> In his secion, we would like o choose wo special equaions, namely, B,, and B,, wih specific iniial condiions o illusrae efficiency of he homoopy analysis mehod. Numerical Mehods for Parial Differenial Equaions DOI /num

19 466 DEHGHAN, MANAFIAN, AND SAADATMANDI Case 1. m = n =. We consider he B, equaion wih iniial condiions 9, 74] D α ux, u xx x, u xxxx x, = 0,, ux,0 = 4 a sin 4 u a x x,0 = sin, where a is an arbirary consan. To solve he above problem wih he HAM mehod we choose he linear non-ineger order operaor Lvx, ; q]=d α vx, ; q Furhermore Eq , suggess o define he nonlinear fracional parial differenial operaor NFDvx, ; q]=d α vx, ; q v xx x, ; q v xxxx x, ; q Using he above definiion, we consruc he zeroh-order deformaion equaion 1 qlvx, ; q u 0 x, ]=qhnfdvx, ; q Obviously, when q = 0 and q = 1 respecively, we have vx, ;0 = u 0 x, = ux,0, vx, ;1 = ux, According o Eqs..9.11, we gain he mh-order deformaion equaion where NFR u m 1 x, = D α Soluion of Eq , for m 1 becomes Lu m x, χ m u m 1 x, ]=hnfr u m 1 x,, m 1 m 1 u m 1 x, u j u m 1 j xx x, u j u m 1 j xxxx x,. j=0 j= u m x, = χ m u m 1 x, + hl 1 NFR u m 1 x, From Eqs and 4.118, we now successively obain u 0 x, = ux,0 + u x,0 = 4 a sin 4 u 1 x, = hd α D α u 0 u 0 u xx 0 xxxx + a, sin = hu0 + hd α Nu 0 Mu 0 = hu 0 + hd α A 1, 4.10 Numerical Mehods for Parial Differenial Equaions DOI /num

20 define A k, as follows NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 467 A k = Nu k 1 + Mu k 1, k = 1,,,..., 4.11 u x, = h + 1u 1 x, + hd α Nu 1 Mu 1, 4.1 u x, = h + 1u 1 x, + hd α fa 1 xx fa 1 xxxx = hh + 1D α A 1 + hd α A, 4.1 u x, = h + 1u x, + hd α Nu Mu = hh + 1 D α A 1 + hh + 1D α A + hd α A, The nonlinear operaors Nu n and Mu n are he nonlinear erms and can be expressed in erms of Adomian s polynomials in he following form m 1 m 1 Nu m 1 = u i u m 1 i xx, Mu m 1 = u i u m 1 i xxxx, 4.15 i=0 Nu 0 = u 0 xx, Mu 0 = u 0 xxxx, Nu 1 = u 0 u 1 xx, Mu 1 = u 0 u 1 xxxx, Nu = u 0 u + u 1, Mu xx = u 0 u + u 1, xxxx A 1 = 1 a 6 a4 sin cos, A = a5 x 4 h α sin a cos Ɣα A = a5 h α 4ah Ɣα + 1 a h + a h + 1. α+1 i=0 Ɣα + + 8h a4 h a α ] Ɣα + sin, 4α cos Ɣα + 1Ɣ4α + 1 4α+1 Ɣα + Ɣ4α + ] sin, Thus we have + a u 0 x, = 4 a sin 4 u 1 x, = a x h sin + a4 h 6 sin a sin Numerical Mehods for Parial Differenial Equaions DOI /num, 4.16 α+1 ] Ɣα + cos α, 4.17 Ɣα + 1

21 468 DEHGHAN, MANAFIAN, AND SAADATMANDI hh + 1a a 5 α+1 u x, = + hh + 1Ɣα + + h a 7 4α+1 sin 48Ɣα + Ɣ4α + a 4 α hh + 1 6Ɣα h a 6 4α ] cos, Ɣα + 1Ɣ4α + 1 a hh + 1 u x, = + a5 hh + 1 α+1 1 Ɣα + + a7 h h + 4α+1 48 Ɣα + Ɣ4α + ] + a5 h h + α+1 1 Ɣα + + a9 h 6α+1 sin 96 Ɣα + Ɣ4α + Ɣ6α + a 4 hh + 1 α + 6 Ɣα a6 h h + 1 4α 4 Ɣα + 1Ɣ4α + 1 ] cos a8 h 96 6α Ɣα + 1Ɣ4α + 1Ɣ6α + 1 By repeaing his procedure for h = 1 and α = 1 we ge o soluion as follows ux, = u i x, = 4 a sin + a 4 i=0 + a a 1 a 4 1 6! 8 4! a 1 a! a 6 Using he Taylor series gives he exac soluions 9] where a is an arbirary consan. Case. m = n =. ux, = 4 a sin + a 4 We consider he B, equaion 6! ! a 7 sin 7! cos , π <x+ a < π, 4.11 D α ux, u xx x, u xxxx x, = 0, 4.1 wih iniial condiions 9] ux,0 = ab sin + ab cos, u x,0 = ab cos ab sin, where a and b are arbirary consans. To solve he general homogeneous nonlinear equaion wih he HAM mehod we choose he linear nonineger order operaor Lvx, ; q]=d α vx, ; q. 4.1 Furhermore Eq. 4.1 suggess o define he nonlinear fracional parial differenial operaor NFDvx, ; q]=d α vx, ; q v xx x, ; q v xxxx x, ; q Numerical Mehods for Parial Differenial Equaions DOI /num

22 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 469 Using he above definiion, we consruc he zeroh-order deformaion equaion 1 qlvx, ; q u 0 x, ]=qhnfdvx, ; q Obviously, when q = 0 and q = 1 respecively, we ge vx, ;0 = u 0 x, = ux,0, vx, ;1 = ux, According o Eqs..9.11, we gain he mh-order deformaion equaion Lu m x, χ m u m 1 x, ]=hnfr u m 1 x,, 4.17 where NFR u m 1 x, = D α u m 1 x, m 1 u m 1 i i=0 m 1 u m 1 i i=0 i u j u i j j=0 xxxx i u j u i j j=0 xx x, x, Therefore we have u m x, = χ m u m 1 x, + hl 1 NFR u m 1 x, From Eqs. 4.1 and 4.19, we now successively obain u 0 x, = ux,0 + u x,0 = u 1 x, = hd α D α u 0 u 0 u xx 0 6 ] ab 1 b sin b cos, xxxx = hu0 + hd α Nu 0 Mu 0 = hu 0 + hd α A Define A k, as given in he following A k = Nu k 1 + Mu k 1, k = 1,,,..., 4.14 u x, = h + 1u 1 x, + hd α Nu 1 Mu 1, 4.14 u x, = h + 1u 1 x, + hd α fa 1 xx fa 1 xxxx = hh + 1D α A 1 + hd α A, u x, = h + 1u x, + hd α Nu Mu = hh + 1 D α A 1 + hh + 1D α A + hd α A, Numerical Mehods for Parial Differenial Equaions DOI /num

23 470 DEHGHAN, MANAFIAN, AND SAADATMANDI The nonlinear operaors Nu n and Mu n are he nonlinear erms and can be expressed in erms of Adomian s polynomials as: Nu m 1 = m 1 u m 1 i i=0 i u j u i j j=0 xx, Mu m 1 = m 1 u m 1 i Nu 0 = u 0, Mu xx 0 = u 0 xxxx, Nu 1 = u 0 u 1, Mu xx 1 = u 0 u 1, xxxx Nu = u 0 u + u 0 u 1, Mu xx = u 0 u + u 0 u 1 6 A 1 = 9 a b 1 + b + b + b cos A = 6 81 a b 4 h { a b + b + b α+1 +a b 1 + b + b + a b b + b Thus we have u 0 x, = α+1 Ɣα + + b α+ Ɣα + 4 i=0 xxxx, i u j u i j j=0 xxxx + 1 b + b b sin α Ɣα b α+ Ɣα+ +b α+ Ɣα+4 α Ɣα α+ b Ɣα + ] 91 b + b, ], Ɣα + ] b +b cos a b 1 b + b } sin. 6 ] ab 1 b sin b cos, u 1 x, = { b 6abh + + b ab + ab u x, = 6abhh + 1 α Ɣα + 1 +b α+1 Ɣα+ +b α+ Ɣα+ +b α+ α Ɣα + 1 b α+1 Ɣα + +b α+ Ɣα + b α+ { b + ab ] + b α+ cos + Ɣα + 4 ] + b α+ Ɣα + α+ } b sin Ɣα + 4 Ɣα+4 Ɣα + 4 α Ɣα b α+1 Ɣα + + α+ b Ɣα + b ab + α Ɣα + 1 b α+1 Ɣα + Numerical Mehods for Parial Differenial Equaions DOI /num ] cos ] } sin, 4.148

24 + NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 471 { 6 81 a b 4 h a b Ɣα a b Ɣα + + a b Ɣα + 4α+1 4α Ɣ4α b 4α+1 Ɣ4α + + b 4α+ Ɣ4α + 4 4α+ Ɣ4α + + b 4α+ Ɣ4α α+4 b Ɣ4α + 5 4α+ Ɣ4α b 4α+4 Ɣ4α α+5 b Ɣ4α + 6 Ɣ4α + + b 4α+ Ɣ4α + + b 4α+ + a b 4 Ɣα + 4 ] α Ɣα + + b α+ Ɣα + + α+ b cos Ɣα + 4 a b 4α + Ɣα + 1 Ɣ4α + 1 b 4α+1 Ɣ4α + + 4α+ b Ɣ4α + a b 4α+1 Ɣα + Ɣ4α + b 4α+ Ɣ4α + + 4α+ b Ɣ4α a b Ɣα + a b 4 Ɣα α+ Ɣ4α + b 4α+ Ɣ4α α+4 b Ɣ4α + 5 4α+ Ɣ4α + 4 b 4α+4 Ɣ4α α+5 b Ɣ4α + 6 α+1 Ɣα + b α+ Ɣα + + b α+ Ɣα + 4 ] sin Ɣ4α + } By repeaing above procedure for h = 1 and α = 1 we ge o soluion as follows ux, = 6 ] 6 ab sin + cos 9 a b 5 + k= 6 81 a5 b + k bk k!! b 4 7 b 5! 4!! 5! { 5 b k k! k= ]} cos + 1 k sin b b 5 4!! 5! + 4! + b 6 4! 6! b 6 6! ! ] cos + 1 k sin b b 8 + b 9 cos 7! 5! 8! 5! 9! b 9 ] sin + 9! b b 8 5! 7! 5! 8! 5! According o he above procedure and by using he Taylor series we have 9] 6 x + b ux, = ab sin where a and b are arbirary consans. 6 + ab cos Numerical Mehods for Parial Differenial Equaions DOI /num + b, 4.151

25 47 DEHGHAN, MANAFIAN, AND SAADATMANDI FIG. 1. The figures show he rd-order approximaion soluion u o Eq when a = b = 1, h = 1. a α = 1, b α = 0.5. Color figure can be viewed in he online issue, which is available a We illusrae he accuracy and efficiency of HAM by applying he mehod o some fracional parial differenial equaions and comparing he approximae soluions wih he exac soluions. For his purpose, we calculae he numerical resuls of he exac soluions for he cases where exac soluions are available and he muli-erms approximae soluions of HAM. A he same ime, he surface graphics of he exac and muli-erms approximae soluions are ploed in Figs One FIG.. The figures show he rd-order approximaion soluion u o Eq when a = b = 1, α = 0.. a h = 0.5, b h = 0.. Color figure can be viewed in he online issue, which is available a Numerical Mehods for Parial Differenial Equaions DOI /num

26 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 47 FIG.. The surface of he exac and he rd-order approximaion soluions u o Eq. 4.7 obained in his work. Color figure can be viewed in he online issue, which is available a FIG. 4. The surface of he exac and he rd-order approximaion soluions w o Eqs for a = 1.5, c = 0.1, b = 0.1 and k = 0.1, obained in his work. Color figure can be viewed in he online issue, which is available a Numerical Mehods for Parial Differenial Equaions DOI /num

27 474 DEHGHAN, MANAFIAN, AND SAADATMANDI FIG. 5. The surface of he exac and he rd-order approximaion soluions v o Eqs for a = 1.5, c = 0.1, b = 0.1, and k = 0.1, obained in his work. Color figure can be viewed in he online issue, which is available a FIG. 6. The surface of he exac and he rd-order approximaion soluions u o Eq for a = obained in his work. Color figure can be viewed in he online issue, which is available a Numerical Mehods for Parial Differenial Equaions DOI /num

28 NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS 475 FIG. 7. The surface of he exac and he rd-order approximaion soluions u o Eq. 4.1 for a = b = 1 obained in his work. Color figure can be viewed in he online issue, which is available a can see ha he approximae soluions obained by HAM are quie close o heir exac soluions. Also Tables I-IV presen some compuaional resuls. V. CONCLUSION In his paper, based on he HAM, a new analyic echnique is proposed o solve he nonlinear fracional parial differenial equaions. Differen from all oher analyic mehods, i provides us wih a simple way o adjus and conrol he convergence region of soluion series by inroducing TABLE I. Numerical resuls of w o Eq for a = 1.5, c = 0.1, b = 0.1, and k = 0.1. x, w approx = i=0 w i w exa w error 0, E E E 0, E E E 10, E E E 0, E E E 10, E E E TABLE II. Numerical resuls of v o Eq for a = 1.5, c = 0.1, b = 0.1, and k = 0.1. x, v approx = i=0 v i v exa v error 0, E E E 0, E E E 10, E E E 0, E E E 5 10, E E E Numerical Mehods for Parial Differenial Equaions DOI /num

29 476 DEHGHAN, MANAFIAN, AND SAADATMANDI TABLE III. Numerical resuls of Eq for a = 1 and m = n =. x, u approx = i=0 u i u exa u error 0, E 1 10, E 1 0, , E 1 0, TABLE IV. Numerical resuls of Eq. 4.1 for a = 1 and m = n =. x, u approx = i=0 u i u exa u error 0, , , E 1 0, E 1 10, E 1 an auxiliary parameer h. This is an obvious advanage of he homoopy analysis mehod. The validiy of he mehod has been successful shown by applying i for differen kinds of nonlinear fracional parial differenial equaions in mahemaical physics, namely KdV, K,, Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV and Boussinesq-like Bm, n equaions. We can simply choose he fracional operaor D α as he auxiliary linear operaor. In his way, we obain soluions in power series. Also, we obained he exac soluions in special case α = 1, h = 1, for some equaions. However, i is well-known ha a power series ofen has a small convergence radius. I should be emphasized ha, in he frame of he homoopy analysis mehod, we have grea freedom o choose he iniial guess and he auxiliary linear operaor L = D α. This work shows ha he homoopy analysis mehod is a very efficien and powerful ool for solving he nonlinear fracional parial differenial equaions. The auhors are very graeful o he hree reviewers for carefully reading he paper and for heir consrucive commens and suggesions which have improved he paper. References 1. M. J. Ablowiz and P. A. Clarkson, Solions, nonlinear evoluion equaions and inverse scaering, Cambridge Universiy Press, New York, K. S. Miller and B. Ross, An inroducion o he fracional calculus and fracional differenial equaions, Wiley, New York, I. Podlubny, Fracional differenial equaions: an inroducion o fracional derivaives, fracional differenial equaions, o mehods of heir soluion and some of heir applicaions, Academic Press, New York, S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fracional inegrals and derivaives: Theory and applicaions, Yverdon, Gordon and Breach, B. J. Wes, M. Bolognab, and P. Grigolini, Physics of fracal operaors, Springer, New York, S. Abbasbandy, An approximaion soluion of a nonlinear equaion wih Riemann Liouville s fracional derivaives by He s variaional ieraion mehod, J Compua Appl Mah , M. Capuo, Linear models of dissipaion whose Q is almos frequency independen, J R Asronomic Soc , L. Debanh, Recens applicaions of fracional calculus o science and engineering, In J Mah Mah Sci 54 00, Numerical Mehods for Parial Differenial Equaions DOI /num

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