Construction of Analytical Solutions to Fractional Differential Equations Using Homotopy Analysis Method

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1 IAENG Inernaional Journal of Applied Mahemaics, 0:, IJAM_0 01 Consrucion of Analical Soluions o Fracional Differenial Equaions Using Homoop Analsis Mehod Ahmad El-Ajou 1, Zaid Odiba *, Shaher Momani 3, Ahmad Alawneh Absrac In his paper, we presen an algorihm of he homoop analsis mehod (HAM) o obain smbolic approximae soluions for linear and nonlinear differenial equaions of fracional order. We show ha he HAM is differen from all analical mehods; i provides us wih a simple wa o adjus and conrol he convergence region of he series soluion b inroducing he auxiliar parameer ħ, he auxiliar funcion H, he iniial guess 0 () and he auxiliar linear operaor. Three examples, he fracional oscillaion equaion, he fracional Riccai equaion and he fracional Lane-Emden equaion, are esed using he modified algorihm. The obained resuls show ha he Adomain decomposiion mehod, Variaional ieraion mehod and homoop perurbaion mehod are special cases of homoop analsis mehod. The modified algorihm can be widel implemened o solve boh ordinar and parial differenial equaions of fracional order. Index Terms Adomian decomposiion mehod, Capuo derivaive, Fracional Lane-Emden equaion, Fracional oscillaion equaion, Fracional Riccai equaion, Homoop analsis mehod. I. INTRODUCTION Fracional differenial equaions have gained imporance and populari during he pas hree decades or so, mainl due o is demonsraed applicaions in numerous seemingl diverse fields of science and engineering. For example, he nonlinear oscillaion of earhquake can be modeled wih fracional derivaives, and he fluid-dnamic raffic model wih fracional derivaives can eliminae he deficienc arising from he assumpion of coninuum raffic flow. The differenial equaions wih fracional order have recenl proved o be valuable ools o he modeling of man phsical phenomena [1-]. This is because of he fac ha he realisic modeling of a phsical phenomenon does no depend onl on he insan ime, bu also on he hisor of he previous ime which can also be successfull achieved b using fracional 1 Deparmen of Mahemaics, Zarqa Privae Universi, Zarqa 1311, Jordan, ( ajou1@ahoo.com) Prince Abdullah Bin Ghazi Facul of Science and IT, Al-Balqa' Applied Universi, Sal 19117, Jordan, ( odiba@bau.edu.jo) 3 Deparmen of Mahemaics, Muah Universi, P. O. Box 7, Al-Karak, Jordan, ( shahermm@ahoo.com) Deparmen of Mahemaics, Universi of Jordan, Amman 119, Jordan, ( alawneh@ju.edu.jo) * Corresponding auhor. addresses: z.odiba@gmail.com; odiba@bau.edu.jo, Tel , Fax calculus. Mos nonlinear fracional equaions do no have exac analic soluions, so approximaion and numerical echniques mus be used. The Adomain decomposiion mehod (ADM) [9-1], he homoop perurbaion mehod (HPM) [15-5], he variaional ieraion mehod (VIM) [-] and oher mehods have been used o provide analical approximaion o linear and nonlinear problems. However, he convergence region of he corresponding resuls is raher small as shown in [9-]. The homoop analsis mehod (HAM) is proposed firs b Liao [9-33] for solving linear and nonlinear differenial and inegral equaions. Differen from perurbaion echniques; he HAM doesn' depend upon an small or large parameer. This mehod has been successfull applied o solve man pes of nonlinear differenial equaions, such as projecile moion wih he quadraic resisance law [3], Klein-Gordon equaion [35], soliar waves wih disconinui [3], he generalized Hiroa-Sasuma coupled KdV equaion [37], hea radiaion equaions [3], MHD flows of an Oldrod -consan fluid [39], Vakhnenko equaion [0], unsead boundar-laer flows [1]. Recenl, Song and Zhang [] used he HAM o solve fracional KdV-Burgers-Kuramoo equaion, Cang and his co-auhors [3] consruced a series soluion of non-linear Riccai differenial equaions wih fracional order using HAM. The proved ha he Adomian decomposiion mehod is a special case of HAM, and we can adjus and conrol he convergence region of soluion series b choosing he auxiliar parameer ħ closed o zero. The objecive of he presen paper is o modif he HAM o provide smbolic approximae soluions for linear and nonlinear differenial equaions of fracional order. Our modificaion is implemened on he fracional oscillaion equaion, he fracional Riccai equaion and he fracional Lane-Emden equaion. B choosing suiable values of he auxiliar parameer ħ, he auxiliar funcion H, he iniial guess 0 () and he auxiliar linear operaor we can adjus and conrol he convergence region of soluion series. Moreover, we illusraed for several examples ha he Adomain decomposiion, Variaional ieraion and homoop perurbaion soluions are special cases of homoop analsis soluion. II. DEFINITIONS For he concep of fracional derivaive we will adop Capuo s definiion [7] which is a modificaion of he Riemann-Liouville definiion and has he advanage of dealing properl wih iniial value problems in which he (Advance online publicaion: 13 Ma 0)

2 IAENG Inernaional Journal of Applied Mahemaics, 0:, IJAM_0 01 iniial condiions are given in erms of he field variables and heir ineger order which is he case in mos phsical processes []. Definiion 1. A real funcion f(x), x > 0, is said o be in he space C μ, μ R if here exiss a real number p, such ha f x = x p f 1 x, where f 1 x C 0, and i is said o be in he space C μ n iff f n x C μ, n N. Definiion. The Riemann-Liouville fracional inegral operaor of order α > 0, of a funcion f x C μ, μ 1, is defined as: J α f x = 1 x x α1 f d, Γ α (1) 0 α > 0, x > 0, J 0 f x = f x. () Properies of he operaor J α can be found in [5-], we menion onl he following: For f C μ, μ 1, α, β 0 and γ 1 J α J β f x = J α+β f x = J β J α f(x), (3) J α x γ = Γ(γ + 1) Γ(α + γ + 1) xα+γ. () Definiion 3. The fracional derivaive of f(x) in he Capuo sense is defined as: D α f x = J nα D n f x = 1 Γ n α 0 x x xα1 f (n) d, n for n 1 < α n, n N, x > 0, f C 1. Lemma 1. If n 1 < α n, n N and f C μ n, μ 1, hen (5) D α J α f x = f x, () n1 J α D α f x = f x f k 0 + x k!, x > 0. (7) k=0 III. HOMOTOPY ANALYSIS METHOD The principles of he HAM and is applicabili for various kinds of differenial equaions are given in [9-3]. For convenience of he reader, we will presen a review of he HAM [9, 3]. To achieve our goal, we consider he nonlinear differenial equaion. N = 0, 0, () where N is a nonlinear differenial operaor, and () is unknown funcion of he independen variable. A. Zeroh- order Deformaion Equaion Liao [9] consrucs he so-called zeroh-order deformaion equaion: 1 q φ ; q 0 = qħh N φ ; q, (9) where q 0,1 is an embedding parameer, ħ 0 is an auxiliar parameer, H 0 is an auxiliar funcion, is an auxiliar linear operaor, N is nonlinear differenial operaor, φ ; q is an unknown funcion, and 0 is an iniial guess of (), which saisfies he iniial condiions. I should be emphasized ha one has grea freedom o choose he iniial guess 0, he auxiliar linear operaor, he auxiliar parameer ħ and he auxiliar funcion H. According o he auxiliar linear operaor and he suiable iniial condiions, when q = 0, we have φ ; 0 = 0, () and when q = 1, since ħ 0 and H 0, he zeroh-order deformaion equaion (9) is equivalen o (), hence φ ; 1 =. (11) Thus, as q increasing from 0 o 1, he soluion φ ; q various from 0 o (). Define 1 m φ ; q m = m! q m q=0. (1) Expanding φ ; q in a Talor series wih respec o he embedding parameer q, b using () and (1), we have: φ ; q = 0 + m =1 m q m. (13) Assume ha he auxiliar parameer ħ, he auxiliar funcion H, he iniial approximaion 0 () and he auxiliar linear operaor are properl chosen so ha he series (13) converges a q = 1. Then a q = 1, from (11), he series soluion (13) becomes = 0 + m =1 m B. High-order Deformaion Equaion Define he vecor:. (1) n = 0, 1,,, n (). (15) Differeniaing equaion (9) m-imes wih respec o embedding parameer q, hen seing q = 0 and dividing hem b m!, we have, using (1), he so-called mh-order deformaion equaion m χ m m 1 = ħh R m m1, m = 1,,, n, (1) (Advance online publicaion: 13 Ma 0)

3 IAENG Inernaional Journal of Applied Mahemaics, 0:, IJAM_0 01 where and 1 m 1 N[φ ; q ] R m m 1 = (m 1)! q m1, (17) q=0 χ m = 0, m 1 1, m > 1. (1) The so-called mh-order deformaion equaion (1) is a linear which can be easil solved using Mahemaica package. IV. AN ALGORITHM OF HOMOTOPY ANALYSIS METHOD The HAM has been exended in [,3] o solve some fracional differenial equaions. The use he auxiliar linear operaor o be d/d or D α, where 0 < α 1. Also he fix he auxiliar funcion H o be 1. In his secion, we presen an efficien algorihm of he HAM. This algorihm can be esablished based on he assumpions ha he nonlinear operaor can involve fracional derivaives, he auxiliar funcion H can be freel seleced and he auxiliar linear operaor can be considered as = D β, β > 0. To illusrae is basic ideas, we consider he following fracional iniial value problem N α = 0, 0, (19) where N α is a nonlinear differenial operaor ha ma involves fracional derivaives, where he highes order derivaive is n, subjec o he iniial condiions k 0 = c k, k = 0,1,,, n 1. (0) The so-called zeroh-order deformaion equaion can be defined as 1 q D β φ ; q 0 = qħh N φ ; q, β > 0, subjec o he iniial condiions (1) φ k 0; q = c k, k = 0,1,,, n 1. () Obviousl, when q = 0, since 0 saisfies he iniial condiions (0) and = D β, β > 0, we have φ ; 0 = 0, (3) and, he so-called mh-order deformaion equaion can be consruced as m (k) 0 = 0, k = 0,1,,, n 1. (5) Operaing J β, β > 0 on boh sides of () gives he mh-order deformaion equaion in he form: m = χ m m 1 + ħ J β H R m m 1. () V. EXAMPLES In his secion we emplo our algorihm of he homoop analsis mehod o find ou series soluions for some fracional iniial value problems. Example 1. Consider he composie fracional oscillaion equaion [1] d d ad α b = 0, 0, n 1 < α n, n = 1,, subjec o he iniial condiions (7) 0 = 0, 0 = 0. () Firs, if we se α = 1, a = b = 1, hen he equaion (7) becomes linear differenial equaion of second order which has he following exac soluion = e / cos sin 3. (9) Since, N α = d d ad α b, according o (1) and (17), we have R m m 1 = m 1 ad α m 1 b m 1 1 χ m. (30) If we ake he auxiliar funcion H = 1, and he parameer β =, hen he auxiliar linear operaor becomes and = d d, (31) m = χ m m 1 + ħ J R m m 1, (3) subjec o he iniial condiions m 0 = m 0 = 0. (33) (I) If we choose he iniial guess approximaion D β m χ m m 1 = ħh R m m 1, β > 0, () hen we have 0 = 0, (3) subjec o he iniial condiions 1 = ħ, (35) (Advance online publicaion: 13 Ma 0)

4 IAENG Inernaional Journal of Applied Mahemaics, 0:, IJAM_0 01 = ħ 1 + ħ + ħ b 3 + ħ a Γ(5 α) α, 3 = ħ 1 + ħ + ħ 1 + ħ b ħ3 b ħ 1 + ħ a α 1ħ3 ab Γ 5 α Γ(7 α) α ħ3 a Γ(7 α) α. Moreover, if we se he auxiliar parameer ħ = 1, hen he resul is he Variaional ieraion soluion obained b Momani and Odiba in [1] and he same homoop perurbaion soluion obained b Odiba in [1]. (II) If we ake he iniial guess hen we will ge he approximaions 0 =, (3) 1 = ħb 3 + ħa Γ(5 α) α, = ħ 1 + ħ b + ħ b ħ 1 + ħ a 3 90 Γ 5 α α + 1ħ ab Γ(7 α) α + ħ a Γ(7 α) α, 3 = ħ 1 + ħ b ħ 1 + ħ b 90 ħ3 b 3 35 ħ 1 + ħ a Γ 5 α + 3ħ 1 + ħ ab Γ 7 α α α + 1ħ 1 + ħ a α Γ 7 α ħ3 ab Γ 9 α α ħ3 a b Γ(9 α) α ħ3 a 3 Γ(9 3α) 3α. (37) Here, if we pu ħ = 1 in (37), hen we will ge he Adomian decomposiion soluion obained b Momani and Odiba in [1]. (III) If we ake he iniial guess +ħ α 30a Γ 7 α +. (VI) If we replace H = 1 b H = 1/ in (III), hen we have 1 = ħ 15 5 ħb 1 9 ħa 9 α (7 α)γ(3 α) 9 α, = ħ 15 5 ħb 1 9 ħa 9 α 7 α Γ 3 α 9 α + ħ () 1+3α 3α. B means of he so-called ħ-curves [9], Fig.1 shows ha he valid region of ħ is he horizonal line segmen. Thus, he valid regions of ħ for he HAM of Eq. (7) a differen values of α are shown in Fig.1. Fig..(a)~(f) shows he approximae soluions for Eq. (7) obained for differen values of α, ħ, H and 0 () using HAM. Fig..(a) shows ha he bes soluion resuls when we use he iniial guess 0 = 3. Fig..(b) shows ha he bes soluion resuls when we use he auxiliar parameer ħ = 0.7. Fig..(c) shows ha he bes soluion resuls when we use he auxiliar funcion H = 1. In Fig..(d)~(f), we compare he approximae soluions for differen values of ħ and H. As shown in Fig..(a)~(f), we can observe ha b choosing a proper value of he auxiliar parameer ħ, he auxiliar funcion H, he auxiliar linear operaor and he iniial guess 0 () we can adjus and conrol convergence region of he series soluions. Moreover, we can observe ha convergence region increases as ħ goes o lef end poin of he valid region of ħ, bu his ma decreases he agreemen wih he exac soluion. Choosing a suiable auxiliar funcion H wih he lef end poin of he valid region of ħ scrimps he disagreemen wih he exac soluion. Fig.3. show he residual error for 15h order approximaion for Eq. (7) a α = 1 obained for differen values of ħ and H. ' = 3, hen we will ge he approximaions 1 = ħ ħb ħa Γ(5 α) α, = ħ ħb ħa Γ 5 α α (3) (39) h 0.0 Fig.1. The ħ-curves of which are corresponding o he15h-order approximae HAM soluion of Eq.(7) when H = 1/, 0 = 3, L = d d. Doed line: α = 0.5, dash.doed line: α = 1, solid line: α = 1.5. (Advance online publicaion: 13 Ma 0)

5 IAENG Inernaional Journal of Applied Mahemaics, 0:, IJAM_ (a) α = 1, H = 1, ħ = 1, L = d d. Dashed line: 0 = 0, dash.doed line: 0 =, doed line: 0 = 3, solid line: Exac soluion (c) α = 1, ħ = 0.7, 0 = 3, L = d d. Dashed line: H =, dash.doed line: H = 1/, doed line: H = 1, solid line: Exac soluion (b) α = 1, H = 1, 0 = 3, L = d d. Dashed line: ħ = 1, dash.doed line: ħ = 0.7, doed line: ħ = 0.3, solid line: Exac soluion (d) α = 1, 0 = 3, L = d d. Doed line: H = 1, ħ = 0.7, dash.doed line: H = 1/, ħ = 0.3, solid line: Exac soluion (e) α = 0.5, 0 = 3, L = d d. 0 (f) α = 1.5, 0 = 3, L = d d. Doed line: H = 1, ħ = 0.7, solid line: H = 1/, ħ = 0.3. Fig.. Approximae soluions for Eq. (7) Doed line: H = 1, ħ = 0.7, solid line : H = 1/, ħ = 0.3. Example. Consider he fracional Riccai equaion [17] D α + 1 = 0, 0 < α 1, 0, (0) subjec o he iniial condiion 0 = 0. (1) According o (17), we have m1 R m m1 = D α m 1 + i m 1i 1 χ m, i=0 () (I) If we choose he iniial guess approximaion 0 =, (3) (Advance online publicaion: 13 Ma 0)

6 IAENG Inernaional Journal of Applied Mahemaics, 0:, IJAM_ Fig. 3. The residual error for Eq. (7) when α = 1, 0 = 3, L = d d. Solid line: H = 1, ħ = 0.7, doed line: H = 1/, ħ = 0.3. he parameer β = 1 and H = 1, hen according o () we find 1 = ħ 3 3 α Γ 3 α he auxiliar linear operaor and he auxiliar funcion hen () gives 0 = 1/, () L = D β, β > 0, (5) H = γ, γ > 1, () 1 = ħ β+γ Γ 1 + γ Γ 1 + γ + β + Γ + γ Γ + γ + β πγ γ α 1 α Γ 3 α Γ 3, + γ α + β = ħ 3 3 α Γ 3 α ħ α Γ 3 α α 3 = ħ 3 3 α Γ 3 α ħ α Γ 3 α Γ 3 α + Γ α 3α Γ α Γ α, 3α Γ α Γ 5 α Γ α α Γ α Γ 5 α +ħ α Γ α + 3Γ 3 α + Γ α + 1 Γ α + 1 Γ 3 α + Γ α + Γ 5 α 5α Γ α 3α Γ α α Γ 7 α Γ 5 α Γ α Γ 5 α 3α Γ α + Γ 5 3α. Now, if we ake ħ = 1, hen we have he homoop perurbaion soluion obained b Odiba and Momani in [17]. (II) If we chose he iniial guess approximaion = ħ β +γ Γ 1 + γ Γ 1 + γ + β + Γ + γ Γ + γ + β πγ γ α 1 α Γ 3 α Γ γ α + β ħ β+γ1, and so on. As poined above, he valid region of ħ is a horizonal line segmen. Thus, he valid region of ħ for he HAM soluion of Eq. (0) a α = 0.5, γ = 0.5, β = 1 is 1.15 < ħ < 0.1 as shown in Fig.. Fig. 5.(a)~(d) show he approximae soluion for Eq. (0) obained for differen values of ħ, β, H() and 0 () using HAM. As observed in Fig..(a)~(f), we can noice ha he convergence region can be adjused and conrolled b choosing proper values of he auxiliar parameer ħ, he auxiliar funcion H, he auxiliar linear operaor and he iniial guess 0 () ', " 0 Fig.. The ħ-curves of & which are corresponding o he HAM soluion of Eq. (0) when α = 0.5, 0 = 1/, γ = 0.5, β = 1. Doed curve: 15h-order approximaion of, solid line: 15h-order approximaion of. 0 h (Advance online publicaion: 13 Ma 0)

7 IAENG Inernaional Journal of Applied Mahemaics, 0:, IJAM_ (a) α = 0.5, L = D α, H = 1, 0 = α /Γ 1 + α. Dash. doed line: ħ = 1, doed line: ħ = 0.5, solid line: ħ = (b) α = 0.5, L = D α, H = 1, ħ = 0.. Dash. doed line: 0 =, doed line: 0 = α /Γ 1 + α, solid line: 0 = 1/ (c) α = 0.5, 0 = 1/, H = 1, ħ = (d) α = 0.5, 0 = 1/, L = D = d d, ħ = 0.. Dash. doed line: L = D 0.5, doed line: L = D 0.75, solid line: L = Dash. doed line: H = 1, doed line: H = 1/, solid D. line: H = 1/. Fig. 5. Approximae soluions for Eq. (0) Example 3. Consider he Lane-Emden fracional differenial equaion [13, 1 hen he mh-order deformaion equaion () gives D α = 0, 0, (7) n 1 < α n, n = 1,, subjec o iniial condiions 0 = 0, 0 = 0. () Hence, according o (17), we have 1 = ħ 3 + 5, " R m m 1 = D α m 1 + m m1 i + m 1i j ij i=0 j =0 (9) χ m h In view of he modified homoop analsis mehod, if we se 0 = 0, H = 1, β =, (50) Fig.. The ħ-curves of which are corresponding o he 5h-order approximae HAM soluion of Eq. (7) for differen values of α. Dash doed line: α =, doed line: α = 1.75, solid line: α = 1.5. (Advance online publicaion: 13 Ma 0)

8 IAENG Inernaional Journal of Applied Mahemaics, 0:, IJAM_ (a) α =, 0 = 0, H = 1, β =. Doed line: ħ = 0.5, dash.doed line: ħ = 0.35, dash.do doed line: ħ = 0.1, solid line: exac soluion. Fig. 7. Approximae soluions for Eq. (7) (b) 0 = 0, H = 1, β =, ħ = Dash. doed line: α = 1.5, doed line: α = 1.75, solid line: α =. = ħ ħ α Γ α Γ 11 α + Γ α + ħ 117 +, and so on. Moreover, if we replace he iniial guess 0 = 0 b 0 =, hen we have m = 0, m 1, hence, = is he exac soluion. The valid region of ħ for he HAM soluion of Eq. (7) a α = is 0. < ħ < 0.15, a α = 1.75 is 0. < ħ < 0.15 and a α = 1.5 is 0. < ħ < 0.15 as shown in Fig.. Fig.7.(a) shows he approximae soluion for Eq. (7) a α = obained for differen values of ħ using HAM. As he previous examples, he convergence region of he series soluion increases as ħ goes o he lef end poin of is valid region. Fig. 7.(b) shows he HAM soluion for Eq. (7) a differen values of α obained for ħ = VI. DISCUSSION AND CONCLUSIONS In his work, we carefull proposed an efficien algorihm of he HAM which inroduces an efficien ool for solving linear and nonlinear differenial equaions of fracional order. The modified algorihm has been successfull implemened o find approximae soluions for man problems. The work emphasized our belief ha he mehod is a reliable echnique o handle nonlinear differenial equaions of fracional order. As an advanage of his mehod over he oher analical mehods, such as ADM and HPM, in his mehod we can choose a proper value for he auxiliar parameer ħ, he auxiliar funcion H, he auxiliar linear operaor and he iniial guess 0 o adjus and conrol convergence region of he series soluions. There are some imporan poins o make here. Firs, we can observe ha he convergence region of he series soluion increases as ħ ends o zero. Second, choosing a suiable auxiliar funcion H or iniial approximaion 0 ma accelerae he rapid convergence of he series soluion and ma increase he agreemen wih he exac soluion. Third, for a cerain value of ħ, choosing a suiable auxiliar linear operaor L = D β, β > 0 ma increase he convergence region. Finall, generall speaking, he proposed approach can be furher implemened o solve oher nonlinear problems in fracional calculus field. REFERENCES [1] G. O. Young, Definiion of phsical consisen damping laws wih fracional derivaives, Z. Angew. Mah. Mech, vol. 75, 1995, pp [] J. H. He, Some applicaions of nonlinear fracional differenial equaions and heir approximaions, Bull. Sci. Technol., vol. 15(), 1999, pp [3] J. H. He, Approximae analic soluion for seepage flow wih fracional derivaives in porous media, Compu. Mehods Appl. Mech. Eng., vol. 17, 199, pp [] F. Mainardi, Fracional calculus: 'Some basic problems in coninuum and saisical mechanics, in Fracals and Fracional calculus in Coninuum Mechanics, A. carpener and F. Mainardi, Eds., New York: Springer-Verlag, 1997, pp [5] I. Podlubn, Fracional Differenial Equaions. New York: Academic Press, [] K. S. Miller, and B. Ross, An inroducion o he fracional calculus and fracional differenial equaions. New York: John Will and Sons, Inc., [7] M. Capuo, Linear models of dissipaion whose Q is almos frequenc independen, Par II. J. Ro Ausral Soc. vol. 13, 197, pp [] K. B. Oldham, and J. Spanier, The Fracional calculus. New York: Academic Press, 197. [9] G. Adomian, Nonlinear sochasic differenial equaions, J. Mah. Anal. Appl., vol. 55, 197, pp [] G. Adomian, A review of he decomposiion mehod in applied mahemaics, J. Mah. Anal. Appl., vol. 135, 19, pp [11] G. Adomian, Solving Fronier Problems of phsics: he decomposiion mehod. Boson: Kluewr Academic Publishers, 199. [1] N. T. Shawagfeh, The decomposiion mehod for fracional differenial equaions, J. Frac. Calc., vol. 1, 1999, pp [13] A. M. Wazwaz, The modified decomposiion mehod for analic reamen of differenial equaions, Appl. Mah. Comp., vol. 173, 00, pp (Advance online publicaion: 13 Ma 0)

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