Laplace Homotopy Analysis Method for Solving Fractional Order Partial Differential Equations

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1 IOSR Journal of Mathematic (IOSR-JM) e-issn: , p-issn: X. Volume 3, Iue 5 Ver. I (Sep. - Oct. 207), PP Laplace Homotopy Analyi Method for Solving Fractional Order Partial Differential Equation Oama H. Mohammed (a), Huda A. Salim (b) a,b AL-Nahrain Univerity, College of Science,Department of Mathematic,Baghdad,Iraq. dr_uama79@yahoo.com (a), huda_ad9@yahoo.com (b) Abtract: In thipaper,we conidered a non-linear ytemof fractional partial differential equation.they have been olved by a computational method which i o-called a modified Laplace Homotopy Analyi method.the fractional derivative are decribed in the Caputo ene. The propoed technique i only a imple modification of the Homotopy Analyi Method.The method wa applied for ome illutrative eample to olve non-linear ytem of fractional partial differential equation. From the reult of the illutrative eample we conclude that the method i computationally efficient. Keyword: Fractional calculu, ytem of fractional order partial differential equation, Laplace tranform, Homotopy Analyi method Date of Submiion: Date of acceptance: I. Introduction Fractional order partial differential equation are popularization of claical partial differential equation. Thee have been of large attention in the recent literature. Thee topic have received a mighty deal of attention epecially in the field of vicoelaticity material, electrochemical procee, dielectric polarization, colored noie, anomalou diffuion, ignal proceing, control theory and other. Increaing etent, thee model are ued in application uch a fluid flow, finance and other. mot nonlinear fractional differential equation do not have eact olution, o approimation and numerical technique mut be applied.the Laplace Homotopy Analyi method (LHAM) i a combination of the Homotopy analyi method propoed by Liao in hi Ph.D. Thei [] and the Laplace tranform [2, 3]. The Homotopy analyi method ha been uccefully employed to olve many type of nonlinear, homogeneou or nonhomogeneou equation and ytem of equation a well a problem in cience and engineering [4-5]. Variou author have propoed everal cheme to olve ytem of fractional partial differential equation with Liouville-Caputo and Caputo-Fabrizio fractional operator. Dehghan in [6] applied the HAM to olve linear partial differential equation, in thi work, fractional derivative are decribed in the Liouville-Caputo ene, Xu in [7] calculated analytically the time fractional wave-like differential equation with a changeable coefficient, the author reduced the governing equation to two fractional ordinary differential equation. Jafari in [8] eercie the HAM to obtain the olution of multi-order fractional differential equation tudied by Diethelmand Ford [9], Goufo et al. [0] developed a mathematical analyi of a model of rock fracture in the ecoytem and applied the Caputo-Fabrizio fractional derivative, where analytical and computational approache are obtained. Other analytical approache that could be of interet are preented in [2-4]. Thi paper i organized a follow: in ection 2 we recall the definition of fractional derivative and fractional integration imply, ection 3 decribe the formulation of Laplace Homotopy analyi method for olving ytem of fractional order P.D.E.Some illutrative eample are given in ection 4 finally a concluion i given in ection 5. II. Fractional Order Derivative And Integral: In thi ection, we review baic definition of fractional order differentiation and fractional order integration uch a: Definition 2.:The Riemann Liouville fractional integral of order α > 0 i defined a follow: I α f() = (t Γ(α) τ)α f(τ)dτ, > 0,α R + 0 Where Γ(α) i the Gamma function. Definition 2.2:The Caputo fractional derivative of order α > 0 i defined a follow: DOI: / Page

2 Laplace Homotopy Analyi Method for Solving Fractional Order Partial Differential Equation f (m) (τ) Γ( α) ( τ) α+ m dτ 0 d m m < α < m c D α f () = f() α = m d m For α > 0, we have the following propertie of the Caputo fractional derivative: c α - D (I α f()) =f(). 2-I α c α ( D f()) = f() c α 3- D (c)=0, c R Γ(γ+) n k=0 f (k) (0 + ) k k! c α 4- D ( γ )= Γ(γ α+) γ α γϵ0,,2,3, }, γ [α] 0 γϵ0,,2,3, }, γ < [α] Where [α] i the floor function of α III. The Approach Let u conider the following ytem of non-linear fractional partial differential equation. C α D i t u i (, t) + R i (u, u 2,, u m ) + N i (u, u 2,, u m ) = g i (, t), n < α i n, i =,2,.. m () With initial data u i (, 0) = f i () (2) Where C α D i t are the Caputo fractional derivative of order α i, R i and N i, i =,2,, m are non-linear operator, repectively, and g i are ource term. In order to olve thi ytem by uing Laplace Homotopy Analyi method, firt we employing the Laplace tranform to the both ide of () yield: L[u i (, t)] = m α i k=0 αi k u i (, 0) α L[R i i (u, u 2,, u m ) + N i (u, u 2,, u m )] + α L[g i i(, t)] (3) The o-called zero-order deformation equation of the Laplace equation (3) ha ( q)[l i (, t; q) Lu i0 (, t)] = L i (, t; q) m α i k=0 αi k f i () + L [g α i t i (, t)] + qh [ L [R i ( (, t, q), 2 (, t, q),, m (, t, q) + N i (( (, t, q), 2 (, t, q),, ] ] (4) α i m (, t, q) Subject to the initial condition: i (, 0, q) = f i (), i =,2, m (5( Where q [0,]i an embedding parameter when q=0 we have L i (, t; 0) =Lu i0 (, t) And when q =, h = the zero-order deformation eq (4) and (5) equivalent to (3) and (2), repectively, provide L i (, t; ) = Lu i (, t). Thu a q increaing from 0 to, L i (, t, q) varie from Lu 0 (, t)tolu(, t). Epanding L i (, t; q) in Taylor erie with repect to q, one ha L i (, t; q) = Lu i0 (, t) + m= Lu im (, t)q m, i =,2,, m (6) Where m Lu im (, t) = L m! i(, t; q) (7) m q = 0 Define the vector Lu im (, t) = Lu i0 (, t), Lu i (, t), Lu i2 (, t),, Lu im (, t)}, i =,2,, m (8) Differentiating equation (4) m time with repect to the embedding parameter q, and etting q=0, h= - and finally dividing them by m!, we have the o-called m th order deformation equation for i =,2, n. Lu im (, t) = m Lu im (, t) R im (Lu im (, t) ) (9) Where DOI: / Page

3 Laplace Homotopy Analyi Method for Solving Fractional Order Partial Differential Equation R im ( Lu im (, t) ) = Lum (, t) + ( α i (m )! m N i (( (, t, q), 2 (, t, q),, m (, t, q)] ) ( m ) ( q = 0 q m [L[R i ( (, t, q), 2 (, t, q),, m (, t, q)) + m α i k=0 αi k f i () + L[g (, t)]) α i i (0) And m = 0, m (), m > Applying the invere Laplace tranform of both ide of (9), then we have a power erie olution of () which can be epreed a: u i (, t)= n=0 u in (, t), i =,2, m... (2) IV. Illutrative Eample In thi ection we will apply the LHAM to ytem of non-linear fractional partial differential equation (FPDE). Eample : Conider the following ytem of linear FPDE: C D α t u(, t) v (, t) + v(, t) + u(, t) = 0 C β D t v(, t) u (, t) + v(, t) + u(, t) = 0 (3) With initial condition a u(, 0) = inh(), v(, 0) = coh(), (0 < α, β < ) (4) The eact olution i given in [3] a u(, t) = inh( t), v(, t) = coh ( t) Taking the Laplace tranform to the both ide of eq (3) and uing (4), we have Lu(, t) inh() L[v (, t)] + α α L[v(, t) + u(, t)] = 0 Lv(, t) coh() L[u (, t)] + (5) β β L[v(, t) u(, t)] = 0} Furthermore, we can contruct the Homotopy a follow R m (u m, v m ) = Lu m (, t) L [(v m (, t)) α ] + L[v m (, t) + u α m (, t)] ( m ) inh()... (6) R 2m (u m, v m ) = Lv m (, t) L [(u m (, t)) β ] + L[v m (, t) u β m (, t)] ( m ) inh() (7) and the m th order deformation equation for m become Lu m (, t) = m Lu m (, t) R m (u m, v m ) (8) Lv m (, t) = m Lv m (, t) R 2m (u m, v m ) (9) From Eq (4), (8) and (9) and ubject to initial condition u m (, 0) = 0, v m (, 0) = 0, m We ucceively obtain Lu 0 (, t) = coh () Lv 0 (, t) = inh (), Lu (, t) = coh() α+, Lv (, t) = inh() β+ Lu 2 (, t) = α+β+2 coh() + inh() + coh() α+β+2 2α+2 Lv 2 (, t) = inh() + inh() + coh() α+β+2 2β+2 α+β+2 And o on. upon applying the invere Laplace tranform to the above equation and by uing (2), one can get: u(, t) =( tα+β+ Γ(α+β+2) tα +... ) inh() + ( tα+β+ + t2α+ + )coh() Γ(α+) Γ(α+β+2) Γ(2α+2) v(, t) =( + tα+β+ tβ + ) coh() + ( tα+β+ + t2β+ + )inh() Γ(α+β+2) Γ(β+) Γ(α+β+2) Γ(2b+2) (20) Following figure () repreent the approimate olution of problem (3) for different value of α and β compared with the eact olution when α = β = at t=0.0, 0.05, 0. DOI: / Page

4 Laplace Homotopy Analyi Method for Solving Fractional Order Partial Differential Equation Fig() the approimate olution of problem (3) for different value of α and β compared with the eact olution when α = β = Eample 2: Conider the nonlinear ytem of FPDE: C D α t u(, t) + v(, t)u (, t) + u(, t) = C D α t v(, t) u(, t)v (, t) v(, t) =, 0 < α (2) With initial condition u(, 0)=e,v(,0)=e (22) Taking the Laplace tranform to the both ide of eq. (2) and uing (22), we have Lu(, t) e + L[v(, t)u (, t)] + α α L[u(, t)] = 0 α+ } (23) Lv(, t) e L[u(, t)v (, t)] α α L[v(, t)] = 0 α+ Furthermore, we can contruct the Homotopy a follow R m (u m, v m ) = Lu m (, t) + α L [v m (, t)(u m (, t)) + u m (, t)] ( m ) ( e + α+) DOI: / Page

5 Laplace Homotopy Analyi Method for Solving Fractional Order Partial Differential Equation R 2m (u m, v m ) = Lv m (, t) α L [u m (, t)(v m (, t)) + v m (, t)] ( m ) ( e + α+) and the m th order deformation equation for m become Lu m (, t) = m Lu m (, t) R m (u m, v m ) (24) Lv m (, t) = m Lv m (, t) R 2m (u m, v m ) (25) From Eq (22), (24) and (25) and ubject to initial condition u m (, 0) = 0, v m (, 0) = 0, m We ucceively obtain e e Lu 0 (, t) = Lu (, t) =, α+ Lv 0 (, t) = e Lv 2 (, t) = e α+ Γ(2α + )t 3α Lu 2 (, t) = Γ(α + )Γ(α + ) 3α+ + e 2α+2 Γ(2α + )t 3α e Lv 2 (, t) = + Γ(α + )Γ(α + )3α+ 2α+2 Upon applying the invere Laplace tranform to the above equation and by uing (2), one can get: u(, t) =( v(, t) =( tα + t2α+ Γ(α+) tα + t2α+ Γ(α+) Γ(2α+)t 3α +... ) e + ( + ) Γ(2α+2) Γ(α+)Γ(α+)Γ(3α+) Γ(2α+)t 3α +... ) Γ(2α+2) e + ( + ) Γ(α+)Γ(α+)Γ(3α+) Following fig (2) repreent the approimate olution of problem (2) for different of α DOI: / Page

6 Laplace Homotopy Analyi Method for Solving Fractional Order Partial Differential Equation Figure (2) the approimate olution of problem (2) for different value of α V. Concluion We preent in thi paper a computational method for olving a ytem of fractional order partial differential equation which i known a Laplace Homotopy analyi method. The method conidered proved that it i a powerful tool which enable u to handle a wide cla of non-linear fractional partial differential equation in a imple way and in order to reach the deired accuracy, all what we have to do i to increae the number of iteration.if the non-linear problem ha an eact olution,then after a certain tage,every iteration lead to the ame eact olution.therefore Laplace Homotopy analyi method i adequate for both linear and non-linear problem Reference [] SJ.Liao, The propoed homotopy analyi technique for the olution of nonlinear problem. PhD thei, Shanghai Jiao Tong Univerity; 992. [2] Gepree.M, Gepreel.MS,Al-Malki. KA,and Al-Humyani. FA. M: Approimate olution of the generalized Abel integralequation uing the etenion Khan homotopy analyi tranformation method. J. Appl. Math. 205, (205) [3] Gupta. VG,Kumar. P: Approimate olution of fractional linear and nonlinear differential equation uing Laplace Homotopy analyi method. Int. J. Nonlinear Sci. 9(2), 3-20 (205) [4] Jafari. H, Seifi S: Homotopy analyi method for olving linear and nonlinear fractional diffuion-wave equation. Commun Nonlinear Sci Numer Simulat 2009; 4: [5] Liao.S, Homotopy analyi method: a new analytical technique for nonlinear problem. Commun Nonlinear Sci Numer Simulat 997;2(2):95 00 [6] Dehghan.M, Manafian. J and Saadatmandi. A: The olution of the linear fractional partial differential equation uing thehomotopy analyi method. Z. Naturforch. A 65(), (200) [7] Xu. H, Cang. J: Analyi of a time fractional wave-like equation with the homotopy analyi method. Phy. Lett. A 372(8), (2008) [8] Jafari. H, Da. S, Tajadodi. H: Solving a multi-order fractional differential equation uing homotopy analyi method. J. King Saud Univ., Sci. 23(2), 5-55 (20) [9] Diethelm. K, Ford, NJ: Multi-order fractional differential equation and their numerical olution. Appl. Math. Comput. 54, (2004) [0] Goufo. EF, Doungmo, MKP, Mwambakana, JN: Duplication in a model of rock fracture with fractional derivative without ingular kernel. Open Math. 3(), (205) [] Youefi. SA, Dehghan. M, Lotfi. A: Generalized Euler-Lagrange equation for fractional variational problem with free boundary condition. Comput. Math. Appl. 62(3), (20). [2] Mohebbi. A, Abbazadeh. M and Dehghan. M: High-order difference cheme for the olution of linear time fractional Klein-Gordon equation. Numer. Method Partial Differ. Equ. 30(4), (204) [3] Atangana. A: On the new fractional derivative and application to nonlinear Fiher reaction-diffuion equation. Appl.Math. Comput. 273, (206) [4] Atangana. A, Alkahtani. BST: Analyi of the Keller-Segel model with a fractional derivative without ingular kernel. Entropy 7(6), (205) Oama H.Mohammed. Laplace Homotopy Analyi Method for Solving Fractional Order Partial Differential Equation. IOSR Journal of Mathematic (IOSR-JM), vol. 3, no. 5, 207, pp DOI: / Page