The Application of Optimal Homotopy Asymptotic Method for One-Dimensional Heat and Advection- Diffusion Equations

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1 Inf. Sci. Le., No., ) 57 Informaion Sciences Leers An Inernaional Journal hp://d.doi.org/1.1785/isl/ The Applicaion of Opimal Homoopy Asympoic Mehod for One-Dimensional Hea and Advecion- Diffusion Equaions Fazle Mabood Deparmen of Mahemaics, Edwardes College Peshawar, KPK, 5, Pakisan Received: 16 Nov. 1, Revised: 3 Mar. 13, Acceped: 4 Mar. 13 Published online: 1 May. 13 Absrac: Aim of he paper is o invesigae approimae analyical soluion of ime-dependen parial differenial equaion using a semi-analyical mehod, he Opimal Homoopy Asympoic Mehod OHAM). To show he efficiency of he proposed mehod, we consider one-dimensional hea and advecion-diffusion equaions. OHAM uses simple compuaions wih prey good enough approimae soluion, which has an ecellen agreemen wih he eac soluion available in open lieraure. OHAM is no only reliable in obaining series soluion for such problems wih high accuracy bu i also saving he volume and ime as compared o oher analyical mehods. Keywords: Opimal Homoopy Asympoic Mehod, Hea Equaion, Advecion-Diffusion Equaion 1 Inroducion Consider he one dimensional advecion-diffusion equaion [4]: u + βu = αu, a b,. 1) subjec o he iniial condiion: u,) = ϕ), [a,b] ) and he boundary condiions are: { ua,) = g ) ub,) = g 1 ), [,T ] where he subscrips and denoe differeniaion wih respec o ime and space respecively, and are supposed o be smooh funcions. In case of g, g 1 he advecion-diffusion equaion will reduced ino one-dimensional hea equaion is considered as hermal diffusion. In case of β =, he advecion-diffusion equaion will reduced ino one-dimensional hea equaion is considered as hermal diffusion. 3) The Advecion-Diffusion Equaion ADE) is of primary imporance in many physical sysems, especially hose involving fluid flow [1], one-dimensional version of he parial differenial equaions which describe advecion-diffusion equaion arise frequenly in ransferring mass, hea, energy and voriciy in chemisry and engineering []. Parlarge [3] used ADE is o model waer ranspor in soils, Caglar e al. [4] have uilized hird-degree B-Spline funcion for he numerical soluion of one dimensional hea equaion, Mohebbi and Dehghan [5] have presened finie difference approimaion and cubic C1-spline collocaion mehod for he soluion wih fourh-order accuracy in boh space and ime variables Oh4, k4). Cubic B-Spline Collocaion Mehod for he numerical soluion of one dimensional hea and advecion-diffusion equaions are well repored by Goh e al. [6]. A newly developed analyical mehod namely he opimal homoopy asympoic mehod has recenly been used o solve a wide class of physical problems. Marinca and Nicolae [7,8,9] used OHAM for solving nonlinear equaions relaed o differen physical phenomena. Also Marinca e al. [1] sudied he hin film flow using OHAM. Iqbal e al. [11] provided he OHAM soluions of Corresponding auhor mabood1971@yahoo.com c 13 NSP

2 58 F. Mabood: The Applicaion of Opimal Homoopy Asympoic... he linear and nonlinear Klein-Gordon equaions, Islam e al. [1] applied OHAM for he asympoic soluions of Couee and Poiseuille flows of a hird grade fluid whils Idrees e al. [13, 14] and Mabood e al. [15, 16] have uilized he proposed mehod OHAM) effecively for differen higher order boundary values problems. According o he bes of auhor s knowledge he hea modeling problem menioned above has no been ye sudied by opimal homoopy asympoic mehod OHAM). Basic Formulaion of OHAM We review he basic principles of OHAM as developed by Marinca e al. [8]. Consider he following differenial equaion and boundary condiion: Lvz,)) + gz,) + Nvz,)) =, z Ω 4) B v, dv ) = 5) where L, N are linear and nonlinear operaors, z, denoe he spaial and ime variables respecively, Ω is he problem domain and vz, ) is an unknown funcion, gz,) is a known funcion and B is a boundary operaor. An equaion known as opimal homoopy equaion is consruced: 1 r)[lϕz,;r) + gz,)] = Hr)[Lϕz,;r) + gz,) + Nϕz,;r))] 6) where r 1 is an embedding parameer, Hr) is auiliary funcion such ha Hr) for r and H) =, we have from Eq.6) r = [Lϕz,;) + gz,)] = 7) r = 1 [Lϕz,;1) + gz,1) + Nϕz,;1))] = 8) Thus, for r = and r = 1 we obain, ϕz,;) = v z,) and ϕz,;1) = vz,) respecively. Hence, as r varies from o 1 he soluion ϕz,;r) varies from v z,) o he soluion vz,), where v z,) is obained from Eq. 6) se r = Lv z,)) + gz,) =, B v, dv ) = 9) The auiliary funcion Hr) is chosen of he form: Hr) = n k=1 r k C k 1) where C i, i N are consans which are o be deermined laer [8]. For soluion, epand ϕz,;r,c i ) in Taylor s series abou r and wrien as: ϕz,;r,c i ) = v z,) + k=1 v k z,;c i ), i = 1,, 11) Subsiuing equaion 11) ino equaion 6), and equaing he coefficiens of he like powers of r equal o zero, gives he linear equaions as described below: The zeroh order problem is given by equaion 9), and he firs and second order problems are given by he equaions 1) and 13), respecively, while he general governing equaion for v k z,) is given in equaion 14): Lv 1 z,)) = C 1 N v z,)), B v 1, dv 1 ) = 1) Lv z,)) Lv 1 z,)) = C N v z,)) +C 1 [Lv 1 z,)) + N 1 v z,),v 1 z,))] 13) B v, dv ) = Lv k z,)) Lv k 1 z,)) = C k N v z,)) k 1 + i=1 C i [Lv k i z,)) + N k i v z,), v 1 z,),...,v k 1 z,)))] 14) B v k, dv ) k =, k = 1,, where N m v z,),v 1 z,),v z,),...,v m z,)) is he coefficien of r m in he epansion of Nϕz,;r,C i )) abou he embeding parameer. Nϕz,;r,C i )) = N v z,)) + N m v,v 1,v,...,v m )r m 15) k 1 The convergence of he series 11) is dependen upon he auiliary consans C 1,C,... If i is convergen a r = 1, one has: ṽz,;c i ) = v z,) + m i=1 v i z,;c i ) 16) Subsiuing equaion 16) ino equaion 4), he general problem resuls in he following residual: Rz,;C i ) = Lṽz,;C i )) + gz,) + Nṽz,;C i )) 17) If Rz,;C i ) =, hen will be he eac soluion. For nonlinear problems, generally his will no be he case. For deermining C i i = 1,,...), a and b are chosen such ha he opimum values for C i are obained, using he mehod of leas squares: JC i ) = where R is he residual, R z,,c i )dz 18) Ω J = J =... = J = 19) C 1 C C m c 13 NSP

3 Inf. Sci. Le., No., ) / Soluion of Hea Equaion via OHAM Consider he hea equaion is as follow [6]: wih iniial condiion is u = u, < < 1, > ) and boundary condiions are u, ) = sinπ) 1) u,) = u1,) = ) The eac soluion is u,) = e π sinπ) wich saisfies equaion ). Applying he proposed mehod OHAM) menioned in Secion, on equaion ) leads o he following: Zeroh order problem: u,) = 3) wih iniial condiion: u,) = sinπ) u,) = sinπ) 4) Firs order problem: u 1,,C 1 ) = 1 +C 1 ) u,) wih iniial condiion: u 1,) = C 1 u,) 5) u 1,,C 1 ) = C 1 π sinπ) 6) Second order problem: wih iniial condiion: u 3,) = u 3,,C 1,C,C 3 ) = 1 6 [6C 1π sinπ) + 1C 1π sinπ) +6C 1π 4 sinπ) + 6C 3 1 π sinπ) +6C 3 1 π4 sinπ) +C 3 1 π6 3 sinπ) +6C πcosπ) + 6C 1 C πcosπ) +3C 1 C π 3 cosπ) + 6C 3 π sinπ) +3C 1 C π 4 sinπ) +6C 1 C π sinπ)] 3) Using equaions 4), 6), 8) and 3), he hird order approimae soluion via OHAM for r = 1 is ũ,,c 1,C,C 3 ) = u,) + u 1,,C 1 ) + u,,c 1,C ) +u 3,,C 1,C,C 3 ) 31) Wih he help of leas square mehod, we can obain he values of unknown consans, for = 1 he values of C 1 = 3639, C = 7461, C 3 = 175 and subsiuing he values of C 1,C,C 3 in equaion 31), we obain he approimae soluion of hea equaion as follow: ũ, ) = [ ]cosπ)[ ))]sinπ) 3) Figs.1, and 3 have been prepared for he comparaive picure of he series soluion obained using OHAM wih he eising eac soluion for differen assigned values of. u,,c 1,C ) C 1 u 1,,C 1 ) = 1 +C 1 ) u 1,,C 1 ) +C u,) wih iniial condiion: u,) = +C u,) u,,c 1,C ) = 1 [C 1π sinπ) + C 1π sinπ) 7) u,.. Fig. 1: Comparison of soluion suing OHAM solid line) wih eac soluion dashed line) for = 1 +C 1π 4 sinπ) + C πcosπ)] 8) Third order problem: u 3,,C 1,C,C 3 ) +C 3 u,) = 1 +C 1 ) u,,c 1,C ) +C u 1,,C 1 ) C 3 u,) C u 1,,C 1 ) C 1 u,,c 1,C ) 9) 4 Soluion of Advecion-Diffusion Equaion via OHAM The advecion-diffusion equaion wih β = 1,α =.1 in equaion 1) is as follow [4]: u + u =.1u, < < 1, > 33) c 13 NSP

4 6 F. Mabood: The Applicaion of Opimal Homoopy Asympoic... 3 uh,l Fig. : Comparison of soluion suing OHAM solid line) wih eac soluion dashed line) for = 15 Fig. 5: Spaial-ime approimaion for Advecion-Diffusion Equaion over a ime period [, 1.] mehod OHAM) on Eq. 33), he zeroh, firs and second order problem are given as: uh,l =. 36) wih boundary u, ) = e5 [cos π ) +.5sin π )]. condiion: Fig. 3: Comparison of soluion suing OHAM solid line) wih eac soluion dashed line) for = Eac OHAM wih iniial condiion: uh,l 1 5. Fig. 4: Comparison of soluion using OHAM wih eac soluion u, ) = e5 [cos π π ) +.5sin )] 34) The eac soluion of equaion 33) is π ) 4 [cos π π ) +.5sin )] 35) The boundary condiions can be obained easily a = and = 1 from he eac soluion. Applying he proposed c 13 NSP wih iniial condiion: u, ) = Solving equaions 36), 37) and 38), we can obain second order approimae soluion using OHAM for r = 1 is u,,c1,c ) = u,) + u1,,c1 ) + u,,c1,c ) 39) wih iniial condiion is u1, ) = u,) = e5 ) e 37) u,,c1,c ) u1,,c1 ) u1,,c1 ) = 1 +C1 ) +C1 u1,) +C.1C1 +C.1C 38) 15 u1,,c1 ) = 1 +C1 ) +C1 u,).1c1 Using he mehod of leas square, he values of unknown consans for = 5 are C1 =.3695,C = 696. Subsiuing he values of C1,C in equaion 39) one can obain he hree erms approimae analyical soluion via OHAM for advecion-diffusion equaion. Fig. 4 has been presened for comparison of OHAM soluion wih eac soluion of advecion-diffusion equaion whils in Fig. 5 we have shown he spaial-ime approimaion.

5 Inf. Sci. Le., No., ) / Conclusion A series soluion based on Opimal Homoopy Asympoic Mehod had been described in secion 3 and 4 for solving one-dimensional hea and advecion-diffusion equaions. The obained soluion using OHAM is hen compared wih he eac soluions. The resuls are in good agreemen wih he eising eac resuls and herefore elucidae he reliabiliy and efficiency of OHAM. The comparisons made sugges ha he OHAM could be a useful and effecive ool for solving one-dimensional hea and advecion-diffusion equaions accuraely. References [13] M. Idrees, S.l Haq, S. Islam, Applicaion of opimal homoopy asympoic mehod o Fourh Order Boundary Values Problems. World Applied Sciences Journal. 9, ). [14] M. Idrees, S. Haq, S. Islam, Applicaion of Opimal Homoopy Asympoic Mehod o special Sih Order Boundary Values Problems, World Applied Sciences Journal. 9, ). [15] F. Mabood, W. A. Khan, A. I. M. Ismail, Soluion of Fifh Order Boundary Values Problems via Opimal Homoopy Asympoic Mehod, Wulfenia Journal. 19, ). [16] F. Mabood, A.I.M. Ismail, I. Hashim. The applicaion of Opimal Homoopy Asympoic Mehod for he Approimae Soluion of Riccai Equaion, Sains Malaysiana. 4, 13) In-Press). [1] M. Dehghan, Weighed finie difference echniques for he one-dimensional advecion-diffusion equaion, Applied Mahemaics and Compuaion. 147, ). [] B.J. Noye, Numerical soluion of parial differenial equaions, Lecure Noes 199). [3] H. Caglar, M. Ozer and N. Caglar, The numerical soluion of he one-dimensional hea equaion by using hird degree B-spline funcions, Chaos, Solions and Fracals. 38, ). [4] A. Mohebbi and M. Dehghan, High-order compac soluion of he one-dimensional hea and advecion-diffusion equaions, Applied Mahemaical Modelling. 34, ). [5] J.Y. Parlarge, Waer ranspor in soils, Ann Rev Fluids Mech.,, ). [6] J. Goh, Ahmad Abd. Majid and Ahmad Izani Md Ismail, Cubic B-Spline Collocaion Mehod for One- Dimensional Hea and Advecion-Diffusion Equaions, Journal of Applied Mahemaics. Aricle ID 45871, 8 1). [7] V. Marinca and N. Herisanu, Applicaion of opimal homoopy asympoic mehod for solving nonlinear equaions arising in hea ransfer, In. Commun. Hea Mass Transfer, 35, ). [8] V. Marinca and N. Herisanu, Opimal homoopy perurbaion mehod for srongly nonlinear differenial equaions, Nonlinear Sci. Le. A, 1, ). [9] N. Herisanu and V. Marinca, Accurae analyical soluions o oscillaors wih disconinuiies and fracional-power resoring force by means of he opimal homoopy asympoic mehod, Compuers and Mahemaics wih Applicaions, 6, ). [1] V. Marinca, N. Herisanu and I. Nemes, Opimal homoopy asympoic mehod wih applicaion o hin film flow, Cen. Eur. J. Phys. 6, ). [11] S. Iqbal, M. Idrees, A.M. Siddiqui, A.R. Ansari, Some soluions of he linear and nonlinear Klein-Gordon equaions using he opimal homoopy asympoic mehod, Applied Mahemaics and Compuaion. 16, ). [1] S. Islam, R. A. Shah, I. Ali, Opimal homoopy asympoic soluions of Couee and Poiseuille flows of a hird grade fluid wih hea ransfer analysis, In. J. Nonlinear Sci. Numer. Simul. 11, ). c 13 NSP