VARIATIONAL ITERATION TRANSFORM METHOD FOR SOLVING BURGER AND COUPLED BURGER S EQUATIONS

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1 VARIATIONAL ITERATION TRANSFORM METHOD FOR SOLVING BURGER AND COUPLED BURGER S EQUATIONS Ali Al-Fayadh ad Haa Ali Khawwa Deparme of Mahemaic ad Compuer Applicaio, College of Sciece, Al-Nahrai Uiveriy, Baghdad, Iraq aalfayadh@yahoo.com ABSTRACT I hi paper, Variaioal ieraio raform mehod i employed o deermie he exac oluio of he Burger equaio which i oe-dimeioal ad coupled Burger equaio oliear parial differeial equaio. Thi mehod i combied form of he Laplace raform ad Variaioal ieraio mehod. The explici oluio obaied were compared o he exac oluio. The mehod fid he oluio wihou ay rericive aumpio ad free from roud-off error ad herefore reduce he umerical compuaio o a grea exe. The mehod i eed o wo example ad coupled Burger equaio. The reul how ha he ew mehod i more effecive ad coveie o ue ad high accuracy of i i evide. Keyword: Laplace raform, variaioal ieraio raform mehod, burger equaio, ad oliear parial differeial equaio. INTRODUCTION Burger equaio i a fudameal parial differeial equaio i fluid mechaic. I occur i variou area of applied mahemaic, uch a modellig of dyamic, hea coducio, ad acouic wave, i i amed for Johae Mari u Burger ( ). Localized i a mall par of pace, play a major role i everal differe field uch a hydrodyamic, plama phyic, oliear opic, ec. The iveigaio of exac oluio of hee oliear equaio i iereig ad impora. I he pa everal decade, may auhor had paid aeio o udy oluio of oliear equaio by uig variou mehod, uch a Backlad raformaio [1,7], Darboux raformaio [34], ivere caerig mehod [13], Hiroa biliear mehod [22], he ah mehod [27], he ie-coie mehod [40,41], he homogeeou balace mehod [35,42], he Riccai expaio mehod wih coa coefficie [43,44]. "Recely, a exeded ah-fucio mehod ad ymbolic compuaio are uggeed i [11] for olvig he ew coupled modified Burger equaio o obai four kid of olio oluio." Thi mehod ha ome meri i cora wih he ah-fucio mehod. I i o oly uig a impler algorihm o produce a algebraic yem, bu alo ca pick up igular oluio wih o exra effor [12,23,28,32,39]. Mo of he developed cheme have heir limiaio like limied covergece, diverge reul, liearizaio, dicreizaio, urealiic aumpio ad o compaibiliy wih he verailiy of phyical problem [8] i he Burger model of urbulece [4]. I i olved aalyically for arbirary iiial codiio [24]. Fiie eleme mehod have bee applied o fluid problem, Galerki ad Perov-Galerki fiie eleme mehod ivolvig a ime-depede grid [6,21]. Numerical oluio uig cubic plie global rial fucio were developed i [31] o obai wo yem or diagoally domia equaio which are olved o deermie he evoluio of he yem. A collocaio oluio wih cubic plie ierpolaio fucio ued o produce hree coupled e of equaio for he depede variable ad i wo fir derivaive [5]. Sice exac oluio of mo of he differeial equaio do o exi, approximaio ad umerical mehod are ued for he oluio of he FDE Ali e al.[3] applied B-plie fiie eleme mehod o he oluio of Burger equaio. The B-plie fiie eleme approach applied wih collocaio mehod over a coa grid of cubic B-plie eleme. Cubic B-plie had a reulig marix yem which i ri-diagoal ad o olved by he Thoma algorihm. Solima [33] ued he imilariy reducio o he parial differeial equaio from develop a cheme for olvig he Burger equaio. The coupled yem i derived by Eipov [10]. I i imple model of edimeaio or evoluio of caled volume coceraio of wo kid of paricle i fluid upeio or colloid, uder he effec of graviy [30]. The Variaioal ieraio mehod wa fir propoed by He [14 17] ad wa uccefully applied o auoomou ODE i [18], o oliear polycryallie olid [29], ad oher field. The combiaio of a Laplace raform mehod, Variaioal ieraio raform mehod, mehod of variaio of coa ad averagig mehod o eablih a approximae oluio of oe degree of freedom weakly oliear yem i [9]. The Variaioal ieraio raform mehod ha may meri ad ha much advaage over he Adomia mehod [37]. The aim of hi paper i o exed he Variaioal ieraio raform mehod propoed by He [14-17,19,20] o olve wo differe ype uch a he oe-dimeioal Burger equaio ad coupled Burger equaio ad made a compario wih he reul obaied by he Adomia decompoiio mehod [2, 25, 26, 36, 38]. Variaioal Ieraio Traform Mehod (VITM) To illurae he baic idea of hi mehod, we coider a geeral oliear o-homogeeou parial differeial equaio wih iiial codiio of he form Dux, + Rux, + ℵux, = gx, ux, = hx,, = Where D i he ecod order liear differeial operaor D = 2 2, Ri liear differeial operaor of le order 6926

2 ha D, ℵ repree he geeral oliear differeial operaor ad g(x,) i he ource erm. Takig Laplace Traform o boh ide of Eq.(1) Thi ca be wrie a Lux, ux, = L [ u u x] L [u x ] L[Dux, ] + L[Rux, ] + L[ℵux, ] = L[gx, ] L[ux, ],, + L[R, ] + L[ℵ, ] = L[, ] Takig Ivere Laplace raform, = + h + L [L(, )] L [L(R, )] L [Lℵ, ] Derivaive by boh ide of Eq.(4) u, = h + L ( L {, }) L ( L{R, }) L ( L{ℵ, }) By he correcio fucio of irraioal mehod u +, =, ξ, ξ + ξ L L{R, ξ} + L ( L{ℵ, ξ}) L ( L{, ξ}) hdξ Fially, he oluio u(x,) i give by ux, = lim, A illuraive paradigm I hi ecio, he (VITM) i implemeed for ackig Burger' equaio wih iiial codiio. We demorae he effecivee of hi mehod wih wo example. Numerical reul obaied by he propoed mehod are compared wih Kow reul. Example 1: Coider oe- dimeioal Burger' equaio of he form u = xx x The iiial codiio i we ge O applyig he above pecified iiial codiio L(ux, ) ( ) = L u u [ x] L [ x ] L[ux, ] = + L [ ] L [ ] Applyig he ivere Laplace raform o boh ide of Eq.(12), We ge ux, = L [ ] + L [ L ] L [ ] L [ L ] ux, = + L [ L ] L [ L ( )] Derivaive by boh ide (13) u, + [L ( L ( ))] [L ( L )] = Makig he correcio fucio i give u +, = u, ξ, ξ + ξ L L[, ξ, ξ] L L[ xx, ξ] dξ We ca ue he iiial codiio o elec u, =, = Uig hi elecio io he correcio fucioal give he followig ucceive approximaio u, = u, =, ξ, ξ + ξ L ( L[, ξ x, ξ]) ux, = + L L[ xx, ξ]dξ Takig Laplace raform o boh ide L[u ] = L[ xx ] L[ x ] 6927

3 u, = ( u, = u, = u, ξ, ξ + ξ L ( L [( ) ( )]) + L ( L [ ])) dξ + ξ (L L[, ξ x, ξ] + L ( L[ xx, ξ])) dξ u, = ( ) + ξ L L [( ξ ) ( ξ + ) ]) L ( L [ ξ ]) u, = ( ), = k k+ k= Fially, he oluio i, = lim, = k k+ k= = Example2: Coider oe dimeioal Burger' equaio of he form = xx x Subjec o he iiial codiio, = L[ ] = L[ xx ] L[ x ] Thi ca be wrie a L[, ], = L [ ] L [ ] we ge O applyig he above pecified iiial codiio L[ux, ] x = L [ ] L [ ] L[ux, ] = x + L [ ] L [ ] Applyig he ivere Laplace raform o boh ide of Eq.(27), we ge ux, = L [ ] + L [ L [ ]] L [ L [ ]] ux, = + L [ L [ ]] L [ L [ ]] Derivaive by boh ide (28) u x, = L [ L ] L [ L ] Or u x, L [ L ] L [ L ( )] = Makig he correcio fucio i give u + x, = u x, ( ξ, ξ ξ L [ L ] L [ L ( )]) dξ We ca ue he iiial codiio o elecu x, =, =. Uig hi elecio io he correcio fucioal give he followig ucceive approximaio. u x, = u x, = u x, ( ξ, ξ ξ L L [ ] L [ L ( )]) dξ = ξ L [ L] L [ L]dξ Takig Laplace raform o boh ide 6928

4 u x, = u x, (u x, ξ ξ L [ L u ] x L [ Lu u ]) dξ x = ξ L [ L] L [ L[ ]]dξ = +, =, ξ, ξ ξ L [ L ] = + + ξ L [ L ( )] dξ ξ L [ L] L [ L( ξ + ξ ξ + ξ ]) dξ Applyig he algorihm of Laplace raform o equaio (37), (38) we have L[ xx x + x ] = L[ xx x + x ] = L[, ], L[ xx ] L[ x ] + L[ x ] = L[, ], L[ xx ] L[ x ] + L[ x ] = Uig he give iiial codiio o equaio (42), (43), we have L[, ] i L[ xx ] L[ x ] + L[ x ] = L[, ] i L[ xx ] L[ x ] + L[ x ] = The applyig he ivere Laplace raform o equaio (44), (45), = L [ i] + L ( L [ xx ]) + L ( L [ x ]) L L[ x ], = L [ i] + L ( L [ xx ]) + L ( L [ x ]) L L[ x ], = +, = k k k= Take, = lim,, = k k, = k= + Derivaive by boh ide o equaio (46),(47), = L ( L[ xx ]) + L ( L[ x ]) L ( L[ x ]), = L ( L[ xx ]) + L ( L[ x ]) L ( L[ x ]) Example 3: Coupled Burger' equaio For he purpoe of illuraio of he Variaioal ieraio raform mehod for olvig he homogeeou form of coupled Burger' equaio, we will coider he yem of equaio. xx x + x =, xx x + x = The oluio of which are o be obaied ubjec o iiial codiio. Makig he correcio fucio are give +, =, + ( L ( L[ xx ]) + L ( L[ x ]) L ( L[ x ])) dξ, = i,, = i 6929

5 +, =, + ( L ( L[ xx ]) + L ( L[ x ]) L ( L[ x ])) dξ, =, + ( L ( L[ xx ]) + L ( L[ x ]) L ( L[ x ])) dξ We ca ue he iiial codiio o elec, =, =, =, = i u x, = u x, + ( L ( L[u xx ]) + L ( L[u u x ]), = i i + ( L ( L[ ix + ξi x]) + L ( L[ix cox ξ ix cox + ξ i xco x]) L ( L[ix cox ξ ix cox L ( L[u v x ])) dξ + ξ i xco x ])) dξ, = i + ( L ( L[ i]) + L ( L[i co]) L ( L[ i co])) dξ, = i ix v x, = v x,, = i + ( L ( L[v xx ]) + L ( L[v v x ]) L ( L[u v x ])) dξ + ( L ( L[ i]) + L ( L[i co]) L ( L[ i co])) dξ, = i ix, = ix ix +! i v x, = ix ix + ( L ( L[ ix + ξi x]) + L ( L[ix cox ξ ix cox + ξ i xco x]) L ( L[ix cox ξ ix cox + ξ i xco x ])) dξ v x, = ix ix +! ix u x, = ix ix +! ix ix +! + ix! v x, = ix ix +! ix ix +! + ix! u x, = ix k k= k 6930

6 v x, = ix k k= Fially, he oluio i ux, = lim u x, = ix k k= k u x, = ix e vx, = lim v x, = ix k k= k v x, = ix e k CONCLUSIONS I hi paper, he Variaioal ieraio raform mehod ha bee uccefully applied o fidig he oluio of a Burger ad coupled Burger equaio. The oluio obaied by he Variaioal ieraio raform mehod i a ifiie power erie for appropriae iiial codiio, which ca, i ur, be expreed i a cloed form, he exac oluio. The reul how ha he Variaioal ieraio mehod i a powerful mahemaical ool o olvig Burger ad coupled Burger equaio; i i alo a promiig mehod o olve oher oliear equaio. REFERENCES [1] M.J. Ablowiz, P.A. Clarko, Solio Noliear Evoluio Equaio ad Ivere Scaerig, Cambridge Uiveriy Pre, Cambridge. [2] G. Adomia Mah. Compu. Modellig. 22: 103. [3] A.H.A. Ali, G.A. Garder, L.R.T. Garder Compu. Mehod Appl. Mech. Eg. 100: [4] J. Burger i: Advace i Applied Mechaic, Academic Pre, New York. pp [5] J. Caldwell, E. Hio, e al (Ed.), Numerical Mehod for Noliear Problem, Pieridge, Swaea. 3: [6] J. Caldwell, P. Wale, A.E. Cook Appl. Mah. Modellig. 5: [7] A. Coely, e al (Ed.), Backlud ad Darboux Traformaio, America Mahemaical Sociey, Providece, RI. [8] J.D. Cole Quar. Appl. Mah. 9: [9] Gh.E. Dragaecu, V. Capalaa Iera. J. Noliear Sci. Numer. Simulaio. 4: [10] S.E. Eipov Phy. Rev. E. 52: [11] E. Fa Phy. Le. A 282 (2001) 18. [12] E.G. Fa, H.Q. Zhag Phy. Le. A. 246: 403. [13] C.S. Garder, J.M. Gree, M.D. Krukal, R.M. Miura Phy. Rev. Le. 19: [14] J.H. He Comm. Noliear Sci. Numer. Simulaio 2 (4) (1997) M.A. Abdou, A.A. Solima / Joural of Compuaioal ad Applied Mahemaic. 181: [15] J.H. He Compu. Mehod Appl. Mech. Eg. 167: [16] J.H. He Compu. Mehod Appl. Mech. Eg. 167: [17] J.H. He Iera. J. No-liear Mech. 34: [18] J.H. He Appl. Mah. Compu. 114(2,3): [19] J.H. He Approximae Aalyical Mehod i Sciece ad Egieerig, Hea Sci. & Tech. Pre, Zhegzhou, (i Chiee). [20] J.H. He Geeralized Variaioal Priciple i Fluid, Sciece & Culure Publihig Houe of Chia, Hog Kog, i Chiee). [21] B.M. Herb, S.W. Schoombie, A.R. Michell Iera. J. Numer. Mehod Eg. 18: [22] R. Hiroa Phy. Rev. Le. 27: [23] R. Hiroa, J. Sauma Phy. Le. A. 85: 407. [24] E. Hopf The parial differeial equaio, Comm. Pure Appl. Mah. 3: [25] D. Kaya Iera. J. Mah. Mah. Sci. 27: 675. [26] D.Kaya Appl. Mah. Compu. 144: [27] W. Malfei Amer. J. Phy. 60: 650. [28] W. Malflie Amer. J. Phy. 60:

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