Discrete Adomian Decomposition Method for. Solving Burger s-huxley Equation

Transcription

1 It. J. Cotemp. Math. Sceces, Vol. 8, 03, o. 3, HIKARI Ltd, Dscrete Adoma Decomposto Method for Solvg Brger s-hxley Eqato Abdlghafor M. Al-Rozbaya Departmet of Mathematcs, College of Compter Sceces ad Mathematcs Uversty of Mosl, Mosl, Iraq Mohammed O. Al-Amr Departmet of Mathematcs, College of Compter Sceces ad Mathematcs Uversty of Mosl, Mosl, Iraq Copyrght 03 Abdlghafor M. Al-Rozbaya ad Mohammed O. Al-Amr. Ths s a ope access artcle dstrbted der the Creatve Commos Attrbto Lcese, whch permts restrcted se, dstrbto, ad reprodcto ay medm, provded the orgal wor s properly cted. Abstract I ths paper, the dscrete Adoma decomposto method (DADM) s appled to a flly mplct scheme of the geeralzed Brger s Hxley eqato. The mercal reslts of two test problems are compared wth the exact soltos. The comparsos reveal that the proposed method s very accrate ad effectve for ths d of problems. Keywords: Dscrete Adoma decomposto method, Fte dfferece scheme, Geeralzed Brger s Hxley eqato Itrodcto Nolear partal dfferetal eqatos (NLPDEs) are ecotered varos felds of scece. Geeralzed Brger s Hxley eqato (GBH) beg oe of the most famos NLPDE s of hgh mportace for descrbg the teracto betwee reacto mechasms, covecto effects, ad dffso trasports. Sce there

2 64 Abdlghafor M. Al-Rozbaya ad Mohammed O. Al-Amr exsts o geeral techqe for fdg aalytcal soltos of olear dffso eqatos so far, mercal soltos of olear dfferetal eqatos are of great mportace physcal problems. The geeralzed Brger s Hxley eqato vestgated by Satsma [6] 987 s of the form ( )( ), a x b, t 0 () t x x Sbject to the tal codto ( x,0) f ( x ), a x b () ad the bodary codtos ( a, t ) f ( t ), ( b, t ) f ( t ), t 0 (3) where,, ad are parameters, 0, 0, (0,). The GBH eqato has receved a great deal of atteto by a wde varety of researchers. I Wag et al. [7] the soltary wave soltos of the GBH eqato s stded. The o-classcal symmetres ad the sglar modfed soltos of the Brger ad Brger s Hxley eqato by Estevez [7]. Recetly, varos powerfl methods have bee appled to solve the GBH eqato sch as spectral methods [6,], Adoma decomposto method [9,0], varatoal terato method [], homotopy aalyss method [3], dfferetal trasform method [], dfferetal qadratre method [4], fte dfferece methods [3,,5], Exp-fcto method [8], Haar wavelet method [5] ad may others. The dscrete verso of Adoma decomposto method was frst proposed by Bratsos et al. [4] appled to dscrete olear Schrödger eqatos. Zh et al. [8] have developed the DADM to D Brgers dfferece eqatos. I ths stdy, the DADM s mplemeted to olear dfferece scheme of geeralzed Brger s Hxley eqato. The obtaed reslts of two test problems are compared wth the exact soltos to verfy the effcecy ad accracy of the proposed method. The paper layot s as follow: Secto deals wth the applcato of the DADM to olear dfferece scheme of the GBH eqato. I Secto 3, we preset two test examples of the GBH eqato wth mercal llstratos. Secto 4 cocldes the paper. Dscrete Adoma decomposto method To apply the DADM to Eq. () wth tal codto (), we formlate the followg flly mplct scheme: h h (4)

3 Dscrete Adoma decomposto method 65 We deote the dscrete approxmato of ( x, t ) at the grd pot ( h, ) by ( 0,,,, N ; 0,,, ), where h N s the spatal step sze ad represets tme cremet. Cosder the above scheme wrtte a operator form as D D h D h (5) wth the tal codto 0 f. (6) The lear operator D deote the forward dfferece approxmato,.e., D (7) The frst ad secod order cetral dfferece approxmatos, deoted by ad D h, respectvely, are gve by Dh, Dh. (8) h h The verse dscrete operator ( D ) s gve by [4] m ( D ) w w, m 0 (9) Usg the above defto, we get 0 ( D ) D. (0) Applyg the verse operator ( D ) to Eq.(5) yelds ( D ) M ( D ) D ( D ) N 0 h ( D ) ( D ) P Followg the DADM, the dscrete approxmato sm of compoets defed by the decomposto seres 0, () D h ca be decomposed to a () The olear operators M, N ad P are related to the olear terms ad ca be decomposed by the fte seres of the so-called Adoma polyomals as follows M D A, N B, h 0 0 P C 0 (3)

4 66 Abdlghafor M. Al-Rozbaya ad Mohammed O. Al-Amr Where A, B ad C are the so-called Adoma polyomals that ca be geerated accordg to the followg algorthms d A M,, 0! d 0 0 d B N,, 0! d 0 (4) 0 d C P,, 0.! d 0 0 Sbstttg () ad (3) to () yelds, f ( D ) A ( D ) D, ( D ) B h ( D ), ( D ) C 0 0 (5) Each term of seres () s gve by the recrrece relato f,0 ( D ) A ( D ) D ( D ) B ( D ), h,, ( D ) C, 0 So, the practcal solto for the -term approxmato s (6),,, (7) 0 ad the exact solto s lm. (8) 0, The frst three compoets of Adoma polyomals A, B ad C read h h h A D 0,0,0 A D D,0,,0,0,,0 h,,0, h,,0 h,0, A D D D ( ) D,0, h,0

5 Dscrete Adoma decomposto method 67 B 0,0 ( ),0, B B ( ) ( ) C,0,,,0 0,0 C ( ),0, C ( ) ( ),0,,,0 3 Nmercal expermets I ths secto, we wll gve two test examples of the GBH eqato to verfy the effcecy ad measre the accracy of the DADM soltos comparso wth the exact solto ad we wll se Maple 6 software to obta the mercal reslts. Cosder the GBH eqato [9]: ( )( ), t x x 0 x, t 0 (9) wth the tal codto ( x,0) tah( x ), ad the bodary codtos ( )( ) (0, t ) tah t, ( ) ( )( ) (, t ) tah t, ( ) where ( ) 4( ) ad ad the exact solto s gve by 4 ( ) ( )( ) ( x, t ) tah x t. ( ) (0) () () (3)

6 68 Abdlghafor M. Al-Rozbaya ad Mohammed O. Al-Amr 4 Example. We cosder Eq. (9) wth 0.00,, 0 ad h 0. The absolte errors for varos vales of,t ad x are gve Table. We plot Fg. the absolte error for the vales 0.00,, 0 ad h 0.. Table : The absolte errors sg 3 for varos vales of,t ad x wth 0.00, t x E E E E E E E E E E E E E E 06.7E E E E E E E E E E E E 05.75E E E E 05.74E E E E E E 04 4 Fg. The absolte error of 3 wth 0.00,, 4 0 ad h 0.

7 Dscrete Adoma decomposto method 69 4 Example. We cosder Eq. (9) wth 0., 0.00, 0 ad h 0.. The absolte errors for varos vales of,t ad x are gve Table. We plot the absolte error Fg. for the vales 0., h ,, 0 ad 0. Table : The absolte errors sg 3 for varos vales of,t ad x wth 0., 0.00 t x E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E 05 Fg. The absolte error of 3 wth 0., h ,, 0 ad

8 630 Abdlghafor M. Al-Rozbaya ad Mohammed O. Al-Amr 4 Coclsos I ths paper, the DADM s proposed for the geeralzed Brger s Hxley eqato. The obtaed reslts, reported Tables ad, reveal that the proposed method s very relable, accrate ad smple tool to solve the GBH eqato ad ca be appled to other olear problems. Refereces [] B. Batha, M.S.M. Noora ad I. Hashm, Applcato of varatoal terato method to the geeralzed Brgers Hxley eqato, Chaos, Soltos ad Fractals, 36 (008), [] J. Bazar ad F. Mohammad, Applcato of dfferetal trasform method to the geeralzed Brgers Hxley eqato, Appl. Appl. Math., 5 (00), [3] A. G. Bratsos, A Forth order mproved mercal scheme for the geeralzed Brgers Hxley eqato, Amer. J. Compt. Math., (0), [4] A. Bratsos, M. Ehrhardt ad I.Th. Famels, A dscrete Adoma decomposto method for dscrete olear Schrödger eqatos, Appl. Math. Compt., 97 (008), [5] I. Çel, Haar wavelet method for solvg geeralzed Brgers Hxley eqato, Arab J. Math. Sc., 8 (0), [6] M. El Kady, S.M. El Sayed ad H.E. Fathy, Developmet of Galer method for solvg the geeralzed Brger s Hxley eqato, Math. Probl. Eg., 03 (03), Artcle ID 6549, 9 pages, do: 0.55/03/6549. [7] P.G. Estevez, No classcal symmetres ad the sglar modfed the Brger s ad Brgers Hxley eqato, J. Phys. A, 7 (994), 3 7. [8] H. Gao ad R. Zhao, New exact soltos to the geeralzed Brgers Hxley eqato, Appl. Math. Compt., 7 (00), [9] I. Hashm, M.S.M. Noora ad M.R.S. Al Hadd, Solvg the geeralzed Brgers Hxley eqato sg the Adoma decomposto method, Math. Compt. Mod., 43 (006),

9 Dscrete Adoma decomposto method 63 [0] H.N.A. Ismal, K. Rasla ad A.A. Abd Rabboh, Adoma decomposto method for Brgers Hxley ad Brgers Fsher eqato, Appl. Math. Compt., 59 (004), [] M. Javd, A mercal solto of the geeralzed Brger s Hxley eqato by spectral collocato method, Appl. Math. Compt., 78 (006), [] S.A. Maaa ad F.M. Saleem, Nmercal solto ad stablty aalyss for Brgers Hxley eqato, Al Rafda J. Compt. Sc. Math., 6 (009), [3] A. Molabahram ad F. Kha, the homotopy aalyss method to solve the Brgers Hxley eqato, Nolear Aal. Real World Appl., 0 (009), [4] M. Sar ad G. Gürarsla, Nmercal soltos of the geeralzed Brgers Hxley eqato by a dfferetal qadratre method, Math. Probl. Eg., 009 (009), Artcle ID , pages, do:0.55/009/ [5] M. Sar, G. Gürarsla ad A. Zeytoğl, Hgh order fte dfferece schemes for mercal soltos of the geeralzed Brgers Hxley eqato, Nmer. Methods Partal Dfferetal Eq., 7 (0), [6] J. Satsma, Topcs Solto Theory ad Exactly Solvable Nolear Eqatos, World Scetfc, Sgapore, 987. [7] X.Y. Wag, Z.S. Zh ad Y.K. L, Soltary wave soltos of the geeralzed Brgers Hxley eqato, J. Phys. A, 3 (990), [8] H. Zh, H. Sh ad M. Dg, Nmercal soltos of two-dmesoal Brgers' eqatos by dscrete Adoma decomposto method, Compt. Math. Appl., 60 (00), Receved: May 8, 03