A Comparative Study of Adomain Decompostion Method and He-Laplace Method

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1 Applied Mahemaic,, 5, 5-6 Publihed Olie December i SciRe. hp:// hp://d.doi.org/.6/am..5 A Comparaive Sudy of Adomai Decompoio Mehod ad He-Laplace Mehod Badradee A. A. Adam, Deparme of Mahemaic, Faculy of Educaio, Uiveriy of Kharoum, Omdurma, Suda ²Deparme of Mahemaic, Norhwe Normal Uiveriy, Lazhou, Chia bdr_uofk@yahoo.com, badradeeabaker@uofk.edu Received 8 Ocober ; revied 9 Ocober ; acceped 8 November Copyrigh by auhor ad Scieific Reearch Publihig Ic. Thi work i liceed uder he Creaive Commo Aribuio Ieraioal Licee (CC BY). hp://creaivecommo.org/licee/by/./ Abrac I hi paper, we pree a comparaive udy bewee he He-Laplace ad Adomai decompoiio mehod. The udy oulie he igifica feaure of wo mehod. We ue he wo mehod o olve he oliear Ordiary ad Parial differeial equaio. Laplace raformaio wih he homoopy mehod i called He-Laplace mehod. A compario i made amog Adomai decompoiio mehod ad He-Laplace. I i how ha, i He-Laplace mehod, he oliear erm of differeial equaio ca be eay hadled by he ue He polyomial ad provide beer reul. Keyword Adomai Decompoiio Mehod, He-Laplace Traform Mehod, Homoopy Perurbaio Mehod, Ordiary Differeial Equaio, Parial Differeial Equaio, He Polyomial. Iroducio Thi paper oulie a reliable Compario bewee wo powerful mehod ha were recely developed. The fir i Adomai decompoiio mehod (ADM) developed by Adomai i [] [], ad ued heavily i he lieraure i []-[] ad he referece herei. The ecod i He-Laplace mehod, a elega combiaio of he Laplace raformaio, he homoopy perurbaio mehod ad He polyomial. The ue of He polyomial i oliear erm wa fir iroduced by Ghorbai []. The propoed algorihm provide he oluio i a rapid coverge erie which may lead o he oluio i a cloed form. The wo mehod give rapidly coverge erie wih pecific igifica feaure for each cheme. Some of he claical aalyic mehod are lyapuov ar- How o cie hi paper: Adam, B.A.A. () A Comparaive Sudy of Adomai Decompoio Mehod ad He-Laplace Mehod. Applied Mahemaic, 5, 5-6. hp://d.doi.org/.6/am..5

2 ificial mall parameer mehod [] perurbaio echique [] [] ad Hiroa biliear mehod [5] [6]. I rece year, may auhor have paid aeio o udy he oluio of oliear parial differeial equaio by uig variou mehod. Variaioal ieraio mehod, He emi ivere mehod [7] ad he differeial raform mehod, ec. are amog hee. The mai objecive i o iroduce a comparaive udy o oliear ordiary differeial ad parial differeial equaio by uig adomai decompoiio mehod ad He-Laplace mehod. Thi paper coai baic idea of homoopy peraurbaio mehod ad He-Laplace mehod i Secio, Adomai decompoiio mehod i, Applicaio i ad cocluio ad dicuio i 5 repecively.. Baic Idea of Homoopy Perurbaio Mehod ad He-Laplace Mehod.. Homoopy Perurbaio Mehod Coider he followig oliear differeial equaio wih boudary codiio of A( y) f ( r) =, r Ω () y B y, =, r Γ where A, B, f ( r ) ad Γ are a geeral differeial operaor, a boudary operaor, a kow aalyic fucio ad he boudary of he domai Ω, repecively. The operaor A ca geerally be divided io a liear par L ad a oliear par M. Equaio () may herefore be wrie a: L( y) + M ( y) f ( r) = () or By he homoopy echique, we coruc a homoopy v( r, p) : [,] R H( vp) ( p) Lv L( y) p Av f( r) where [,] Ω which aifie:, = + = () H v, p = L v L y + pl y + pm v f r = (5) p i a embeddig parameer, while y i a iiial approimaio of Equaio (), which aifie he boudary codiio. Obviouly, from Equao () ad (5), we will have: H v, = L v L y = (6) The chagig proce of PP from zero o uiy i ju ha of (, ) called deformaio, while ad Av f( r) H v, = L v L y = (7) y o v r p from y r. I opology, hi i L v L y are called homoopy. If he embeddig parameer pi coidered a a mall parameer, applyig he claical perurbaio echique, we ca aume ha he oluio of Equaio () ad (5) ca be wrie a a power erie i p : v = v + pv + p v + p v + + (8) () Seig p = i Equaio (8), we have y = lim v = v + v + v + (9) p The combiaio of he perurbaio mehod ad he homoopy mehod i called he HPM, which elimiae he drawback of he radiioal perurbaio mehod while keepig all i advaage. The erie (9) i coverge for mo cae. However, he coverge rae deped o he oliear operaor Av. Moreover, He [8] made he followig uggeio: 5

3 ) The ecod derivaive of large, i.e. p. ) The orm of M L v M v wih repec o mu be mall becaue he parameer may be relaively mu be maller ha oe o ha he erie coverge... He-Laplace Mehod Coider he followig oliear differeial equaio (IVP): y + py + py+ p f y = f () y ( ) α, y = = β () where,,,, f i he ource erm. Takig Laplace raformaio (deoed hroughou hi paper by L ) o boh ide of Equaio (), we have p p p αβ are coa. f ( y ) i a oliear fucio ad [ ] + [ ] + [ ] + = L y L py L py L pf y L f () By uig lieariy of Laplace raformaio, he reul i [ ] + [ ] + [ ] + = L y Lp y Lp y Lp f y L f () Applyig he formula o Laplace raform, we obai { [ ] } [ ] L[ y] y y + p L L y y + p L y + p L f y = L f Uig iiial codiio i Equaio (), we have Or ( + p) L[ y] = α+ β + αp pl( y) pl f ( y) + L f ( ) ( α + β + α ) () (5) p p p L[ y] = L[ y] L f ( y) + L f ( ) p p p Takig ivere Laplace raform, we have (6) p p y = F L L[ y] L L f ( y) p p + + where F( ) repree he erm ariig from he ource erm ad he precribed iiial codiio. Now, we apply homoopy perurbaio mehod [], (7) where he erm y = p y (8) = y are o recurively calculaed ad he oliear erm f y ca be decompoed a for ome He polyomial f y = ph y (9) = H (ee [] [9]) ha are give by H y, y, y,, y = f py =,,,, i ( ) i! p i= P= Subiuig Equaio (8) ad (9) i (7), we ge 55

4 p p p p + p y = F p L L p y ( ) + L L p H ( y) () = = N= which i he couplig of he Laplace raformaio ad he homoopy perurbaio mehod uig He polyomial. Comparig he coefficie of like power of p, he followig approimaio are obaied: = :, p y F p p p : y( ) = L L y ( ) + L L H ( y) p p + + p p p : y( ) = L L y ( ) + L L H ( y) p p + + p p p : y( ) = L L y ( ) + L L H ( y) + p + p. A Domai Decompoiio Mehod A domai decompoiio mehod [] [] defie he ukow fucio u( ) by a ifiie erie u u = F u ca be de- where he compoe, compoed io a ifiie erie of polyomial give by where or equivalely () =, () u are uually deermied recurrely. The oliear operaor F u = A are he o-called Adomai polyomial of A = F u, A = uf u, = A () u, uu,, u defied by d i A = F( λ ui), =,,, ()! dλ λ = A = uf ( u) + uf ( u), A = uf ( u) + uu F ( u) + uf ( u), A = uf ( u) + uu + u F ( u) + uuf ( u) + uf u ( iv ) I i ow well kow ha hee polyomial ca be geeraed for all clae of oliear accordig o pecific algorihm defied by (). Recely, a aleraive algorihm for corucig Adomai polyomial ha bee developed by Wazwaz [6]. Thi powerful echique hadle boh liear ad oliear equaio i uified maer wihou ay eed for he o-called Adomai polyomial. However, Adomi decompoiio mehod provide he compoe of he eac oluio, where hee compoe hould follow he ummaio give i (), wherea ADM require he evaluaio of he Adomai polyomial ha moly require ediou algebraic work.. (5) 56

5 . Applicaio.. Eample Coider he followig oliear PDE []: wih he followig codiio: u u+ = y+ y = = ( + ) (, ) =, (, ) = + u, a, u, a, u y u y y a (6) (7)... Uig He-Laplace Mehod Equaio () ca be wrie a u u u + + = y+ y y (8) By applyig he Laplace raform o boh ide of Equaio () ubjec o he iiial codiio, we have The ivere of he Laplace raform implie ha ( ) ( ) Lu [ ] = L[ y] + L Lu u + (9) 6 u(, y) = y+ pl L u yy + L u y Now, we apply he homoopy perurbaio mehod, we have () where H 6 pu( y, ) = y+ pl L pu + L ph ( u) () = = = u are He polyomial. The fir few com-poe of He polyomial are give by H u = y = y H u = y y y y H u = y + y y = y y y Comparig he coefficie of like power of p, we have 6 u ( y, ) = y+ + α { yy } 6 p: u( y, ) = L L y L H( y) + = p : u( y, ) = L { L yyy L H( y) } + = p : u( y, ) = L { L y yy + L H( y) } = () 6 p : u y, = y+, bu we coider () 57

6 So ha he oluio uy (, ) i give by which i he eac oluio of he problem. 6 6 u (, y) = u + u+ u + u + = y + + a = y + a ()... Adomai Decompoiio Mehod We fir rewrie Equaio (6) i a operaor L i u u u + + = y+ y y yy y where he differeial operaor L, Lyy & L y are L u+ L u+ Lu = y+ L u= y+ L u Lu (5) yy y (,) =, (,) = ( + ) u a u a (, ), (, ) u y = u y = y+ a yy y (.) (., ) (.) (.) & (.) (.) L = L = L = (6) y y The ivere L are aumed a a iegral operaor give by (.) L. = dd, (7) Applig he ivere operaor L Subiuig () io he fucio Equaio (8) give Thi ca be rewrie a he form o boh ide of (5) ad uig iiial codiio we fid 6 u (, y) = y + + a L Lyyu + ( Lyu) (8) 6 u ( y, ) = y + + a L Lyy u ( y, ) + Ly u ( y, ) (9) = = = u + u + u + ( ( ) ( ( ) ) ) 6 = y+ + a L Lyy u + u + u + + Ly u + u + u + I view of (9), he followig recurive relaio follow immediaely. Coequely, we obai 6 u = y + + a u y, = L L u y, + Lu y,, k ( ) k + yy k y k () () 58

7 6 u = y + + a ( (, ) ( (, )) ) ( (, ) ( (, )) ) u= L L u y+ Lu y yy y u = L L u y+ Lu y yy y Accordig o Adomai [9], ad approimae oluio ca be obaied []. y+ 5 u (, y) = ( y + a) + y ( y ) + + ( ) he eac oluio i give by u ( y) = ( y + a),. () ().. Eample Coider he followig o-homogeeou oliear PDE []: wih he followig codiio: u u = () u(,) = (5)... Uig He-Laplace Mehod By applyig he Laplace raform mehod ubjec o he iiial codiio, we have (, ) y The ivere of he Laplace raform implie ha L = L u u(, ) = L L u Now, we apply he homoopy perurbaio mehod, we have where H (6) (7) pu(, ) = p L L pu( u) (8) = = u are He polyomial. The fir few compoe of He polyomial are give by H u = u = 8 H( u) = uu = 6 H( u) = u + uu = Comparig he coefficie of like power of p we have p : u, = 6 p : u(, ) = L { L H( u) } = p : u(, ) = L { L H( u) } = 5 5 (9) (5) 59

8 Proceedig i a imilar maer, we have So ha he oluio u(, ) i give by p : u, = (, ) = = + + (5) 5 5 u u u u u... Adomai Decompoiio Mehod We fir rewrie Equaio () i a operaor L i where he differeial operaor are defie a; Lu = Lu (5) u(,) = Ad he ivere operaor L, L ( ) = ( ), L ( ) = ( ) provided ha i ei, i defied a: L ( ) = ( ) d (5) Applig he ivere operaor o boh he ide of (5) ad uig he iiial codiio, yield: L Lu L L Lu = u = L Lu (5) (, ) Now, we decompoe he ukow fucio u(, ) a a um of compoe defied by he erie (): where Or (, ) (, ) u = u (55) = u i ideified a u( ;). The compoe u (, ) are obaied by he recurive formula: u(, ) = L L u(, ) (56) = =, u = (57) u ( ( )) k+, = L L uk,, k (58) We oe ha he recurive relaiohip i coruced o he bai ha he compoe u, i defied by all erm ha arie from he iiial codiio ad from iegraig he ource erm. The remaiig compoe uk (, ), ca be compleely deermied recurively. Accordigly, coiderig he fir few erm, Equaio () ad (5) give: 6

9 u = u = L ( L( u(, ) )) = L L( ) = 7 u = L ( L( u(, ) )) = 6 Fially, uig (55) we obai he oluio i erie form: Tha i:.. Eample (, ) u = u + u + u + u + (59) (, ) u 6 Coider he followig fir order oliear differeial equaio [9] Wih he followig codiio: 7 = + (6) + =, (6) y y y... Uig He-Laplace Mehod By applyig he aforeaid mehod ubjec o he iiial codiio, we have The ivere of Laplace raform implie ha Now we apply he homoopy perurbaio mehod, we have where H y = (6) L y( ) = L y (6) y( ) = L L y (6) py( ) = p L L ph( y) (65) = = y are He polyomial. The fir few compoe of He polyomial are give by H y = y = H y = yy= Comparig he coefficie of like power of p, we have H y = y + yy = : = p y p : y( ) = L { L H( y) } = p : y( ) = L { L H( y) } = p : y ( ) = L { L H ( y) } = (66) (67) 6

10 So ha he oluio y( ) i give by which i covergig o + i.e. eac oluio. y = y + y + y + y + = + + y (68)... Adomai Decompoiio Mehod We fir rewrie Equaio (6) i a operaor L i where he differeial operaor are defie a; Ad he ivere operaor L Ly = y (69) y = L( ) = ( ) provided ha i ei, i defied a L ( ) = ( ) (7) d (7) Applig he ivere operaor o boh he ide of (69) ad uig he iiial codiio yield: = ( ) L Ly L y y = L y (7) Now, we decompoe he ukow fucio y( ) a a um of compoe defied by he erie (): where Or u u = (7) = y i ideified a y. The compoe y k are obaied by he recurive formula: y( ) = L y (7) = = y = ( k ) yk+ = L y (75) We oe ha he recurive relaiohip i coruced o he bai ha he compoe y i defied by all erm ha arie from he iiial codiio ad from iegraig he ource erm. The remaiig compoe yk ( ), ca be compleely deermied recurively. Accordigly, coiderig he fir few erm, Equaio (7) ad (7) give: y = y = L y = L = y( ) = L ( y ) = L ( ) = 6 7 y( ) = L ( y) = L = 9 6 6

11 Fially, uig (55) we obai he oluio i erie form: Tha i: 5. Dicuio y( ) = y + y+ y + y + (76) 7 y( ) = 6 The mai goal of hi work i o coduc a comparaive udy bewee Adomai decompoiio mehod ad he He-Laplace mehod. The wo mehod are powerful ad efficie mehod ha boh give approimaio of higher accuracy ad cloed form oluio if eiig. A impora cocluio ca be made here. Adomai decompoiio mehod for olvig oliear ordiary ad parial differeial equaio, he ame problem are olved by He-Laplace mehod. Adomai decompoiio mehod provide he compoe of eac oluio, where hee compoe hould follow he ummaio give i (). However, He-Laplace i a elega combiaio of he Laplace raformaio, he homoopy perurbaio mehod ad He polyomial. Moreover, he ADM require he evaluaio of he Adomai polyomial ha moly require ediou algebraic calculaio. The ADM provide he oluio i ucceive compoe ha will be added o ge he erie oluio. Referece [] Adomia, G. (988) A Review of he Decompoiio Mehod i Applied Mahemaic. Joural of Mahemaical Aalyi ad Applicaio, 5, 5-5. hp://d.doi.org/.6/-7x(88)97-9 [] Adomia, G. (99) Solvig Froier Problem of Phyic: The Decompoiio Mehod. Kluwer Academic Publiher, Boo. [] Wazwaz, A.M. (997) Neceary Codiio for he Appearace of Noie Term i Decompoiio Soluio Serie. Applied Mahemaic ad Compuaio, 8, hp://d.doi.org/.6/s96-(95)7- [] Wazwaz, A.M. (997) A Fir Coure i Iegral Equaio. World Scieific, Sigapore. hp://d.doi.org/./ [5] Wazwaz, A.M. (999) Aalyical Approimaio ad Padé Approima for Volerra Populaio Model. Applied Mahemaic ad Compuaio,, -5. hp://d.doi.org/.6/s96-(98)8-6 [6] Wazwaz, A.M. () A New Techique for Calculaig Adomia Polyomial for Noliear Polyomial. Applied Mahemaic ad Compuaio,, -5. hp://d.doi.org/.6/s96-(99)6-6 [7] Wazwaz, A.M. () A New Algorihm for Calculaig Adomia Polyomial for Noliear Operaor. Applied Mahemaic ad Compuaio,, [8] Wazwaz, A.M. () The Decompoiio Mehod for Solvig he Diffuio Equaio Subjec o he Claificaio of Ma, Iera. Applied Mahemaic ad Compuaio,, 5-. [9] Wazwaz, A.M. () Parial Differeial Equaio: Mehod ad Applicaio. Balkema Publiher, The Neherlad. [] Wazwaz, A.M. () A New Mehod for Solvig igular Iiial Value Problem i he Secod Order Differeial Equaio. Applied Mahemaic ad Compuaio, 8, hp://d.doi.org/.6/s96-()- [] Ghorbai, A. (9) Beyod Adomia Polyomial: He Polyomial. Chao, Solio Fracal, 9, hp://d.doi.org/.6/j.chao.7.6. [] Lyapuov, A.M. (99) The Geeral Problem of he Sabiliy of Moio. Taylor & Fraci, Lodo. [] Saberi-Nadjafi, J. ad Ghorbai, A. (9) He Homoopy Per-urbaio Mehod: A Effecive Tool for Solvig Noliear Iegral ad Iegro-Differeial Equaio. Compuer ad Mahemaic wih Applicaio, 58, 5-5. [] Sweilam, N.H. ad Khadar, M.M. (9) Eac Soluio of Some Coupled Noliear Parial Differeial Equaio Uig he Homoopy Perurbaio Mehod. Compuer ad Mahemaic wih Applicaio, 58, -. hp://d.doi.org/.6/j.camwa [5] Hiroa, R. (97) Eac Soluio of he Koreweg-de Vrie Equaio for Muliple Colliio of Solio. Phyic Review Leer, 7,

12 [6] Wazwaz, A.M. () O Muliple Solio Soluio for Coupled KdV-mkdV Equaio. Noliear Sciece Leer A,, [7] Wu, G.C. ad He, J.H. () Fracioal Calculu of Variaio i Fracal Space Time. Noliear Sciece Leer A,, [8] He, J.H. () A Simple Perurbaio Approach o Blaiu Equaio. Applied Mahemaic ad Compuaio,, 7-. hp://d.doi.org/.6/s96-()89- [9] Liu, G.L. (995) Weighed Reidual Decompoiio Mehod i Noliear Applied Mahemaic. Proceedig of 6h Cogre of Moder Mahemaic ad Mechaic, Suzhou, 995, [] He, J.H. (997) A New Approach o Noliear Parial Differeial Equaio. Commuicaio i Noliear ad Numerical Simulaio,, -5. hp://d.doi.org/.6/s7-57(97)97-6

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