A Comparative Study of Variational Iteration Method and He-Laplace Method

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1 Applied Mahemaic,,, 9- hp://d.doi.org/./am..7 Publihed Olie Ocober (hp:// A Comparaive Sud of Variaioal Ieraio Mehod ad He-Laplace Mehod Hradeh Kumar Mihra Deparme of Mahemaic, Japee Uiveri of Egieerig & Techolog, Gua, Idia hk.mihra@jue.ac.i Received Augu, ; revied Sepember, ; acceped Sepember, ABSTRACT I hi paper, variaioal ieraio mehod ad He-Laplace mehod are ued o olve he oliear ordiar ad parial differeial equaio. Laplace raformaio wih he homoop perurbaio mehod i called He-Laplace mehod. A compario i made amog variaioal ieraio mehod ad He-Laplace. I i how ha, i He-Laplace mehod, he oliear erm of differeial equaio ca be eail hadled b he ue of He polomial ad provide beer reul. Keword: Variaioal Ieraio Mehod; He-Laplace Traform Mehod; Homoop Perurbaio Mehod; Ordiar Differeial Equaio; Parial Differeial Equaio; He Polomial. Iroducio Nolieari ei everwhere ad aure i oliear i geeral. The earch for a beer ad ea o ue ool for he oluio of oliear equaio ha illumiae he oliear pheomea of real life problem of ciece ad egieerig ha recel received a coiuig iere. Variou mehod, herefore, were propoed o fid approimae oluio of oliear equaio. Some of he claical aalic mehod are Lapuov arificial mall parameer mehod [], perurbaio echique [,], -epaio mehod [] ad Hiroa biliear mehod [5-]. I rece ear, ma auhor paid aeio o ud he oluio of oliear parial diffeeial equaio b uig variou mehod. Amog hee are he Adomia decompoiio mehod (ADM) [7], He Semiivere mehod [8], he ah mehod, homoop perurbaio mehod(hpm), Sih-Coh mehod, he differeial raform mehod ad he variaioal ieraio mehod (VIM) [9-7]. Several echique icludig he Adomia decompoiio mehod, he variaioal ieraio mehod, he weighed fiie differece echique ad he Laplace decompoiio mehod bee ued o olve oliear differeial equaio [8-]. J. H. He developed he homoop perurbaio mehod (HPM) [7-] b mergig he adard homoop ad perurbaio for olvig variou phical problem. The Laplace raform i oall icapable of hadlig oliear equaio becaue of he difficulie ha are caued b he oliear erm. Variou wa bee propoed recel o deal wih hee olieariie uch a he Adomia decompoiio mehod [] ad he Laplace decompoiio algorihm [-8]. Furhermore, he homoop perurbaio mehod i alo combied wih he well-kow Laplace raformaio mehod [9] which i kow a He-Laplace mehod. I hi paper, he mai objecive i o iroduce a comparaive ud o oliear ordiar differeial equaio ad parial differeial equaio b uig variaioal ieraio mehod ad He-Laplace mehod. I i worh meioig ha He-Laplace mehod i a elega combiaio of he Laplace raformaio, he homoop perurbaio mehod ad He polomial. The ue of He polomial i he oliear erm wa fir iroduced b Ghorbai [5]. The propoed algorihm provide he oluio i a rapid coverge erie which ma lead o he oluio i a cloed form. Thi paper coai baic idea of homoop perurbaio mehod i Secio, variaioal ieraio mehod i Secio, Laplace homoop perurbaio mehod i Secio ad cocluio i Secio 5 repecivel.. Baic Idea of Homoop Perurbaio Mehod ad He-Laplace Mehod.. Homoop Perurbaio Mehod Coider he followig oliear differeial equaio A f r, r () wih he boudar codiio of B,, r () Coprigh SciRe.

2 9 H. K. MISHRA where A, B, f r ad are a geeral differeial operaor, a boudar operaor, a kow aalic fucio ad he boudar of he domai, repecivel. The operaor A ca geerall be divided io a liear par L ad a oliear par N. Equaio () ma herefore be wrie a: LN f r () B he homoop echique, we coruc a homoop v r, p :, R which aifie: or H v, p plvl pav f r where,, pl pnv f r () H v p L v L (5) p i a embeddig parameer, while i a iiial approimaio of Equaio (), which aifie he boudar codiio. Obvioul, from Equao () ad (5), we will : H v, L v L () H v, A v f r (7) The chagig proce of p from zero o ui i ju ha of v(r, p) from o (r). I opolog, hi i called deformaio, while Lv L ad Av f r are called homoop. If he embeddig parameer p i coidered a a mall parameer, applig he claical perurbaio echique, we ca aume ha he oluio of Equaio () ad (5) ca be wrie a a power erie i p : Seig p v v pv p v p v (8) i Equaio (8), we lim v v v v (9) p The combiaio of he perurbaio mehod ad he homoop 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 [5] made he followig uggeio: ) The ecod derivaive of Nv wih repec o v mu be mall becaue he parameer ma be relaivel large, i.e. p. N ) The orm of L mu be maller ha oe v o ha he erie coverge... He-Laplace Mehod Coider he followig oliear differeial equaio (IVP): p p p f f (), () where p, p, p,, are coa. f() i a oliear fucio ad f() i he ource erm. Takig Laplace raformaio (deoed hroughou hi paper b L) o boh ide of Equaio (), we L f L L p L p Lp f () B uig lieari of Laplace raformaio, he reul i L pl p L p Lf () Lf or Applig he formula o Laplace raform, we obai () () L p L p L p Lf Lf Uig iiial codiio i Equaio (), we p L p p L pl f L f L p p p p L p Lf Lf p Takig ivere Laplace raform, we p FL L p p L Lf p () (5) () (7) F repree he erm ariig from he ource where erm ad he precribed iiial codiio. Now, we appl homoop perurbaio mehod [5], where he erm oliear erm p (8) are o recurivel calculaed ad he ca be decompoed a f f p H (9) Coprigh SciRe.

3 H. K. MISHRA 95 for ome He polomial give b H,,,, i i i p H (ee [5,5]) ha are f p,,,! p Subiuig Equaio (8) ad (9) i (7), we ge p F p p L L p ( ) ( p) p L L p H ( ) ( p) () which i he couplig of he Laplace raformaio ad he homoop perurbaio mehod uig He polomial. Comparig he coefficie of like power of p, he followig approimaio are obaied: p : F, p p : L L ( ) ( p) p L L H ( ) ( p) p p : L L () p p L LH p p p : L L ( p) L p LH ( ) ( p ) Eample.. Coider he followig oliear PDE [5]: u u wih he followig codiio: u, a, u, a, u,, u, a. Equaio () ca be wrie a u u u () () () B applig he Laplace raform o boh ide of Equaio () ubjec o he iiial codiio, we Lu LL Lu u (5) The ivere of he Laplace raform implie ha u, L Lu Lu () Now, we appl he homoop perurbaio mehod, we pu, (7) pl L p u L p Hu where H u are He polomial. The fir few compoe of He polomial are give b Hu Hu (8) H u Comparig he coefficie of like power of p, we p : u,, bu we coider u, a p : u, L L L H p : u, (9) L L L H p : u, L L LH So ha he oluio u, i give b u, u uu u a () a which i he eac oluio of he problem. Eample.. Coider he followig o-homogeeou Coprigh SciRe.

4 9 H. K. MISHRA oliear PDE [5]: wih he followig codiio: u u () u, () B applig he Laplace raform mehod ubjec o he iiial codiio, we, L Lu The ivere of he Laplace raform implie ha u, L Lu () () Now, we appl he homoop perurbaio mehod, we pu, (5) p L L p H ( u) where H u are He polomial. The fir few compoe of He polomial are give b H u u 8 Hu uu () Huu uu 9 5 Comparig he coefficie of like power of p, we p : u, p : u, L LHu (7) 5 p : u, L LHu 5 Proceedig i a imilar maer, we p : u, So ha he oluio u,, i give b u u u u u (8). Variaioal Ieraio Mehod (VIM) To illurae he baic cocep of he echique, we coider he followig geeral differeial equaio LuNu g where L i a liear operaor, N i a oliear operaor ad g() i he forcig erm. Accordig o VIM, we ca coruc a correc fucioal a follow u u Lu Nu g d where i a Lagrage muliplier. The ubcrip deoe he h approimaio, u i coidered a a rericed variaio i.e. u. I hi mehod, i i required fir o deermie he Lagrage muliplier opimall. The ucceive approimaio u, of he oluio u will be readil obaied upo uig he deermied Lagrage muliplier ad a elecive fucio u, coequel, he oluio i give b u lim u. Now, we coider he followig eample: Eample.. Coider he followig fir order oliear differeial equaio [5]:, (9) () If i a iiial approimaio or rial-fucio he we ca wrie dow followig epreio for correcio: d () where he la erm of righ i called correcio, i a geeral Lagrage muliplier. The above fucioal i called correcio fucioal, he Lagrage muliplier i he fucioal hould be uch choe ha i correcio oluio i uperior o i iiial approimaio (rialfucio) ad i he be wihi he fleibili of he rialfucio, accordigl we ca ideified he muliplier b variaioal heor [5,55]. Makig he above correcio fucioal aioar wih () = o ha, we ca obai followig aioar codiio: () () The Lagrage muliplier, herefore, ca be ideified a follow: ep d () To implif he muliplier, we approimae Equaio () a follow: Coprigh SciRe.

5 H. K. MISHRA 97 (5) ep d Subiuig Equaio (5) i Equaio () ield followig variaioal ieraio formula ep d d () We ar wih b above ieraio formula, we ca obai followig reul, ( ) d e e e e e e e e e 8 e e e 8 e e e 8 ( ) e e e e ( ) d d (7) (8) if, uppoe, i ufficie, he approimaio a =. i..78, while i eac oe i (.) =.7, he.7% accurac i remarkabl good i view of he crudee of i iiial approimaio. The proce ca, i priciple, be coiued a far a deired, however, he reulig iegral quickl become ver cumberome, o ome implificaio i he proce of ideificaio of Lagrage muliplier will be dicued a below: We re-coider he correcio fucioal Equaio () a follow: d (9) Where he oliear erm i coidered a ovariaioal variaio or coraied variaio [5], i.e.. The Lagrage muliplier, herefore, ca be readil ideified ad he followig variaio ieraio formula ca be obaied: d (5) Puig =,, i Equaio (5), we ca obai followig reul. d d Similarl puig =,,,, he h approimaio ca be obaied, which coverge o i eac oluio, a lile more lowl due o he approimae ideificaio of he Lagrage muliplier. Remark The variaioal ieraio echique meioed above ca be readil eeded o parial differeial equaio (PDE). Here he auhor will illurae i proce. Eample.. Coider he followig equaio [5]:,,,,,, u u u u a u a u a (5) which ha he eac oluio u a. Accordig o Adomia [5], a approimae oluio ca be obaied [57]. u a 5 (5) I i obviou ha he approimaio doe o aif i boudar codiio. I 995, Liu [57] propoe a modified Adomia mehod called weighed reidual decompoiio mehod, wih uch mehod, he obaied followig approimaio: u a (5) which aifie all i boudar codiio ad ha more higher accurac ha Adomia. I 978, Iokui e al. [55] propoed a geeral Lagrage muliplier mehod o olve oliear mahemaical phic which wa fir applied o quaum mechaic. I hi mehod, a more accurae oluio, depedig upo i rial-fucio ca be obaied for ome pecial poi, bu o a approimae aalical oe. J. H. He [5] rie o olve i b variaioal ieraio mehod a follow: Suppoig he iiial approimaio of Equaio (5) i u, i correcio variaioal fucioal i -direcio ad -direcio ca be epreed repecivel a follow: u, u, u, uˆ, ˆ u, d (5) Coprigh SciRe.

6 98 H. K. MISHRA,, u u, ˆ, u, uˆ u d (55) where u ˆ i a ovariaioal variaio. Their aioar codiio are wrie dow repecivel a follow ad,, (5),, (57) The Lagrage muliplier ca be eail ideified:, (58) The ieraio formulae i -direcio ad -direcio ca be, herefore, epreed repecivel a follow u,,, u u,, u u d u u u,,,,, u u d (59) () To eure he approimaio aif he boudar codiio a = ad =, we modif he variaioal ieraio formulae i -direcio ad -direcio a follow: u u u,,,,, u u d u u u,,,,, u u d () () Now we ar wih a arbirar iiial approimaio: u AB, where A ad B are coa o be deermied, b he variaioal ieraio formula i -direcio (59), we, u A B d A B () B impoig he boudar codiio a = ad = ield A = ad B = a /, hu we 5 u, a () B (), we u a 5, d (5) a which i a eac oluio. The approimaio ca alo be obaied b -direcio. Eample.. Coider he followig oliear PDE [5]: u u u, () I -direcio correcio fucioal ca be coruced a u, u,, ˆ, u u d (7) I which u ˆ i ovariaioal variaio. The muliplier ca be ideified ad i variaioal ieraio formula -direcio ca be obaied,, u u,, u u d (8) We ar wih iiial approimaio u, b above ieraio formula, we ca obai ucceivel i approimaio:, d, u u, d, Coprigh SciRe.

7 H. K. MISHRA 99 u, d d which i he ame a Adomia [5,58].. Compariio of Variaioal Ieraioal Mehod ad He-Laplace Mehod Eample.. Coider he followig fir order oli- ear differeial equaio [5]: wih he followig codiio:,. (9) (7) B applig he aforeaid mehod ubjec o he iiial codiio, we L L The ivere of he Laplace raform implie ha L L (7) (7) Now, we appl he homoop perurbaio mehod, we p p L L p H (7) where H are He polomial. The fir few compoe of He polomial are give b H H H (7) Comparig he coefficie of like power of p, we p : p : L LH p : L LH p : L LH (75) Table. Numerical reul of Eample.. app. () He-Laplace mehod app. () VIM () eac Relaive error of He-Laplace Mehod Relaive error of VIM E 8.7E E 5.8E E.8E E.8E E 9.E E.599E E. 58E E.E So ha he oluio i give b app. (7) whic h i covergig o i.e. eac oluio. The compuaioal reul are preeed i Table. 5. Cocluio I hi paper, variaioal ieraio mehod i emploed for olvig oliear ordiar ad parial differeial equaio. The ame problem are olved b He-Laplace mehod. I i worh meioig ha he He-Laplace mehod i capable of reducig he volume of he compuaioal work a compared o he variaioal ieraio mehod while ill maiaiig he high accurac of he umerical reul. REFERENCES [] A. M. Lapuov, The Geeral Problem of he Sabili of Moio, Talor & Fraci, Lodo, 99 [] J. Saberi-Nadjafi ad A. Ghorbai, He Homoop Perurbaio Mehod: A Effecive Tool for Solvig Noliear Iegral ad Iegro-Differeial Equaio, Compuer ad Mahemaic wih Applicaio, Vol. 58, No. -, 9, pp doi:./j.c amwa.9.. [] N. H. Sweilam ad M. M. Khadar, Eac Soluio of Some Coupled Noliear Parial Differeial Equaio Uig he Homoop Perurbaio Mehod, Compuer ad Mahemaic wih Applicaio, Vol. 58, No. -, 9, pp. -. doi:./j.camwa [] A. V. Karmihi, A. I. Zhukov ad V. G. Koloov, Mehod of Damic Calculaio ad Teig for Thi- Walled Srucure, Mahioroeie, Mocow, 99. [5] R. Hiroa, Eac Soluio of he Koreweg-deVrie Equaio for Muliple Colliio of Solio, Phic Review Leer, Vol. 7, No. 8, 97, pp Coprigh SciRe.

8 H. K. MISHRA doi:./phrevle.7.9 [] A. M. Wazwaz, O Muliple Solio Soluio for Coupled KdV-mkdV Equaio, Noliear Sciece Leer A, Vol.,, pp [7] G. Adomia, Solvig Froier Problem of Phic: The Decompoiio Mehod, Kluwer Academic Publicaio, Boo, 99. [8] G. C. Wu ad J. H. He, Fracioal Calculu of Variaio i Fracal Space Time, Noliear Sciece Leer A, Vol.,, pp doi:.55/ijnsns...s.8 [9] M. A. Abdou ad A. A. Solima, New Applicaio of Variaioal Ieraio Mehod, Phica D: Noliear Pheomea, Vol., No. -, 5, pp. -8. doi:./j.phd.5.8. [] C. Chu, Fourier-Serie Baed Variaioal Ieraiomehod for a Reliable Treame of Hea Equaio wih Variable Coefficie, Ieraioal Joural of Noliear Sciece ad Numerical Simulaio, Vol., 9, pp doi:.55/ijnsns [] N. Faraz, Y. Kha ad A. Yildirim, Aalical Approach o Two-Dimeioal Vicou Flow wih a Shrikig Shee via Variaioal Ieraio Algorihm-II, Joural of Kig Saud Uiveri, Vol., No.,, pp. -. doi:./j.jku... [] J. H. He, Variaioal Ieraio Mehod A Kid of Noliear Aalical Techique: Some Eample, Ieraioal Joural of Noliear Mechaic, Vol., No., 999, pp doi:./s-7(98)8- [] J. H. He ad X. H. Wu, Variaioal Ieraio Mehod: New Developme ad Applicaio, Compuer ad Mahemaic wih Applicaio, Vol. 5, 7, pp doi:./j.camwa...8 [] J. H. He, G. C. Wu ad F. Aui, The Variaioal Iera- io Mehod Which Should Be Followed, Noliear Sci- Sciece ad ece Leer A, Vol., No., 9, pp. -. [5] E. Heameddii ad H. Laifizadeh, Recorucio of Variaioal Ieraio Algorihm Uig he Laplace Traform, Ieraioal Joural of Noliear Numerical Simulaio, Vol., 9, pp doi:.55/ijnsns [] L. A. Solai ad A. Shirzadi, A New Modificaio of he Variaioal Ieraio Mehod, Compuer ad Mahemaic wih Applicaio, Vol. 59, No.,, pp doi:./j.camwa... [7] G. C. Wu ad E. W. M. Lee, Fracioal Variaioal Ieraio Mehod, ad I Applicaio, Phic Leer A, Vol. 7, No. 5,, pp doi:./j.phlea... [8] G. Adomia, Soluio of Phical Problem b Decompoiio, Compuer ad Mahemaic wih Applicaio, Vol., No. 9-, 99, pp doi:./898-(9)9-5 [9] D. Bahugua, A. Ujlaa ad D. N. Pade, Acomparaive Sud of Numerical Mehod for Solvig a Iegro- Differeial Equaio, Compuer ad Mahemaic wih Applicaio, Vol. 57, 9, pp doi:./j.camwa [] M. Dehgha, Weighed Fiie Differece Techique for he Oe Dimeioal Advecio-Diffuio Equaio, Applied Mahemaic ad Compuaio, Vol. 7, No.,, pp doi:./s9-()7-7 [] D. D. Gaji ad A. Sadighi, Applicaio of He Homo- Coupled Sem op Perurbaio Mehod o Noliear of Reacio-Diffuio Equaio, Ieraioal Joural of Noliear Sciece ad Numerical Simulaio, Vol. 7,, pp. -8. doi:.55/ijnsns..7.. [] D. D. Gaji, The Applicaio of He Homoop Perur- Ariig i Hea baio Mehod o Noliear Equaio Trafer, Phic Leer A, Vol. 5, No.,, pp. 7-. doi:./j.phlea...5 [] H. K. Mihra ad A. K. Nagar, He-Laplace Mehod for Liear ad Noliear Parial Differeial Equaio, Joural of Applied Mahemaic, Vol.,, pp. -. doi:.55//85 [] S. T. Mohud-Di, M. A. Noor ad K. I. Noor, Travellig Wave Soluio of Seveh-Order Geeralized KdV Equaio Uig He Polomial, Ieraioal Joural of Noliear Sciece ad Numerical Simulaio, Vol., 9, pp. 7-. doi:.55/ijnsns [5] A. M. Wazwaz, A Compario bewee he Variaioal Ieraio Mehod ad Adomia Decompoeio Mehod, Joural of Compuaioal ad Applied Mahemaic, Vol. 7, No., 7, pp. 9-. doi:./j.cam..7.8 [] E. Yuufoglu, Numerical Soluio of Duffig Equaio b he Laplace Decompoiio Algorihm, Applied Mahemaic ad Compuaio, Vol. 77, No.,, pp [7] J. H. He, Homoop Perurbaio Mehod: A New Noliear Aalical Techique, Applied Mahemaic ad Compuaio, Vol. 5, No.,, pp doi:./s9-()-5 [8] J. H. He, Homoop, Compario of Homoop Perurbaio Mehod ad Homoop Aali Mehod, Applied Mahemaic ad Compuaio, Vol. 5,, pp doi:./j.amc..8.8 [9] J. H. He, Homoop, he Homoop Perurbaio Mehod for Noliear Ocillaor wih Dicoiueie, Applied Mahemaic ad Compuaio, Vol. 5,, pp doi:./s9-()- [] J. H. He, Rece Developme of he Homoop Perurbaio Mehod, Topological Mehod i Noliear Aali, Vol., 8, pp [] J. H. He, New Ierpreaio of Homoop Perurbaio Mehod, Ieraioal Joural of Moder Phic, Vol., No. 8,, pp doi:./s [] J. H. He, A Couplig Mehod of Homoop Techique ad a Perurbaio Techique for Noliear Problem, Ieraioal Joural of Noliear Mechaic, Vol. 5,, pp. 7-. doi:./s-7(98)85-7 [] J. H. He, Variaioal Ieraio Mehod for Auoomou Ordiar Differeial Sem, Applied Mahemaic ad Compuaio, Vol.,, pp. 5-. Coprigh SciRe.

9 H. K. MISHRA doi:./s9-(99)- [] J. H. He, Homoop Perurbaio Techique, Compuer mehod i Applied Mechaic ad Egieerig, Vol. 78, No. -, 999, pp doi:./s5-785(99)8- [5] J. H. He, A Simple Perurbaio Approach o Blaiu Equaio, Applied Mahemaic ad Compuaio, Vol., No. -,, pp. 7-. doi:./s9-()89- [] J. H. He, Applicaio of Homoop Perurbaio Mehod o Noliear Wave Equaio, Chao, Solio, Fracal, Vol., No., 5, pp doi:./j.chao.5.. [7] J. H. He, Homoop Perurbaio Mehod for Solvig Boudar Value Problem, Phic Leer A, Vol. 5, No. -,, pp doi:./j.phlea.5..5 [8] E. Heameddii ad H. Laifizadeh, A Opimal Choice of Iiial Soluio i he Homoop Perurbaio Me- Sciece ad hod, Ieraioal Joural of Noliear Numerical Simulaio, Vol., 9, pp doi:.55/ijnsns [9] E. Heameddii ad H. Laifizadeh, A New Viio of he He Homoop Perurbaio Mehod, Ieraioal Joural of Noliear Sciece ad Numerical Simulaio, Vol., 9, pp. 5-. doi:.55/ijnsns [] M. Rafei ad D. D. Gaji, Eplici Soluio of Helm- hoz Equaio ad Fifh-Order KdV Equaio Uig Homoop Perurbaio Mehod, Ieraioal Joural of Noliear Sciece ad Numerical Simulaio, Vol. 7,, pp. -8. doi:.55/ijnsns..7.. [] A. M. Siddiqui, R. Mohmood ad Q. K. Ghori, Thi Film Flow of a Third Grade Fluid o a Movig Bel b He Homoop Perurbaio Mehod, Ieraioal Joural of Noliear Sciece ad Numerical Simulaio, Vol. 7, No.,, pp. 7-. doi:.55/ijnsns [] L. Xu, He Homoop Perurbaio Mehod for a Boudar Laer Equaio i Ubouded Domai, Compuer ad Mahemaic wih Applicaio, Vol. 5, No. 7-8, 7, pp doi:./j.camwa...5 [] J. Biazar, M. Gholamiporhokuhi ad B. Ghabari, Eracig a Geeral Ieraive Mehod from a Adomia Decompoiio Mehod ad Comparig I o he Variaioal Ieraio Mehod, Compuer ad Mahemaic wih Applicaio, Vol. 59, No.,, pp. -8. doi:./j.camwa.9.. [] S. Ilam, Y. Kha, N. Faraz ad F. Aui, Numerical Soluio of Logiic Differeial Equaio b Uig he Laplace Decompoiio Mehod, World Applied Sciece Joural, Vol. 8,, pp. -5. [5] Y. Kha ad F. Aui, Applicaio of he Laplace De- compoiio Mehod o Noliear Homogeeou ad No-Homogeeou Advecio Equaio, Zeichrif fuer Naurforchug, Vol. 5,, pp. -5. [] Y. Kha ad Q. B. Wu, Homoop Perurbaio Traform Mehod for Noliear Equaio Uig He Polomial, Compuer ad Mahemaic wih Applicaio, Vol., No. 8,, pp doi:./j.camwa..8. [7] Y. Kha, A Effecive Modificaio of he Lap-Lace Decompoiio Mehod for Noliear Equaio, Ieraioal Joural of Noliear Sciece ad Numerical Simulaio, Vol., 9, pp doi:.55/ijnsns [8] S. A. Khuri, A Laplace Decompoiio Algorihm Applied o a Cla of Noliear Differeial Equaio, Joural of Applied Mahemaic, Vol., No.,, pp doi:.55/s757x8 [9] M. Madai ad M. Fahizadeh, Homoop Perurbaio Algorihm Uig Laplace Traformaio, Noliear Sciece Leer A, Vol.,, pp. -7. [5] A. Ghorbai, Beod Adomia Polomial: He Polomial, Chao, Solio, Fracal, Vol. 9, 9, pp doi:./j.chao.7.. [5] J. H. He, Homoop Perurbaio Techique, Compuer Mehod i Applied Mechaic ad Egieerig, Vol. 78, No. -, 999, pp doi:./s5-785(99)8- [5] S. T. Mohud-Di ad A. Yildirim, Homoop Perurb- Baio Mehod for Advecio Problem, Noliear Sciece Leer A, Vol., No.,, pp. 7-. [5] J. H. He, A New Approach o Noliear Parial Differeial Equaio, Commuicaio i Noliear Sciece ad Numerical Simulaio, Vol., 997, pp. -5. doi:./s7-57(97)97- [5] B. A. Filao, The Mehod of Weighed Reidual ad Variaioal Priciple, Academic Pre, Lodo, 97. [55] M. Iokui, e al., Geeral Ue of he Lagrage Muliplier i Noliear Mahemaical Phic, I: S. Nema- Naer, Ed., Variaioal Mehod i he Mechaic of Sol- id Pergamo, 978, pp. 5-. [5] G. Adomia, Sochaic Sem, Academic Pre, Lodo, 98. [57] G. L. Liu, Weighed Reidual Decompoiio Mehod i Noliear Applied Mahemaic, Proceedig of h Cogre of Moder Mahemaic ad Mechaic, Suzhou, 995, pp. -8. [58] G. Adomia, A Review of he Decompoiio Mehod i Applied Mahemaic, Joural of Mahemaical Aali ad Applicaio, Vol. 5, No., 988, pp Coprigh SciRe.