Applications of Homotopy Analysis Transform Method for Solving Various Nonlinear Equations

Transcription

1 World Applied Scieces Joral 8 (): , ISSN IDOSI Pblicaios, DOI:.589/idosi.wasj Applicaios of Hooopy Aalysis Trasfor Mehod for Solvig Varios Noliear Eqaios V.G. Gpa Si Gpa Depare of Maheaics, Uiversiy of Rajasha, Jaipr-355, Idia Jaga Nah Gpa Isie of Egieerig Techology, Siapra, Jaipr, Idia Absrac: I his paper, we apply Hooopy Aalysis Trasfor Mehod (HATM) for solvig varios oliear eqaios. This ehod is he cobied for of he hooopy aalysis ehod Laplace rasfor ehod. HATM is applied wiho ay discreizaio or resricive asspio avoids rodoff errors which ay lead he solio i closed for. The resls reveal ha he HATM is very effecive, coveie qie accrae o syse of oliear eqaios. Also i is show ha he Adoai decoposiio ehod hooopy perrbaio ehod Variaioal ieraio ehod are special case of hooopy aalysis rasfor ehod = Key words: Hooopy aalysis ehod Laplace rasfor ehod copled eqaios approiae solio eac solio INTRODUCTION Aalyical ehods have ade a coeback i research ehodology afer akig a backsea o he erical echiqes for he laer half of he precedig cery. The advaage of aalyical ehods are aifolds, he ai beig ha hey give a ch beer isigh ha he bers crched by a coper sig a prely erical algorih. Mos ew oliear eqaios do-o have a precise aalyic solio; so erical ehods have largely bee sed o hle hese eqaio. I rece years, ay ahors have paid aeio o sdyig he solios of oliear parial differeial eqaios by varios ehods. Aog hese are Adoai Decoposiio ehod [-4], he ahehod [5], he sie-cosie ehod [6, 7], he differeial rasfor ehod [8, 9], he Variaioal ieraio ehod [-7] he Laplace decoposiio ehod [8-]. Oe of he well kow perrbaio echiqe is Hooopy Perrbaio Mehod (HPM), firs proposed by Ji Ha He by cobiig he sard hooopy classical perrbaio echiqe for solvig varios liear, oliear iiial bodary vales probles [3-33] has bee odified laer by soe scieiss o obai ore accrae resls, rapid covergece redce he ao of copaio [34-37]. Also hooopy perrbaio ehod is cobied wih Laplace rasfor ehod [38], Sd rasfor [39] Variaioal ieraio ehod [4] o prodce highly effecive echiqes for oliear probles. Recely, Liao [4] proposed a powerfl aalyical ehod, aely he Hooopy Aalysis Mehod (HAM), for solvig varios liear oliear parial differeial eqaios. Differe fro perrbaio echiqes, he hooopy aalysis ehod does o deped po ay sall or large paraeers. The HAM was sccessflly applied o solve ay oliear probles [4-6]. Drig he las wo decades; M.Kha [56] proposed a echiqe by cobiig he hooopy aalysis ehod Laplace rasfor ehod, firs ie i lierare, aely Hooopy Aalysis Trasfor Mehod (HATM) for solvig varios oliear probles. The ai advaage of his proble is ha we ca accelerae he covergece rae, iiize ieraive ies, accordigly save he copaio ie evalae he efficiecy. Several eaples are give o access he reliabiliy of HATM. HOMOTOPY ANALYSIS METHOD Cosider he followig differeial eqaio N[( τ )] = () N is a oliear operaor, τ is he idepede variable (τ) is a kow fcio. For sipliciy, we igore all bodary or iiial codiios, which ca be reaed i he siilar way. By eas of geeralizig he radiioal hooopy ehod, cosrcs he zero-order deforaio eqaio, give by Liao. Correspodig Ahor: V.G. Gpa, Depare of Maheaics, Uiversiy of Rajasha, Jaipr-355, Idia 839

2 World Appl. Sci. J., 8 (): , ( q)l[ φ( τ;q) ( τ )] = q H()N[ τ φ( τ;q)] () q [,] is a ebeddig paraeer, is a o zero ailiary paraeer, H(τ) is a o zero ailiary fcio, L is a ailiary liear operaor, (τ) is a iiial gess of (τ) φ(τ;q) is a kow fcio. I is ipora ha oe has grea freedo o choose ailiary higs i HAM. Obviosly, whe q = q =, i holds φτ ( ;) = ( τ) φτ ( ;) = ( τ ) (3) Ths, as q icreases fro o, he solio φ(τ;q) varies fro he iiial gess (τ) o he solio (τ). Epig φ(τ;q) by Taylor series wih respec o q, we ge φτ ( ;q) = ( τ+ ) ( τ)q (4) = φτ ( ;q) ( τ ) = q=! q If he ailiary liear operaor, he iiial gess, he ailiary paraeer he ailiary fcio are so properly chose, he series (5) coverges a q =, he we have (r,) (r,) (r,) = (5) = + (6) χ =,, > () i shold be ephasized ha (τ) for is govered by he liear bodary codiios ha coe fro origial proble, which ca be easily solved by sybolic copaio sofware sch as Maple, Maheaica Malab. If eqaio () adis iqe solio, he his ehod will prodce he iqe solio. If eqaio () does o posses a iqe solio, he HAM will give a solio aog ay oher possible solios. HOMOTOPY ANALYSIS TRANSFORM METHOD (HATM) Cosider eqaio N[()] = g(), N represes a geeral oliear ordiary or parial differeial eqaio icldig boh liear oliear ers. The liear ers are decopose io L+R, L is he highes order liear operaor R is he reaiig of he liear operaor. Ths, he eqaio ca be wrie as [56] L+ R + N = g() () N, idicaes he oliear ers. By applyig Laplace rasfor o boh sides of Eq. (3), we ge L[L + R + N = g()] (3) which s be oe of he solios of he origial oliear eqaio, as prove by Liao []. As = H(τ) =, eqaio (3) becoes ( q)l[ φ( τ;q) ( τ )] + qn[ φ( τ ;q)] = (7) The goverig eqaio ca be dedced fro he zero-order deforaio eqaio (3). Defie he vecor = ( τ),(), τ ( τ) { } (8) Usig he differeiaio propery of Laplace rasfor, we ge k ( k) s L[] s () + L[R] + L[N] = L[g()] (4) k= O siplifyig k ( k) s k= s L[] s () + L[R] + L[N] = (5) Differeiaig eqaio (3) -ies wih respec o he ebeddig paraeer q, he seig q = fially dividig he by!, we obai he h -order deforaio eqaio. L ( τ) χ ( τ ) = H( τ)r ( ) (9) [ ] N[ φτ ( ;q)] R ( ) = q =! q () 84 We defie he oliear operaor k (k) N[ φ (,;q)] = L φ(,;q) s φ (,;q)() s k= + L[ φ (,;q)] + L[R φ (,;q)] (6) s φ(,;q) is a real fcios of, q. We cosrc a hooopy as

3 World Appl. Sci. J., 8 (): , ( q)l[ φ(,;q) (,)] = q H(,)N[(,)] (7) q [,] is a ebeddig paraeer, is a o zero ailiary paraeer, H(,τ) is a o zero ailiary fcio, L is a ailiary liear operaor, (,τ) is a iiial gess of (τ φ(,;q) is a kow fcio. I is ipora ha oe has grea freedo o choose ailiary higs i HAM. Obviosly, whe q = q =, i holds φ (,;) = (,) φ (,;) = (,) (8) respecively. Ths, as q icreases fro o, he solio φ(;q) varies fro he iiial gess (τ) o he solio (τ). Epig φ(;q) by Taylor series wih respec o q, we ge (,;q) (,) (,)q φ = + (9) = φ(,;q) (,) = q =! q () χ =,, > APPLICATIONS Eaple 4.: Le s cosider he followig proble wih he iiial codiios (6) + = (7) (,) = To solve eqaio (7) by eas of he hooopy aalysis ehod Le s cosider he followig liear operaor wih he propery ha φ(,;q) L φ (,;q) = (8) If he ailiary liear operaor, he iiial gess, he ailiary paraeer he ailiary fcio are so properly chose, he series (9) coverges a q =, he we have which iplies ha Lc L() = (9) = ()d (3) (r,) (r,) (r,) = + () = Takig Laplace rasfor of eqaio (7) boh of sides sbjec o he iiial codiio, we ge which s be oe of he solios of he origial oliear eqaio. The goverig eqaio ca be dedced fro he zero-order deforaio eqaio (7). Defie he vecor = (,), (,), (,) () { } Differeiaig eqaio (7) -ies wih respec o he ebeddig paraeer q, he seig q = fially dividig he by!, we obai he h -order deforaio eqaio. L (,) χ (,) = H(,)R ( ) (3) Applyig iverse Laplace rasfor we ge (,) = χ + L H(,)R ( ) (4) N[ φ(,;q)] R ( ) = q = (5)! q 84 L(,) L s s + + = (3) We ow defie he oliear operaor as φ(,;q) N[ φ (,;q)] = L φ (,;q) + + L φ (,;q) = s s (3) he he h -order deforaio eqaio is give by L (,) χ (,) = H(,)R ( ) (33) Takig iverse Laplace rasfor of Eq. (33), we ge (,) = χ + L H(,)R ( ) i R( ) = L + ( χ ) + L i (34) s s i=

4 World Appl. Sci. J., 8 (): , le s ake he iiial approiaio as (,) = he oher copoes are as follows ( )( ) (,) = { } ( ) ( ) (,) = + (35) ( (,;q) ) ( φ ) φ + φ = φ + = s s (,;q) N (,;q) L (,;q) L (4) he he h -order deforaio eqaio is give by L (,) χ (,) = H(,)R ( ) (43) { } ( ) ( ) (,) = + + so o he he approiae solio a = is give by R( ) = L ( χ) s 3 + L i i + 3 i i s i= i= Le s ake he iiial approiaio as (44) (,) =, < (36) which is a eac solio is sae as obaied by HAM [57] VIM [58]. Eaple 4.: Le s cosider he followig proble wih he iiial codiio + ( ) + ( ) = (37) (,) = To solve eqaio (37) by eas of he hooopy aalysis ehod Le s cosider he followig liear operaor wih he propery ha which iplies ha φ(,;q) L φ (,;q) = Lc L() (38) = (39) = ()d (4) Takig Laplace rasfor of eqaio (37) boh of sides sbjec o he iiial codiio, we ge (,) = he oher copoes are give by (,) = ( ) (,) = ( ) (,) = + + (45) 4 ( ) 3 (,) = + + so o he he approiae solio a = is give by (,) = (46) + which is a eac solio is sae as obaied by HAM [57] VIM [58]. Eaple 4.3: Le s cosider he followig copled syse of eqaio wih he iiial codiios ( ) + v = (47) ( ) v v vv + v = L(,) L ( ) ( ) + + = s s we ow defie he oliear operaor as (4) (,) = si, v(,) = si To solve eqaio (47) by eas of he hooopy aalysis ehod 84

5 World Appl. Sci. J., 8 (): , Le s cosider he followig liear operaor φ(,;q) L φ (,;q) = (48) wih he propery ha which iplies ha Lc = (49) L() = ()d (5) Takig Laplace rasfor of eqaio (47) boh of sides sbjec o he iiial codiio, we ge si L(,) + L + ( v) = s si L v(,) + L v vv + ( v) = s (5) we ow defie he oliear operaor as si φ (,;q) φ (,;q) N φ (,;q) = L φ(,;q) + L L φ (,;q) s si φ (,;q) φ (,;q) N φ (,;q) = L φ(,;q) + L L φ (,;q) s ( φ φ ) (,;q) (,;q) (5) ( φ φ ) (,;q) (,;q) (53) he he h -order deforaio eqaio is give by L (,) χ (,) = H(,)R ( ) (54) L v (,) χ v (,) = H(,)R (v ) (55) Takig iverse Laplace rasfor of Eq. (54) Eq. (55), we ge (,) = χ + L H(,)R ( ) (56) v (,) =χ v + L H(,)R (v ) (57) ( v ) si i i i R( ) = L (,) ( χ ) + L i s i= i= ( v i i) si i R( ) = L (,) ( χ ) + L i s i= i= ( v i i) si v v i R( v ) = L v (,) ( χ ) + L v i s i= i= (58) (59) Le s ake he iiial approiaio he oher copoes are give by (,) = si v (,) = si (6) 843

6 (,) = si v(,) = si (,) = si + ( + )! v (,) = si + ( + ) (6)! (,) = si + ( + + ) + ( + ) 3! v(,) 3 = si + ( + + ) + ( + ) 3! so o he he approiae solio a = is give by (,) World Appl. Sci. J., 8 (): , = e si (6) v(,) = e si which is a eac solio is sae as obaied by HAM [57] VIM [58]. CONCLUSIONS I his paper, he hooopy aalysis rasfor ehod (HATM) is sccessflly applied o solve ay oliear probles. I is apparely see ha HATM is very powerfl efficie echiqe i fidig aalyical solios for wider class of probles. They also do o reqire large coper eory discreizaio of variable. The resls show ha HATM is powerfl aheaical ool for solvig oliear eqaios. Also i has bee show ha VIM is he special case of HATM HAM. REFERENCES. Adoai, G., 994. Solvig Froier proble of Physics: The Decoposiio Mehod, Klwer Acad.Pbl., Boso.. Wazwaz, A.M.,. A ew algorih for calclaig Adoai polyoials for oliear operaors. Applied Maheaics Copaio, : Wazwaz, A.M.,. Cosrcig of soliary wave solios raioal solios for he KdV eqaio by Adoai Decoposiio Mehod. Chaos Solios Fracals, : Sadighi, A., D.D. Gaji Y. Sabzeheidai, 8. A Decoposiio ehod for Vole Fl Average Velociy of hi fil flow of a hird grade fil dow a iclied plae. Advace i Theoreical Applied Mechaics, : Wazwaz, A.M., 4. A sie-cosie ehod for hlig oliear wave eqaios. Maheaical Coper Modelig, 4: Wazwaz, A.M., 5. The ah sie-cosie ehods for he cople odified geeralized KdV eqaios. Copers Maheaics wih Applicaios, 49: Fa, E.,. Eeded ah-fcio is applicaios o oliear eqaios. Physics Leers A 77: Keski, Y. G. Orac, 9. Redced Differeial Trasfor Mehod for Parial differeial eqaios, Ieraioal Joral of Noliear Scieces Nerical Silaio, (6): Keski, Y. G. Orac,. Redced Differeial Trasfor Mehod for Fracioal Parial Differeial eqaios. Noliear Scieces Leers, A (): He, J.H., 999. Variaioal Ieraio ehod-a kid of oliear aalyical echiqes: soe eaples. Ieraioal Joral of Noliear Mechaics, 34: He, J.H. X.H. W, 7. Variaioal Ieraio Mehod: ew develope applicaios. Copers Maheaics wih Applicaios, 59: Solai, L.A. A. Shirzadi,. A ew odificaio of he variaioal ieraio ehod. Copers Maheaics wih Applicaios, 59: Faraz, N., Y. Kha A. Yildiri,. Aalyical Approach o wo diesioal viscos flows wih a shrikig shee via variaioal ieraio algorih-ii. Joral of Kig Sad Uiversiy, doi:.6j.jkss W, G.C. E.W.M. Lee,. Fracioal variaioal ieraio ehod is applicaio. Physics Leers A doi:.6 / j.physlea Ch, C., 9. Forier-Series based variaioal ieraio ehod for a reliable reae of hea eqaios wih variable coefficies. Ieraioal Joral of Noliear Scieces Nerical Silaio, : Mohyd-Di, S.T.,. Modified variaioal ieraio ehod for iegro-differeial eqaios copled syses. Zeischrif fr Narforschg A. A Joral of Physical Scieces, 65 a:

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