Approximate Solutions for the Coupled Nonlinear. Equations Using the Homotopy Analysis Method

Transcription

1 Applied Maheaical Scieces, Vol. 5,, o. 37, Approxiae Solios for he Copled Noliear Eqaios Usig he Hooopy Aalysis Mehod Spig Qia a, b a Facly of Sciece, Jiags Uiersiy, Zhejiag, Jiags 3, Chia b Depare of Maheaics, Chagsh Isie of Techology, Chagsh, Jiags 55, Chia qsp@cslg.ed.c Li Wei a a Facly of Sciece, Jiags Uiersiy, Zhejiag, Jiags 3, Chia Absrac I his aricle, he hooopy aalysis ehod (HAM) is ipleeed o obai he approxiae solios of he oliear eolio eqaios i aheaical physics. The resls obaied by his ehod hae a good agreee wih oe obaied by oher ehods. I illsraes he alidiy ad he grea poeial of he hooopy aalysis ehod i solig parial differeial eqaios. Maheaics Sbjec Classificaio: 58B5 Keywords: Hooopy aalysis ehod; oliear eolio eqaio; approxiae solios. Irodcio I is well kow ha oliear dyaical syses arise i arios fields. A

2 8 Spig Qia ad Li Wei wealh of ehods hae bee deeloped o fid hese exac physically sigifica solios of a parial differeial eqaio hogh i is raher difficl. Soe of he os ipora ehods are Backld rasforaio [], he CK direc ehod [,3], ad he hoogeeos balace ehod [4]. The hooopy aalysis ehod (HAM) [5], is a powerfl ehod o sole o-liear probles. Based o hooopy of opology, he alidiy of he HAM is idepede of wheher or o here exis sall paraeers i he cosidered eqaio. Therefore, he HAM ca oercoe he foregoig resricios ad liiaios of perrbaio echiqes [6]. I rece years, his ehod has bee sccessflly eployed o sole ay ypes of o-liear probles [7,8,9]. I his paper, he hooopy aalysis ehod is sed o sole he syse of he approxiae eqaios for log waer waes [,], as follows: + /= x x xx { ( ) /= x wih he soliary wae solios [4] { xx [ ] = α [( α + β + γ) ] x, = α ah αx+ β + γ / / + α /+ c x, sec h x / /4 () () where αγ,,c are arbirary cosas ad β = cα + α /.. Aalysis of he ehod For he coeiece of he reader, we will firs prese a brief acco of he HAM. Le s cosider he followig differeial eqio : [ ( x ) ] Ν, = (3) where Ν is a oliear operaor, ( x, ) is a kow fcio, ad x ad deoe spaial ad eporal idepede ariables. 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, we ca cosrc he so-called zero-order deforaio eqaio as: ( q) L[ φ( x, ; q) ( x, ) ] = q Η( x, ) Ν[ φ( x, ; q) ] h (4) where q [,] is he ebeddig paraeer, h is a o-zero axiliary paraeer,

3 Approxiae solios 8 Η( x, ) is a axiliary fcio, L is a axiliary liear operaor, a iiial gess of ( x, ) ad ( x,; q) whe q = ad q =, i holds φ ( x,; ) = ( x, ), φ ( x,; ) = ( x, ), x is φ is a kow fcio. Obiosly, respeciely. Ths, as q icreases fro o, he solio ( x,; q) φ aries fro he iiial gess ( x, ) o he solio ( x, ). Expadig φ ( x,; q) i Taylor series wih respec o q, we hae + = + (5) (,; ) (,) φ x q x q = where ( x ) ( xq,; ) φ, =! q q= (6) If he axiliary liear operaor, he iiial gess, he axiliary paraeer h, ad he axiliary fcio are properly chose, he series (5) coerges a q =, he we hae (, ) (, ) x = + x (7) + = which s be oe of solios of origial oliear eqaios. Accordig o he defiiio (6), he goerig eqaio ca be dedced fro he zero-order deforaio (4). Defie he ecor x x x x (, ) { (, ), (, ),, (, )} = L (8) Differeiaig eqaio (4) ies wih respec o he ebeddig paraeer q ad he seig q = ad fially diidig he by!, we hae he so-called h-order deforaio eqaio L [ ( x, ) χ ( x, )] = h H( x, ) R (9)

4 8 Spig Qia ad Li Wei where R ( ) = ( )! Ν [ φ ( xq,; )] q q= () ad {, χ =, > I order o assess he adaages ad accracy of HAM, we cosider he followig applicaio. () 3. Applicaio we shall deal wih he syse of he approxiae eqaios for log waer waes (). For sipliciy, α =, γ =, c = are sed i he aalyses. For applicaio of hooopy aalysis ehod, we choose he iiial codiios { (,) sec = x, = ah x + = x = h x ad he axiliary liear operaors ( x q) ( xq) φ,; ϕ,; L [ φ( x,; q) ] =, L [ ϕ( x,; q )] = () (3) wih he propery L [ C ] L [ C ] =, = where C ad C are cosa coefficies, φ ad ϕ are real fcios. Frherore, we defie he oliear operaors φ x,; q φ xq,; ϕ xq,; φ xq,; Ν [ φ( xq,; ), ϕ( xq,; )] = φ( xq,; ) + x x x (4), ϕ ϕ φ ϕ Ν [ φ( xq,; ), ϕ( xq,; )] = φ( xq,; ) ϕ( xq,; ) x x x x,; q xq,; xq,; xq,; where q [, ], φ ( x,; q) ad ( x,; q) ϕ are real fcios of x, ad q. Le (5)

5 Approxiae solios 83 h, h deoe he ozero axiliary paraeers. Usig he aboe defiiio, wih asspio H ( x, ) =, eqaios as follows, H x, =, we cosrc he zero-order deforaio ( q) L[ φ( xq, ; ) ( x, )] q N ( φ( xq, ; ), ϕ( xq, ; )) = h, (6) ( q) L[ ϕ( xq, ; ) ( x, )] q N ( φ( xq, ; ), ϕ( xq, ; )) = h (7) obiosly, whe q = ad q =, i is clear ha ( x, ; ) ( x, ), ( x, ;) ( x, ), ( x, ;) ( x, ), ( x, ;) ( x, ) φ = ϕ = φ = ϕ = (8) Boh of h ad h are properly chose so ha he ers ( x ) ( xq,; ) φ, =! q q= ad ( x ) ( xq,; ) ϕ, =! q q= (9) exis for ad he power series of q i he followig fors () = = (,; ) = (,) + (,), ϕ(,; ) = (,) + (,) φ x q x x q xq x x q are coerge a q =. So sig (9), we obai + +. () = = (, ) = + (, ), (, ) = + (, ) x x x x Accordig o he fdaeal heore of HAM, we hae he h-order deforaio eqaio [ χ ] χ [ ] L x, x, = h R,, L x, x, = h R,,() where i (3) x, x, x, x, R x x x x i,, = + i= x, x, x, x, R x x i i, (, ) (, = i i ) i= x i= x x (4)

6 84 Spig Qia ad Li Wei ad χ is defied by (). Now, he solio of he h-order deforaio eqaio () for becoes ( x, ) = χ (, ),, (, ) (, ), x + h L R x χ x L R = + h. (5) Noe ha he solios series () coai wo axiliary h ad h. For sipliciy, le = = h h h, he he approxiaios of ( x, ) ad (, ) x are oly depede o h. We wrie he differeial eqaios eed o calclae,,, L, ad 3,, 3, L, as follows x x x = h h + h h x x = h + h h h = ( h+ ) h + h h + x x x x = ( h+ ) h h x x x x x = 3 ( + h) h + h h + + x x x x x = 3 ( + h ) h h x x x x x x x M (6) i = ( h+ ) h + h h i x x i= x i i = ( h+ ) h h + i i x i= x i= x The ad i i,( i =,, 3, ) L copoes hae bee obaied sig he aple package. I is ipora o esre ha he series solios are coerge. We iesigae he iflece of he axiliary paraeer h o he coergece of he

7 Approxiae solios 85 series by ploig he so-called h -cres. The h cres of,,,,, ad x xx x a he poi ( x, ) = ( 3.5,5.5) are depiced i Fig. ad Fig.. The alid xx regio of h is a horizoal lie sege. I is obsered he alid regio for h is.5 < h <.5 as show i Fig. ad Fig.. Fig.. h -cres of ad is deriaies wih differe h i he case of ( x, ) = ( 3.5, 5.5) ad 8h-order approxiaio. Fig.. h -cres of ad is deriaies wih differe h i he case of ( x, ) = ( 3.5, 5.5) ad 8h-order approxiaio. I order o erify he alidiy of he HAM solio, he hree-diesioal plos of he absole error bewee he exac solios ad he solios series obaied by HAM for h =.8 are show i Fig. 3 ad Fig. 4. As show i hese figres, he behaior of he approxiae solios obaied by hooopy aalysis ehod agree well wih oe obaied by he exac solios (). Fig. 3 Absole error for he 8h-order approxiaio by HAM for (, ) Fig. 4 Absole error for he 8h-order approxiaio by HAM for (, ) x ad h =.8. x ad h =.8.

8 86 Spig Qia ad Li Wei 4. Coclsio I his paper, he hooopy aalysis ehod (HAM) is sed o obai he approxiae solios of he syse of he approxiae eqaios for log waer waes. The resls obaied by his ehod hae a good agreee wih oe obaied by oher ehods. The adaages of he HAM are illsraed. I is easy o see ha he HAM is a ery powerfl ad efficie echiqe i fidig aalyical solios of wide classes of oliear parial differeial eqaios. Refereces [] S.P. Qia, L.X. Tia, Nolocal Lie Bäckld syeries of he copled KdV syse, Physics Leer A. 364 (7), [] S.Y. Lo, A oe o he ew siilariy redcios of he Bossiesq eqaio, Chiese Physics Leer A, 5 (99), [3] P.J. Oler, Applicaio of Lie Grop o differeial Eqaios, New York,spriger, 993. [4] M.L. Wag, Y.B. Zho, Z.B. Li, Applicaio of a hoogeeos balace ehod o exac solios of oliear eqaios i aheaical physics, Physics Leer A, 6 (996), [5] S.J. Liao, The proposed hooopy aalysis echiqes for he solio of oliear probles, Ph.D. disseraio, Shaghai Jiao Tog Uiersiy, Shaghai, 99 [i Eglish]. [6] J.H. He, Hooopy perrbaio ehod, Coper Mehods i Applied Mechaics Egieerig 78 (999), [7] S.J. Liao, O he hooopy aalysis ehods for oliear probles, Applied Maheaics Copaio, 47 (4), [8] S. Abbasbady, The applicaio of hooopy aalysis ehod o sole a geeralized Hiroa Sasa copled KdV eqaio, Physics Leer A, 36 (7), [9] T. Haya, M. Sajid, O aalyic solio for hi fil flow of a forh grade flid dow a erical cylider, Physics Leer A,36 (7), [] G.B. Whiha, Variaioal ehods ad applicaios o waer wae, Proceedigs of he Royal Sociey A, 99 (967), 6-5. [] L.T.F. Broer, Approxiae eqaios for log waer waes, Applied Scieific Research, 3 (975), Receied: Noeber,