An Efficient Approach for Fractional Harry Dym Equation by Using Homotopy Analysis Method

Transcription

1 hp:// ISSN: 78-8 Vol. 5 Issue, January-6 An Efficien Approach for Fracional Harry Dy Equaion by Using Hooopy Analysis Mehod Bhuvaneshwari Shunugarajan VIT Universiy,Cennai. Tail Nadu,India. Absrac - In his aricle, a srucured approach based on Hooopy analysis ehod is ipleened o resolve nonlinear fracional Harry Dy equaion. The fracional derivaive is described in Capuo sense. The uliae goal of his paper is o find error analysis which shows our final concluded resul converge o he eac and approiae soluion. The soluion which was derived fro convergen series for proves he effeciveness of he ehod in resolving differen ype of fracional differenial equaion. Henceforh his echnique is efficien and less ie consuing when copare o oher ehods of finding approiae and eac soluions for nonlinear parial differenial equaions. Keywords: Harry Dy Equaion, Capuo derivaive, Hooopy analysis ehod, approiae soluion. INTRODUCTION Fracional calculus is a separae sudy of invesigaing he characerisics of inegral and derivaives of non inegers order. This fracional calculus is specifically popular for solving differenial equaion involving fracional derivaives of unknown funcion which is also called fracional differenial equaion. The nuber of probles in science and engineering involving in fracional derivaives is growing coninuously and perhaps he fracional calculus of he weny five cenury[-6].the idea appeared in a leer of Leibniz o L Hospial in 965.The analyical resuls relaed o fracional differenial equaion have been sudied by any auhors[7,8].during he las decades, uliple nuber of nuerical and analyical ehods have been suggesed o resolve fracional differenial equaions. The os acceped ehods are fracional difference ehod[9,],adoin decoposiion ehod[],variaional ieraion ehod[,3].recenly he Hooopy perurbaion ehod and Lagrange uliplier ehod were used eensively o solve uliorder fracional differenial equaion[4]. In his paper, we refer he below nonlinear ie fracional Harry Dy equaion of he for D U(, ) = U 3 (, )D 3 U(, ), < Wih he iniial condiion U(, ) = (a ) Where denoes he order of he fracional derivaive and U(, ) refers o a funcion of and.the fracional derivaive is presen in he Capuo sense. In case of =, he equaion(fracional Harry Dy equaion) siplify ino classical nonlinear Harry Dy equaion. The eac soluion of he Harry Dy equaion is given by[5] U(, ) = (a ( + b)) Where a and b are suiable consan. This Harry Dy Dynaical equaion is used o idenify applicaions in several physical syses. I is also appeared(harry Dy equaion) in Kruskal and Moser[6] and is aribued in an unpublished paper by Harry Dy in Harry Dy is a copleely inegrable nonlinear evoluion equaion. Meanwhile i obeys an infinie nuber of conversion laws; bu i doesn possess he Painleve propery. The priary developen of fracional calculus was achieved wih he help of book by Oldhanand Spanier[7].The essenial resuls relaed wih he soluion of fracional differenial equaion ay be found in book[8-].the Hooopy analysis ehod was developed by liao[-3].i ade an analyical sudy on fracional Kdv, K(,),Burgers, BBM-Burgers, Cubic Boussineq, Boussinesq-like B(,n) equaion and coubled Kdv. In he presen paper, we ipleened he Hooopy analysis ehod o deerine he approiae soluion of nonlinear fracional differenial equaion. Our ai of his paper is o eend he applicaion of Hooopy analysis ehod o obain analyic and approiae soluion o he nonlinear ie fracional Harry Dy equaion. HAM provides he soluions in ers of convergen series wih easily copuable coponens in a direc way which is no achievable using linearizaion, perurbaion. The error analysis will be ade by coparing he approiae and he eac soluion. The resuls are represened in he for of graph.. BASIC DEFINITIONS OF FRACTIONAL CALCULUS In his par, we give soe definiions and properies of he fracional calculus[3] Definiion.: A real funcion f(), >, is said o be in he space C μ, μ R,if here eiss a real nuber p > μ, such ha f() = p f () where f () C(, ) and i is said o be in he space C μ n, if and only if f (n) C μ, n N [3]. IJERTV5IS444 (This work is licensed under a Creaive Coons Aribuion 4. Inernaional License.) 56

2 hp:// ISSN: 78-8 Vol. 5 Issue, January-6 Definiion.: The Rieann-Liouville fracional inegral operaor (J α )of order α, of a funcion f C λ, λ if defined as [3] J α f() = D α f() = ( Γ(α) τ)α f(τ)dτ, (α > ) J f() = f() Where Γ(z) is he well known Gaa funcion.soe of he properies of he operaor (J α ), according o our requireen are given below. For f C λ, λ, α, β and γ : () J α J β f() = J α+β f() () J α J β f() = J β J α f() (3) J α J γ = Γ(γ+) Γ(α+γ+) α+γ Definiion.3: The fracional derivaive D α of f() in Capuo sense defined as D α f() = Γ( α) f () (τ) ( α) α+ dτ, where < α, N, >, f C The wo basic properies of he Capuo s fracional derivaive are given below Lea.: If < α, N, and f C μ, μ hen (D α J α )f() = f(), (J α D α )f() = f() f i ( + ) i 3. BASIC HOMOTOPY ANALYSIS METHOD The Hooopy analysis ehod (HAM) has been proposed by Liao in []. The applicaion of HAM is use o calculae he eac soluion and he approiae soluions of linear and nonlinear Differenial equaions. Consider he nonlinear differenial equaion i= N[X()]=; (3.) Where N is a nonlinear auiliary operaor and X() is an unknown funcion. The boundary and iniial condiions are ignored, which can be reaed in he siilar way. The zeroorder deforaion equaion is consruced as (-q)l[x n()- ()]=q ħn[x(),a], (3.) Where q [,] he ebedding paraeer, ħ a nonzero auiliary paraeer, L an auiliary linear operaor, X() an unknown funcion. Also we can choose whaever auiliary unknown in HAM because of is properies. By Taylor s heore, X() can be epanded wih respec o he ebedding paraeer q as i! X() = X o() + n= X n ()q n (3.3) X n()= n! n X() q n q= (3.4) Differeniaing he Zeroh-Order deforaion equaion n- ies wih respec o q a q = and hen dividing i by n,we have he following nh-order deforaion equaion L[Xn()- χ nx n-()] = ħr n(x n ()); (3.5) R n(x n ()) = (n )!, n and χ n = {, > n N(X()) q n q= (3.6) (3.7) Here X n = [X (), X (), X (), X 3 ().. X ()] If he series (3) converges a q = we have X() = X o() + X n () n= (3.8) 4.HAM SOLUTION OF HARRY DYM EQUATION: Consider he Harry Dy equaion: D U(, ) = U 3 (, )D 3 U(, ), < Wih he iniial condiion U(, ) = (a ) Where denoes he order of he fracional derivaive and U(, ) refers o a funcion of and. Soluion: Sep : Le us consider he iniial approiaion fro he iniial condiion () U (, ) = (a ) Sep : Le us consider he nonlinear operaor N(φ(, ; q)) = D φ(, ; q) φ 3 (, ; q)d 3 φ(, ; q)- Sep 3: Le us consider he Linear operaor L(φ(, ; q)) = D φ(, ; q) Sep 4: Using he above definiion, we consruc he zeroh order deforaion equaion ( q)l[φ(, : q) U (, )] = qħn[φ(, ; q)] Obviously when q= and q=, we can wrie fro equaion (4.6) φ(; ) = U () = U() and IJERTV5IS444 (This work is licensed under a Creaive Coons Aribuion 4. Inernaional License.) 56

3 hp:// ISSN: 78-8 Vol. 5 Issue, January-6 φ(; ) = U() Sep 5: According o equaion 3.4 and 3.7 we obain h order deforaion equaion L[U (, ) χ U ()] = hr [U ()] Where R [U () ] = φ 3 (, ; q)d 3 φ(, ; q)}] q= [ N[φ(;q)] ] ( )! q= = [ ( )! =[(D α U k n= U k n n k= s= U n s U s D 3 U k ] q {D φ(, ; q) Sep 6: Fro equaion 4.7 and 4.8 we can wrie L[U (, ) χ U ()] = h[( D α U k U k n U n s U s D 3 U k ] k= n= n s= U = χ U () + L [hd α U ] k L [h U k n U n s U s D 3 U k ] k= n= n s= U = χ U () + J α [hd α U ] k J α [h U k n U n s U s D 3 U k ] k= n= n s= a Sep 7: For =, equaion 4.8a becoes U = h J α [U 3 D 3 U ] For =, equaion 4.8a becoes Sep 8: Taking h =, We ge fro equaion 4.3,4.7,4.8 and 4.9 afer soe rigorous calculaion, U (, ) = (a ) 3 U (, ) = U (, ) = α Γ(α+) b3 α b 3 Γ(α+) (a ) 3 (a ) 4 3 U 3 (, ) = 3α Γ(3α+) b9 ) 7 5 Γ(α+) 3[ 6] (Γ(α+) ) and so on. Sep 9: By using equaion 3.3, we obain (a U() = U () + = U (), for q= (i) The second order approiaion of U() is given U HAM (, ) = U () + = =U () + U () =(a ) 3 b 3 (a ) α 3 Γ(α+) (ii) U () The hird order approiaion of U() is given U HAM (, ) = U () + ) α 3 b3 Γ(α+) = U () =U () + U () + U () =(a ) 3 b 3 (a (a ) 4 α 3 Γ(α+) U = U + [hu ] h J α [U 3 D 3 U + 3U U D 3 U ] For =3, equaion 4.8a becoes (iii) The hird order approiaion of U() is given U HAM (, ) = U () + 3 = U () U 3 = U + [hu ] h J α [U 3 D 3 U + 3U U D 3 U + (3U U + 3U U )D 3 U ] For 3,equaion 4.8 becoes U = U + [hu ] h J α [ k n= U k n n k= s= U n s U s D 3 U k ] =U () + U () + U () + U 3 () =(a ) 3 b 3 (a ) α 3 Γ(α+) b 3 (a ) 4 α 3 + Γ(α+) b9 (a ) 7 5 Γ(α+) 3[ 6] 3α (Γ(α+) ) Γ(3α+) IJERTV5IS444 (This work is licensed under a Creaive Coons Aribuion 4. Inernaional License.) 563

4 hp:// ISSN: 78-8 Vol. 5 Issue, January-6 TABLE : Coparison sudy beween HPSTM,ADM,HAM and Eac soluion, when α= and for consan values of a=4 and b= T HPSTM[33] ADM[33] HAM Eac soluion HAM soluion U(,) HAM soluion U(,).5.5 U(,).5 U(,).5 Figure : HAM soluion of U(,) when alpha= Figure : HAM soluion of U(,) when alpha= Absolue error Eac soluion U(,) E(,) U(,).5.5 Figure 3: Absolue error E(,) Figure 4: Eac soluion of U(,) IJERTV5IS444 (This work is licensed under a Creaive Coons Aribuion 4. Inernaional License.) 564

5 hp:// ISSN: 78-8 Vol. 5 Issue, January alpha= alpha=/ alpha=/3 alpha=. alpha=.5 alpha= U(,).4. U(,) Figure 5&6: Plo of U(, )versus a = for differen value of alpha 5. RESULTS & DISCUSSION In his aricle, deail sudy has been ade on error analysis beween eac and approiae soluion which is specified by hrough figure 3 wih high accuracy. The easies and accuracy of he suggesed ehod is depiced by calculaing he absolue error E 5 (, ) = U(, ) U 5 (, ) a he consan value (a=4,b=) where U(, ) and U 5 (, ) are he eac soluion and approiae soluion of equaion (4.) respecively. Figure 3 shows he error analysis beween he eac and approiae soluion which indicaes significanly sall convergence of he series soluion very quickly. Hence during all nuerical evoluion we are going o ake only hird order of he series soluion By inroducing ore ers of approiae soluion, he accuracy of error analyical resul will be iproved. The characerisics of he approiae soluion is specified in he figure 5 for U 5 (, ) for differen value of =, ½,/3 for sandard Harry Dy equaion. As per he Table and graphs we inferred ha HAM soluion ade a good agreeen wih he HPSTM and ADM soluion. The figure shows ha he accurae soluion can only be iprove inroducing ore ers of HAM soluion. This is due o quick convergence of he HAM. 6. CONCLUSION In his paper HAM is efficienly applied o ge approiae soluion of ie fracional Harry Dy equaion. In HAM a Hooopy wih ebedding sall paraeer q [,] which is helpful o ipleen full advanages of radiional perurbaion ehod and Hooopy echnique.ham is differen fro oher analyical ehod by providing he way o as o conrol he convergence region of soluion series by wih auiliary paraeer ħ.this is he ain useful feaures of HAM.HAM does no require any oher ehods or ransforaion echniques. The resuls of he copare soluion (HAM hird order soluion, HPSTM, ADM) are specified in he able.fro he able i is observed ha here is a good undersanding beween he HSPTM, ADM and eac soluion. Fro he derived resuls i shows he reliabiliy of he algorih and i is grealy suiable for Non-linear fracional parial differenial equaion. ACKNOWLDGEMENTS The auhors are graeful o he reviewers for inerpreaion his paper and for heir valuable coens and suggesion for he iproveen of he aricle. REFERENCES [] J. S. Leszczynski, An Iorducion o Fracional Mechanics, Czesochowa Universiy of Technology, Częsochowa, Poland,. [] K. S. Miller and B. Ross, An Inroducion o he Fracional Calculus and Fracional Differenial Equaions, John Wiley & Sons Inc., New York, NY, USA, 993. [3] A. Ousaloup, La Derivaion Non Eniere, Heres, Paris, France, 995. [4] J. A. T. Machado, Analysis and design of fracionalorder digial conrol syses, Syses Analysis Modelling Siulaion, vol. 7, no. -3, pp. 7, 997. View a Google Scholar [5] A. A. Kilbas, H. M. Srivasava, and J. J. Trujillo, Theory and Applicaions of Fracional Differenial Equaions, vol. 4, Elsevier Science B.V., Aserda, The Neherlands, 6. [6] J. T. Machado, V. Kiryakova, and F. Mainardi, Recen hisory of fracional calculus, Counicaions in Nonlinear Science and Nuerical Siulaion, vol. 6, no. 3, pp. 4 53,. View a Publisher View a Google Scholar [7] K. Deihel, N. J. Ford, e al., Analysis of fracional differenial equaion, Nuerical Analysis Repor 377, The Universiy of Mancheser, 3. View a Google Scholar [8] R. W. Ibrahi and S. Moani, On he eisence and uniqueness of soluions of a class of fracional differenial equaions, Journal of Maheaical Analysis and Applicaions, vol. 334, no., pp., 7. View a Publisher View a Google Scholar [9] I. Podlubny, Fracional Differenial Equaions, vol. 98, Acadeic Press Inc., San Diego, Calif, USA, 999. [] K. B. Oldha and J. Spanier, The Fracional Calculus, Acadeic Press, New York. NY, USA, 974. IJERTV5IS444 (This work is licensed under a Creaive Coons Aribuion 4. Inernaional License.) 565

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