Homotopy Analysis Method for Nonlinear Jaulent-Miodek Equation

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1 ISSN , England, UK Journal of Inforation and Coputing Science Vol. 5, No.,, pp Hootopy Analysis Method for Nonlinear Jaulent-Miodek Equation J. Biazar, M. Eslai Departent of Matheatics, Faculty of Sciences, University of Guilan (Received October, 9, accepted October, 9) Abstract. The hootopy analysis ethod (HAM) has been developed by Liao [-] to obtain series solutions of controllable convergence to various nonlinear probles. In this work, we propose this ethod (HAM), for solving Jaulent-Miodek (JM) equation [9-]. Nuerical solutions obtained by the hootopy analysis ethod are copared with the exact solutions. The results for soe values for the variables are shown in the tables and the solutions are presented as plots as well, showing the ability of the ethod Keywords: Hootopy analysis ethod, Jaulent-Miodek equation. Introduction Large varieties of physical, cheical, and biological phenoena are governed by nonlinear evolution equations. Except a liited nuber of these probles, ost of the do not have precise analytical solutions so that they have to be solved using other ethods. The hootopy analysis ethod (HAM) is a powerful analytical tool for nonlinear probles. This technique provides us with a siple way to ensure the convergence of the solution series, so that we can always get accurate enough approxiations. In recent years the application of analysis theory has appeared in any researches [-8]. In this paper, we propose hootopy analysis ethod (HAM), for solving Jaulent-Miodek (JM) equation [9-] 9 t x x x x With the initial condition of ux (,) = g(),(,) x vx = g(). x u u v v v u v u + + v + -6u -6uv - v = t x x x x x x x v v u v 5 v + -6 v-6u - v =.. Basic ideas hootopy analysis ethod To describe the basic ideas the hootopy analysis ethod, we consider the following differential equation, Nurt ((,)] =, () N is a nonlinear operator, r and t are independent variables, urt (,) is an unknown function, respectively. For siplicity, we ignore all boundary or initial conditions, which can be treated in the siilar way. By eans of generalizing the traditional hootopy ethod, Liao [] constructs the so-called zero - order deforation equation (-pl ) j(,; rtp) - u(,) rt = p HrtNrtp (,) (,; ), () ( ) p Î [,] is the ebedding paraeter, is a nonzero auxiliary paraeter, L is an auxiliary linear operator, u (,) r t is an initial guess of urt (,), j (,; rt p) is a unknown function on independent variables rtp,,., Corresponding author.. P.O. Box 4594, P.C , Rasht, Iran Tel: , Fax: E-ail address: biazar@guilan.ac.ir, eslai_ostafa@yahoo.co. Published by World Acadeic Press, World Acadeic Union

2 84 J. Biazar, et al: Hootopy Analysis Method for Nonlinear Jaulent-Miodek Equation It is iportant that one has great freedo to choose auxiliary paraeter in HAM. If p = and p =, it holds j(,;) rt = u(,), rt j(,;) rt = urt (,), () Thus, as p increases fro to, the solution (,; rtp) varies fro the initial guesses u (,) r t to the j solution ur (, t ). Expanding j(,; rtp), in Taylor series with respect to p, we have j(,; rtp) = u(,) rt +å u (,) rtp, (4) If the auxiliary linear operator, the initial guess, the auxiliary paraeter, and the auxiliary function are so properly chosen, the series (4) converges at p =, then we have = j(,; rtp) u (,) r t =. p=! p = urt (,) = u(,) rt +å u (,). rt (6) Define the vector u = { u, u, u }. Differentiating equation () ties with respect to the ebedding n n paraeter p and then setting p = and finally dividing the by!, we obtain the th -order deforation equation Lu [ - c u ] = H ( r, t ) R ( u -), (7) and Applying R - L - on both side of equation (7), we get - ( u-) =, - p= c ( - )! p ì, = >. î Nrtp (,; ) u r t = u r t + L H r t R u )]. () - (,) c (,) [ (,) ( - - In this way, it is easily to obtainu for ³,, at Mth -order, we have M urt (,) = å u (,). rt () = When M. we get an accurate approxiation of the original equation (). For the convergence of the above ethod we refer the reader to Liao []. If equation () adits unique solution, then this ethod will produce the unique solution. If equation () does not possess unique solution, the HAM will give a solution aong any other (possible) solutions.. Applications Consider the following Jaulent-Miodek equation [9-] 9 u u v v v u v u + + v + -6u - 6uv - v =, t x x x x x x x 5 v v u v v + -6 v-6u - v =. t x x x x With the following initial condition (5) (8) (9) () JIC eail for contribution: editor@jic.org.uk

3 Journal of Inforation and Coputing Science, Vol. 5 () No., pp u(,) x = c - b - b c sec h( c x)- c sec h ( c x), vx (,) = b + c sec h( cx). With the exact solution æ ö æ ö u ( c b ) b c sec h c ( x (6 b c ) t) c sec h = - - c ( x (6 b c ) t), ç è ø 4 è ç ø æ ö v = b + c sec h c ( x (6 b c ) t), ç + + çè ø b, c are arbitrary constants. To solve the equation () by eans of hootopy analysis ethod, according to the initial conditions denoted in equation (), it is natural to choose: æ ö æ ö u ( c b ) b c sec h c ( x (6 b c ) t) c sec h = - - c ( x (6 b c ) t), ç è ø 4 è ç ø (4) æ ö v = b + c sec h c ( x (6 b c ) t). ç + + çè ø We choose the linear operator with the property Lc [] =. ì j (,; xtp) L = t j (,; xtp) L = î t ( j ( x, t; p)], ( j ( x, t; p)]. Where c is constant. We now define a nonlinear operator as ì j (,; xtp) j (,; xtp) j (,; xtp) (,; ) (,; ) 9 j xtp j xtp N[ j ( x, t; p)] = + + j ( x, t; p) + t x x x x j (,; xtp) j (,; xtp) j (,; xtp) x x x j (,; xtp) j (,; xtp) j (,; xtp) j (,; xtp) N[ j ( x, t; p)] = + -6 j ( xtp, ; )-6 j ( xtp, ; ) t x x x 5 j (,; xtp) - j (,; xtp). î x 6 j ( xtp, ; ) 6 j ( xtp, ; ) j ( xtp, ; ) j ( xtp, ; ), Using above definition, with assuption Hxt (,) =. We construct the zeroth -order deforation equations ì ( -pl ) ( j ( xtp, ; ) - u( xt, ) ) = pn( j ( xtp, ; )), (7) ( -pl ) ( j (,; xtp) - v(,) xt ) = pn( j (,; xtp)). î Obviously, when p = and p =, j (,;) xt = u(,), xt j (,;) xt = v(,), xt (8) j (,;) xt = uxt (,), j (,;) xt = vxt (,). Thus, we obtain theth -order deforation equations Lu [ - c u ] = R ( u - Lv [ - c v ] = R ( v ), ), () (5) (6) (9) JIC eail for subscription: publishing@wau.org.uk

4 86 J. Biazar, et al: Hootopy Analysis Method for Nonlinear Jaulent-Miodek Equation ì U U - V - 9 V V - U k k -- k -- k R ( u- ) = åv + k å - åuk t x x k= x x k= x - -i- - -i- V k i U i -6 ååuv - VV, i k åå k -k-i- i= k= x i= k= x V V - U - V - -i- - - k 5 V --k i R ( v- ) = + -6 V 6 U V. å - --k å - k å å V k -k-i- î t x k= x k= x i= k= x Now, the solution of the th -order order deforation equation (9) - u (,) x t = c u (,) x t + L [ R ( u-)], - - v (,) x t = c v (,) x t + L [ R ( v-)]. Finally, we have - uxt (,) = u(,) xt + u (,), xt = vxt (,) = v(,) xt + v (,). xt å å = Fro equations (4) and () and subject to initial condition We obtain u = c - b - b c sec h( cx) - c sec h( cx), v = b + c sec h( c x). u (,) x = v (,) x =, ³. () 5 u =- c sec h( cx)tanh( cxt ) - 6bc sec h( cx)tanh ( cxt ) + bc sec h( cx)tanh( c) xt ) tbc sec h( cx)tanh( cx) - tb c sec h( cx)tanh( cx) -6tbc sec h( cx)tanh( cx), v =- 6c sec h( c x)tanh ( c x) t + c sec h( c x)tanh( c x) t + tb csec h( c x)tanh( c x) - 6tc sec h ( c x)tanh( c x), Hence, u= u + u + u +, v = v + v + v +. When =-, suppose u = åu, and v =, the results are presented in Table and Fig.. j åvj j= j= JIC eail for contribution: editor@jic.org.uk

5 Journal of Inforation and Coputing Science, Vol. 5 () No., pp Table The nuerical results, whenb = c =. and for solutions of Eqs. () for initial conditions (). x t u ( x, t) u u v ( x, t ) v v (a) (b) (c) Fig.. The nuerical results for u ( x, t), v ( x, t) are, respectively (a) and (c) in coparison with the analytical solutions u( x, t ) and v( x, t) are, respectively (b) and (d) with the initial conditions () of Eq (). when, b., c., and. 4. Conclusions In this article, we have applied hootopy analysis ethod for the solving the nonlinear Jaulent-Miodek (JM) equation. The approxiate solutions obtained by the hootopy analysis ethod are copared with (d) JIC eail for subscription: publishing@wau.org.uk

6 88 J. Biazar, et al: Hootopy Analysis Method for Nonlinear Jaulent-Miodek Equation exact solutions. The results show that the hootopy analysis ethod is a powerful atheatical tool for solving systes of nonlinear partial differential equations having wide applications in sciences and engineering. In our work, we use the aple package to carry the coputations. 5. References [] S.J. Liao. The proposed hootopy analysis technique for the solution of nonlinear probles. PhD Thesis, Shanghai Jiao Tong University, 99. [] S.J. Liao. Beyond perturbation: an introduction to hootopy analysis ethod. Boca Raton: Chapan Hall/CRC Press,. [] S.J. Liao. A kind of approxiate solution technique which does not depend upon sall paraeters: a special exaple. Int J Non-Linear Mech. 995, (7): 8. [4] S.J. Liao, A.T. Chwang. Application of hootopy analysis ethod in nonlinear oscillations. ASME J Appl Mech. 998, 65(9):. [5] S.J. Liao. An analytic approxiate technique for free oscillations of positively daped systes with algebraically decaying aplitude. Int J Non-Linear Mech., 8(7): 8. [6] S.J. Liao. On the analytic solution of agneto hydrodynaic flows of non-newtonian fluids over a stretching sheet. J Fluid Mech., 488(89):. [7] S.J. Liao. On the hootopy analysis ethod for nonlinear probles. Appl Math Coput. 4, 47(499): 5. [8] S. Abbasbandy. The application of hootopy analysis ethod to solve a generalized Hirota Satsua coupled KdV equation. Phys Lett A. 7, 6(478): 8. [9] D. Kaya, S.M. El-Sayed. A nuerical ethod for solving Jaulent-Miodek equation. Physics Letters A., 8(45): 5. [] Engui Fan. Uniforly constructing a series of explicit exact solutions to nonlinear equations in atheatical physics. Chaos, Solitons and Fractals., 6(89): 9. JIC eail for contribution: editor@jic.org.uk