Series Solutions with Convergence-Control. Parameter for Three Highly Non-Linear PDEs: KdV, Kawahara and Gardner equations

Transcription

1 Applied Matheatical Sciences, Vol. 5, 211, no. 21, Series Solutions with Convergence-Control Paraeter for Three Highly Non-Linear PDEs: KdV, Kawahara and Gardner equations Saeed Dinarvand 1,2, Soroush Khosravi, Hasan Khoosheh 4 and Mohsen Nasrollahzadeh 5 1. Mechanical Engineering Departent, Engineering Faculty of Bu-Ali Sina University, Haedan, Iran 2. Islaic Azad University, Central Tehran Branch Young Researchers Club, Tehran, Iran. Inforation Technology Manageent Departent Manageent Faculty of Mehralborz University, Tehran, Iran 4. Islaic Azad University, Arak Branch Young Researchers Club, Arak, Iran 5. Electrical Engineering Departent Islaic Azad University, South Tehran Branch, Tehran, Iran Abstract Here, an analytic ethod, naely the hootopy analysis ethod (shortly HAM), is applied to solve the KdV, Kawahara and Gardner equations. The HAM is a strong and easy-to-use analytic tool for nonlinear probles and dose not need sall paraeters in the equations. This ethod contains the auxiliary paraeter h, which provides us with a siple way to adjust and control the convergence region of solution series. The obtained solutions, in coparison with the exact solutions adit a rearkable accuracy. E-ail address: saeed_dinarvand@yahoo.co

2 18 Saeed Dinarvand et al Keywords: Series solution, Non-linear PDEs, KdV equation; Kawahara equation; Gardner equation 1. Introduction Finding explicit analytic solutions of nonlinear differential equations is extreely iportant in atheatical physics. In recent years, any powerful ethods have been developed to construct explicit analytic solution of nonlinear differential equations. In 1992, Liao [1] eployed the basic ideas of the hootopy in topology to propose ethod for nonlinear probles, naely hootopy analysis ethod (HAM), [2 6]. This ethod has any advantages over the classical ethods, ainly, it is independent of any sall or large quantities. So, the HAM can be applied no atter if governing equations and boundary/initial conditions contain sall or large quantities or not. The HAM also avoids discretization and provides an efficient nuerical solution with high accuracy, inial calculation, and avoidance of physically unrealistic assuptions. Furtherore, the HAM always provides us with a faily of solution expressions in the auxiliary paraeter h, the convergence region and rate of each solution ight be deterined conveniently by the auxiliary paraeter h. This ethod has been successfully applied to solving any types of nonlinear probles [7 11]. In this Letter, we extend the application of the hootopy analysis ethod to construct approxiate solutions for the KdV, Kawahara and Gardner equations. A substantial aount of research work has been directed for the study of the nonlinear KdV, Kawahara and Gardner equations given by and u + 6uu + u =, (1) t x xxx ut + uux + uxxx uxxxxx = (2) 2 t x x xxx u + 6uu 6u u + u =, () respectively. The well known KdV equation had been derived in 1885 by the two scientists Korteweg and de Vries to describe long wave propagation on shallow water. Although this equation was known since the last century, but its physical behaviour is still ysterious. This equation has any other direct physical applications to solids, liquids, gases and plasa: agnetohydrodynaic waves in a cold plasa [12], longitudinal waves propagating in a one diensional lattice of equal asses coupled by nonlinear springs, the Feri et al. proble [1,14], ion acoustic waves in a cold plasa [15], rotating flow in tube [16] and longitudinal dispersive waves in elastic rods [17].

3 Series solutions with convergence-control paraeter 19 The Kawahara equation is a fifth order KdV equation, which occurs in the theory of agneto-acoustic waves in a plasas [18] and in the theory of shallow water waves with surface tension [19]. The Gardner equation (cobined KdV-KdV or ekdv equation ) is a useful odel for the description of internal solitary waves in shallow water [2] and have been widely studied by the various ethods. The copetition aong dispersion, quadratic and cubic nonlinearities constitutes the ain interest. This Letter has been organized as follows. In Section 2, the basic idea of the HAM is introduced. In Section, we extend the application of the HAM to construct approxiate solutions for the KdV, Kawahara and Gardner equations. Results are presented in Section Basic idea of the HAM Let us consider the following differential equation [ ] N u ( τ ) =, (4) where N is a nonlinear operator,τ denotes independent variable, u ( τ ) 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 [5] constructs the so-called zero-order deforation equation [ ϕτ τ ] τ [ ϕτ ] (1 p ) L ( ; p ) u ( ) = phh ( ) N ( ; p ), (5) where p [,1] is the ebedding paraeter, h is a non-zero auxiliary paraeter, H ( τ ) is an auxiliary function, L is an auxiliary linear operator, u ( τ ) is an initial guess of u ( τ ), ϕ( τ ; p ) is a unknown function, respectively. It is iportant, that one has great freedo to choose auxiliary things in HAM. Obviously, when p = and p = 1, it holds ϕ( τ;) = u ( τ), ϕ( τ;1) = u( τ), respectively. Thus as p increases fro to 1, the solution ϕ( τ ; p) varies fro the initial guess u ( τ ) to the solution u ( τ ). Expanding ϕ( τ ; p) in Taylor series with respect to p, we have + ϕτ ( ; p ) = u ( τ) + u ( τ) p, (6) where u = 1 1 ϕτ ( ; p ) ( τ ) =.! p If the auxiliary linear operator, the initial guess, the auxiliary paraeter h, and the auxiliary function are so properly chosen, the series (4) converges at 1, p = then we have p = (7)

4 14 Saeed Dinarvand et al + u( τ ) = u ( τ) + u ( τ), (8) = 1 which ust be one of solutions of original nonlinear equation, as proved by Liao [5]. As h = 1 and H ( τ ) = 1, equation () becoes [ ϕτ τ ] [ ϕτ ] (1 p) L ( ; p) u ( ) + pn ( ; p) =, (9) which is used ostly in the hootopy perturbation ethod, where as the solution obtained directly, without using Taylor series [21,22]. According to the definition (5), the governing equation can be deduced fro the zeroorder deforation equation (). Define the vector r u = u ( τ ), u ( τ), K, u ( τ). n { } 1 Differentiating equation () ties with respect to the ebedding paraeter p and then setting p = and finally dividing the by!, we have the so-called th-order deforation equation L u r ( τ) χ u ( τ) = hh ( τ) R ( u ), (1) where and R [ ] n [ ϕτ p ] r 1 N ( ; ) ( u 1) =, 1 ( 1)! p χ {, 1, = 1, >1. It should be ephasized that u ( τ ) for 1 is governed by the linear equation (8) with the linear boundary conditions that coe fro original proble, which can be easily solved by sybolic coputation software such as Maple and Matheatica. p = (11). Application.1. The solution with HAM In this Section, we apply the hootopy analysis ethod for the KdV, Kawahara and Gardner equations. We start with initial approxiation u ( x, t) = u( x,) and the linear operator ϕ( x, t; p) L [ ϕ( x, t; p) ] =, (12) t possesses the property L (c ) =, (1) 1

5 Series solutions with convergence-control paraeter 141 where c 1 is an integral constant to be deterined by initial condition. Furtherore, equations (1), (2) and () suggest to define the nonlinear operators N Kawahara ϕ( x, t; p) ϕ( x, t; p) ϕ( x, t; p) N KdV [ ϕ( x, t; p) ] = + 6 ϕ ( x, t; p) +, (14) t 5 ϕ( x, t; p) ϕ( x, t; p) ϕ( x, t; p) ϕ( x, t; p) [ ϕ( x, t; p) ] = + ϕ ( x, t; p) + 5 t (15) and ϕ( x, t; p) ϕ( x, t; p) N Gardner [ ϕ( x, t; p) ] = + 6 ϕ( x, t; p) t 2 ϕ( x, t; p) ϕ( x, t; p) 6 ( ϕ ( x, t; p) ) +, respectively. Using the above definition, with assuption H ( τ ) = 1, we construct the zeroorder deforation equation Obviously, when p = and p = 1, [ ϕ ] [ ϕ ] (16) (1 p) L ( x, t; p) u ( x, t) = phn ( x, t; p). (17) ϕ( x, t;) = u ( x, t), ϕ( x, t;1) = u( x, t). (18) Differentiating the zero-order deforation equation (17) ties with respect to p, and finally dividing by!, we have the th-order deforation equation r L u ( x, t) χ u ( x, t) = hr ( u ), (19) subject to initial condition [ ] 1 1 u ( x,) =, (2) where 1 r u 1( x, t) u 1 n( x, t) u 1( x, t) R ( u 1) 6 un( x, t), KdV = + + t n = u ( x, t) u ( x, t) R u u x t 1 r 1 1 n ( 1) n(, ) Kawahara = + t n = u ( x, t) u ( x, t) +, 5 1 r 1 1 n ( 1) 6 n(, ) Gardner = + t n = u ( x, t) u ( x, t) R u u x t 1 n u 1 n( x, t) u 1( x, t) 6 u j ( x, t) un j ( x, t) + n= j= And (21) (22) (2)

6 142 Saeed Dinarvand et al χ {, 1, = 1, >1. Obviously, the solution of the th-order deforation equation (19) for 1 becoes r u x t u x t h R u [ ] 1 (, ) = χ 1(, ) + L ( 1 ). (24) In the following parts, we consider the initial approxiations and deterine other coponents of the solution series for the KdV, Kawahara and Gardner equations The KdV equation For the KdV equation, we choose the initial approxiation 2 2 u ( x, t) = u( x,) = 2k sech [ kx], (25) where k is arbitrary constant and k. Using previous forulae to deterine other coponents of the solution series. Fro (21), (24) and (25), by the Matheatica package, we have 5 2 u1( x, t) = 16hk t sech [ kx ]tanh[ kx ], u2 ( x, t) 8hk tsech [ kx ]( 4 hk t( 2 cosh[2 kx]) (1 h)sinh[2 kx] ), (26) 5 4 = + + (27) u( x, t) = hk tsech [ kx] ( 24 h(1 + h) k t cosh[2 kx] ( + h(6+ h + 2 hk t )) sinh[2 kx] + 48 hk t(1+ h 4hk t tanh[ kx]), M ) (28) We used 15 ters in evaluating the approxiate solution u app KdV = 14 i = ui The Kawahara equation For the Kawahara equation, we choose the initial approxiation

7 Series solutions with convergence-control paraeter x u ( x, t) = u( x,) = sech (29) Fro (22), (24) and (29), we can obtain the following coponents 756ht x x u1( x, t) sech tanh, = 4 () 78ht 6 x x x u2 ( x, t) = sech 54ht 8ht cosh 169 1( 1 h) sinh, (1) 756ht x x u( x, t) = sech h(1 + h) t (1 + h) + 864h t tanh sech2 x x ht 2197(1 h) 18 1ht tanh, M 4 ( 2 2 2) We used 1 ters in evaluating the approxiate solution u app Kawahara = 9 i = ui. (2).1.. The Gardner equation For the Gardner equation, we choose the initial approxiation 1 1 x u ( x, t) u( x,) tan h () Therefore, fro (2), (24) and (), we can obtain the following coponents ht u1( x, t) =, 21 ( + cosh[ x ]) (4) ht x u2 ( x, t) = 2 2h ht tanh +, 41 ( cosh[ x ]) 2 + (5) 1 4 x u( x, t) = ht sech ( 6(1 h) 2h t ( 6 h( 12 h(6 t ))) cosh[ x] 6 h(1 h) t sinh[ x] ), M We used 1 ters in evaluating the approxiate solution u 4. Results and discussion app Gardner = 9 i = ui. The series solutions contain the auxiliary paraeter h. The validity of the ethod is based on such an assuption that the series (6) converges at p = 1. It is the auxiliary paraeter h which ensures that this assuption can be satisfied. In general, by eans of (6)

8 144 Saeed Dinarvand et al the so-called h-curve, it is straightforward to choose a proper value of h which ensures that the solution series is convergent. Figs. 1 show the h-curves obtained fro the 14thorder approxiate solution of the KdV equation and 9th-order approxiate solutions of the Kawahara and Gardner equations. Fro these figures, the valid regions of h correspond to the line segents nearly parallel to the horizontal axis. u ttt (1, ) h Fig. 1. The h-curve of u ttt (1, ) given by the 14th-order approxiate solution of KdV equation, when k =.275. u tttt (1, ) h Fig. 2. The h-curve of u tttt (1, ) given by the 9th-order approxiate solution of Kawahara equation.

9 Series solutions with convergence-control paraeter 145 u tt (1, ) h Fig.. The h-curve of u tt (1, ) given by the 9th-order approxiate solution of Gardner equation. For approxiate solution of the KdV equation, we choose the auxiliary paraeter h =.96 and for both approxiate solutions of the Kawahara and Gardner equations, we choose the auxiliary paraeter h = 1 by h-curves. In continuation, we copare approxiate solutions of the KdV, Kawahara and Gardner equations, with exact solutions ( ) 2 2 u (, ) 2 sech 4 2 ex KdV x t = k k x k t, k, (7) ( ) u ex ( x, t ) = kawahara sech x 169 t + (8) and u ex ( x, t ) = Gardner tanh ( ), x t (9) respectively. Tables 1 show the absolute errors for differences between the exact solutions and the approxiate solutions obtained by HAM at soe points. Besides, the behavior of the exact and approxiate solutions are illustrated in Figs Note that when h = 1, the solution series obtained by the HAM is as the sae solution series obtained by the hootopy perturbation ethod (shortly HPM), which proposed in 1998 by Dr. He [2]. x Table 1 Absolute errors for the 14th-order approxiate solution of the KdV equation given by HAM for h =.96, when k =.2. t

10 146 Saeed Dinarvand et al x Table 2 Absolute errors for the 9th-order approxiate solution of the Kawahara equation given by HAM for h = 1. t x Table Absolute errors for the 9th-order approxiate solution of the Gardner equation given by HAM for h = 1. t Conclusions In this Letter we solved soe probles by the hootopy analysis ethod. It can be concluded: 1. The HAM is very powerful and efficient ethod in finding analytical solutions for wide classes of nonlinear probles. 2. The HAM provides us with a convenient way to control the convergence of approxiation series, which is a fundaental qualitative difference in analysis between the HAM and other ethods.. The HAM can be applied no atter if governing equations and boundary/initial conditions contain sall or large quantities or not. 4. The HAM provides an efficient nuerical solution with high accuracy, inial calculation, avoidance of physically unrealistic assuptions.

11 Series solutions with convergence-control paraeter 147 (a) (b) Fig. 4. The surfaces show the solutions obtained by: (a) HAM for h =.96; solution, when k =.2, for the KdV equation. (b) exact (a) (b) Fig. 5. The surfaces show the solutions obtained by: (a) HAM for h = 1; (b) exact solution, for the Kawahara equation. (a) (b) Fig. 6. The surfaces show the solutions obtained by: (a) HAM for h = 1; (b) exact solution, for the Gardner equation.

12 148 Saeed Dinarvand et al References [1] S.J. Liao, The proposed hootopy analysis technique for the solution of nonlinear probles, Ph.D. Thesis, Shanghai Jiao Tong University, [2] S.J. Liao, A uniforly valid analytic solution of 2D viscous flow past a seiinfinite flat plate, J. Fluid Mech. 85 (1999) [] S.J. Liao, An explicit, totally analytic approxiation of Blasius viscous flow probles, Int. J. Nonlinear Mech. 4 (1999) [4] S.J. Liao, On the analytic solution of agnetohydrodynaic flows of non- Newtonian fluids over a stretching sheet, J. Fluid Mech. 488 (2) [5] S.J. Liao, Beyond perturbation: Introduction to the Hootopy Analysis Method, CRC Press/chapan Hall, Boca Raton, 2. [6] S.J. Liao, On the hootopy analysis ethod for nonlinear probles, Appl. Math. Coput. 147 (24) [7] S. Dinarvand, A reliable treatent of the hootopy analysis ethod for viscous flow over a non-linearly stretching sheet in presence of a cheical reaction and under influence of a agnetic field, Cent. Eur. J. Phys. 7(1) (29) [8] S. Dinarvand, On explicit, purely analytic solutions of off-centered stagnation flow towards a rotating disc by eans of HAM, Nonlinear Anal. Real World Appl. 11 (21) [9] S. Dinarvand, M. M. Rashidi, A reliable treatent of hootopy analysis ethod for two-diensional viscous flow in a rectangular doain bounded by two oving porous walls, Nonlinear Anal. Real World Appl. 11 (21) [1] S. Dinarvand, The lainar free-convection boundary-layer flow about a heated and rotating down-pointing vertical cone in the presence of a transverse agnetic field, Int. J. Nuer. Meth. Fluids, In Press (21). [11] S. Dinarvand, A. Doosthoseini, E. Doosthoseini, M. M. Rashidi, Series solutions for unsteady lainar MHD flow near forward stagnation point of an ipulsively rotating and translating sphere in presence of buoyancy forces, Nonlinear Anal. Real World Appl. 11 (21) [12] C.S. Gardner, G.K. Morikawa, Siilarity in the asyptotic behavior of collision-free hydroagnetic waves and water waves. New York Univ, Courant Inst. Math. Sci. Res. Rep. NYO-982, 196. [1] E. Feri, J. Pasta, S. Ula, Studies of nonlinear probles, I. Los Alaos Report LA 194; [14] M.D. Kruskal, N.J. Zabusky, Progress on the Feri Pasta Ula nonlinear string proble. Princeton Plasa Physics Laboratory Annual Rep. MATT-Q- 21, Princeton, NJ, (196) 1 8. [15] T. Taniuti, H. Washii, Propagation of ion-acoustic solitary waves of sall aplitude, Phys. Rev. Lett. 17 (1966) [16] S. Leibovich, Weakly non-linear waves in rotating fluids, J Fluid Mech. 42 (197) [17] G.A. Nariboli, Nonlinear longitudinal dispersive waves in elastic rods. Iowa State Univ. Engineering Res. Inst Preprint, p. 442.

13 Series solutions with convergence-control paraeter 149 [18] T.J. Kawahara, Oscillatory solitary waves in dispersive edia, Phys. Soc. Jpn. (1972) [19] J.K. Hunter, J. Scheurle, Existence of perturbed solitary wave solutions to a odel equation for water waves, Physica D 2 (1988) [2] R. Grishaw, D. Pelinovsky, T. Talipova, A. Kurkin, Siulation of the transforation of internal solitary waves on oceanic shelves, J. Phys. Oceanogr. 4 (24) [21] J.H. He, A coupling ethod for hootopy technique and perturbation technique for non-linear proble, Int. J. Nonlinear. Mech. 5 (2) 7 4. [22] J.H. He, Hootopy perturbation ethod for solving boundary value probles, Phys. Lett. A 5 (26) Received: Septeber, 21