Homotopy perturbation method for the Wu-Zhang equation in fluid dynamics

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1 Journal of Physics: Conference Series Homotopy perturbation method for the Wu-Zhang equation in fluid dynamics To cite this article: Z Y Ma 008 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - Application of the homotopy perturbation method for solving second-order nonlinear wave equations A Janalizadeh, A Barari and D D Ganji - Application of homotopy perturbation method to the Zakharov-Kuznetsov equation A Barari, A Janalizadeh, D D Ganji et al. - Application of homotopy perturbation method to the MHD pipe flow of a fourth grade fluid A Ranjbar Noiey, N Haghparast, M Miansari et al. This content was downloaded from IP address on 4/08/08 at 07:

2 007 International Symposium on Nonlinear Dynamics (007 ISND) IOP Publishing Journal of Physics: Conference Series 96 (008) 08 doi:0.088/ /96//08 Homotopy perturbation method for the Wu-Zhang equation in fluid dynamics Z.Y. Ma ¹Department of Mathematics, Zhejiang Lishui University, Zhejiang Lishui 000, China Abstract:He's homotopy perturbation method, which does not need a small parameter in an equation, is implemented to finding the soliton solutions of the Wu-Zhang equation arising in fluid dynamics. In this method, a homotopy is first constructed for the equation. An initial approximation can be chosen with possible unknown constants that can be determined by imposing the boundary and initial conditions. The results reveal that the method is very effective, convenient and quite accurate to systems of nonlinear equations.. Introduction It is well known that nonlinear phenomena are very important in a variety of scientific fields, especially in fluid mechanics, solid state physics, plasma physics, plasma waves and chemical physics. In many different fields of science and engineering, it is very important to obtain exact or numerical solutions of nonlinear partial differential equations. Searching for exact and numerical solutions, especially, for traveling wave solutions, of nonlinear equations in mathematical physics plays an important role in soliton theory [,]. Recently, many new approaches to nonlinear equations were proposed, such as Bäcklund transformation [], Hirota's bilinear method [], the homogeneous balance method [4], the Riccati expansion method [5] and the variational iteration method [6,7]. Among all of the analytical methods in open literature, the homotopy perturbation method (HPM) [8], proposed first by He in 998 and was further developed and improved by He [9-]. The method yields a very rapid convergence of the solution series in the most cases. The main application of the HPM shows miraculous exactness and convenience compared to other methods [-5]. In this paper, we apply He's homotopy perturbation method to find the approximation solution and numerical solutions for the following Wu-Zhang (WZ) equation [6]: ut uux vuy wx = 0, v uv vv w = 0, t x y y wt ( uw) x ( vw) y ( uxxx uxyy vxxy vyyy ) = 0, () To whom any correspondence should be addressed. c 008 IOP Publishing Ltd

3 007 International Symposium on Nonlinear Dynamics (007 ISND) IOP Publishing Journal of Physics: Conference Series 96 (008) 08 doi:0.088/ /96//08 which is also called as the ()-dimensional dispersive long wave equation (here u u( x, y, t v v( x, y, t) and w w( x, y, t) ). The WZ equation describes nonlinear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth. In Eq. ( w is the elevation of the water wave, u is the surface velocity of water along the x direction, and v is the surface velocity of water along the y direction. The explicit solutions of Eq. () are very helpful for costal and civil engineers to apply the nonlinear water wave model in harbor and coastal design. Therefore, finding explicit solutions and numerical results of Eq. () are of fundamental interest in fluid dynamics. In [7], Chen, Tang and Lou have proven that Eq. () has no Painlevé property, and presented two types of periodic wave solutions as well as the corresponding solitary wave solutions by using extended Painlevé truncated expansion method.. Basic idea of He's homotopy perturbation method To illustrate the basic ideas of this method, we consider the following nonlinear differential equation [0]: Au ( ) f( r) = 0, r Ω, () with the boundary conditions Bu (, u/ n) = 0, r Γ, () where A is a general differential operator, B is a boundary operator, f () r is a known analytical function, Γ is the boundary of the domain Ω and / n denoted differential along the normal drawn outwards from Ω. Generally speaking, the operator A can be divided into two parts L and N, where L is linear, but N is nonlinear. Equation () can therefore be rewritten as follows: Lu ( ) Nu ( ) f( r) = 0. (4) or By the homotopy technique, we construct a homotopy 0 V(, r p): Ω [0,] which satisfies HV (, p) = ( p)[ LV ( ) Lu ( )] pav [ ( ) f( r)] = 0, r Ω, (5) HV (, p) = LV ( ) Lu ( ) plu ( ) pnv ( ( ) f( r)) = 0, (6) 0 0 where p [0,] is an embedding parameter, u0 is an initial approximation of Eq. ( which satisfies the boundary conditions. Hence, it is obvious that HV (,0) = LV ( ) Lu ( ) = 0, (7) 0 HV (,) = AV ( ) f( r) = 0, (8) and the changing process of p from zero to unity is just that of HV (, p) from LV ( ) Lu ( 0) to AV ( ) f( r). In topology, this is called deformation, and L( V) Lu ( 0) and Av () f() r are called homotopy. According to the HPM, we can first use the embedding parameter p as a small parameter, and assume that the solution of Eqs. (5) or (6) can be written as a power series in p :

4 007 International Symposium on Nonlinear Dynamics (007 ISND) IOP Publishing Journal of Physics: Conference Series 96 (008) 08 doi:0.088/ /96//08 V = V pv p V 0. Setting p results in the approximate solution of Eq. (): u = lim V = V V V. p 0 (9) (0) The combination of the perturbation method and the homotopy method is called the homotopy perturbation method (HPM which has eliminated the limitations of the traditional perturbation methods. On the other hand, this technique can have full advantage of the traditional perturbation techniques.. Analysis of the method To investigate the solitary wave solution of Eq. () through the HPM, we first construct a homotopy as follows: ( p)( v u ) p( v vv v v v ) = 0,, t 0, t, t, x, y, x ( p)( v, t v0, t) p( v, t vv, x vv, y v, y) = 0, () ( p)( v, t w0, t ) p( v, t ( vv ) x ( vv) y ( v, xxx v, xyy v, xxy v, yyy )) = 0, and the initial approximations are as follows: v ( x, y, t) = u ( x, y, t) = u( x, y,0 0 0 v ( x, y, t) = v ( x, y, t) = v( x, y,0 0 0 () and v ( x, y, t) = w ( x, y, t) = w( x, y,0 0 0 v = v pv p v p v p v 4 0 4, v = v pv pv pv pv 4 0 4, () v = v pv p v p v p v 4 0 4, where v ij ( i =,,, j =,,,4, ) are functions of the independent variables {, x yt,} yet to be determined. Substituting Eqs. () and () into Eq. () and arranging the coefficients of p powers, we have: ( v, t v0v0, x v0v0, y v0, t v0, x) p ( vv vv vv v vv v ) p 0, x 0, y 0, x, t 0, y, x ( vv0, x vv, y v, t v0v, x v, x v0v, y vv, x vv0, y) p = 0,

5 007 International Symposium on Nonlinear Dynamics (007 ISND) IOP Publishing Journal of Physics: Conference Series 96 (008) 08 doi:0.088/ /96//08 ( v v v v v v v ) p, t 0 0, x 0 0, y 0, t 0, y ( vv vv vv v vv v ) p 0, x 0, y 0, x, t 0, y, y ( vv0, x vv, y v, t v0v, x v, y v0v, y vv, x vv0, y) p = 0, (4) ( v0, xv0 v0v0, y v0, yv0 v, t v0, t v0v0, x ( v0, xyy v0, yyy v0, xxx v0, xxy )) p ( v0, yv v, t v0v, x v0v, y vv0, x v, yv0 vv0, y v0v, x v0, xv ( v, xxx v, xyy v, yyy v, xxy )) p ( v0, xv v, yv0 vv, y v, t vv0, x v, yv0 vv, x v0v, x v0, yv v, xv v0, yv vv, y v, xv0 ( v, xxx v, yyy v, xyy v, xxy )) p = 0. In order to obtain the unknowns of vij ( x, y, t ) (, i j =,, we must construct and solve the following system which includes nine equations with nine unknowns, considering the initial conditions of u ( i0 x, y,0) ( i =,, and having the initial approximations of Eq. (): v, t v0v0, x v0v0, y v0, t v0, x = 0, vv vv vv v vv v = 0, 0, x 0, y 0, x, t 0, y, x v v v v v v v v v v v v v v 0, x, y, t 0, x, x 0, y, x 0, y = 0, v v v v v v v = 0,, t 0 0, x 0 0, y 0, t 0, y vv vv vv v vv v = 0, 0, x 0, y 0, x, t 0, y, y v v v v v v v v v v v v v v = 0, 0, x, y, t 0, x, y 0, y, x 0, y (5) v0, xv0 v0v0, y v0, yv0 v, t v0, t v0v0, x ( v0, xyy v0, yyy v0, xxx v0, xxy ) = 0, v v v v v v v v v v v v v v v v v 0, y, t 0, x 0, y 0, x, y 0 0, y 0, x 0, x ( v, xxx v, xyy v, yyy v, xxy ) =0, v v v v v v v v v v v v v v v v v v v 0, x, y 0, y, t 0, x, y 0, x 0, x 0, y, x v0, yv vv, y v, xv0 ( v, xxx v, yyy v, xyy v, xxy ) = 0. From Eq. ( if the first three approximations are sufficient, we will obtain: uxyt (,, ) lim v( xyt,, ) v ( xyt,, = k p k = 0 4

6 007 International Symposium on Nonlinear Dynamics (007 ISND) IOP Publishing Journal of Physics: Conference Series 96 (008) 08 doi:0.088/ /96//08 vxyt (,, ) lim v( xyt,, ) v ( xyt,, p = (6) k k = 0 wxyt (,, ) lim v( xyt,, ) v ( xyt,, ). = p k k = 0 4. Application Firstly, we consider the solutions of Eq. () with the initial conditions: k k b uxy k kx ky 0 (,,0) = tanh( k vxy (,,0) = b0 k tanh( kx ky (7) wxy (,,0) = ( k k)sech ( kx ky where b0, k, k and k are arbitrary constants. To calculate the terms of the homotopy series (6) for ux (, yt, vxyt (,, ) and wxyt (,, we substitute the initial conditions (7) into the system (5) and using Maple, the solutions of the equations can be obtained as follows: k k b v ( x, y, t) = u( x, y,0) = k tanh( k x k y 0 0 k v( x, y, t) = kk sech ( kx k y) t, v( x, yt, ) = kk tanh( kx ky)sech ( kx kyt ), v( x, yt, ) = kk (tanh ( kx k y) )sech ( kx k yt ), 9 v0( x, y, t) = v( x, y,0) = b0 k tanh( kx k y v( x, y, t) = kk sech ( kx k y) t, v( x, yt, ) = kk tanh( kx ky)sech ( kx kyt ), v( x, yt, ) = kk (tanh ( kx k y) )sech ( kx k yt ), 9 5

7 007 International Symposium on Nonlinear Dynamics (007 ISND) IOP Publishing Journal of Physics: Conference Series 96 (008) 08 doi:0.088/ /96//08 v0( x, y, t) = w( x, y,0) = ( k k )sech ( kx k y 4 v( x, y, t) = ( k k ) k tanh( kx k y)sech ( kx k y) t, v( x, y, t) = ( k k ) k (tanh ( kx k y) )sech ( kx k y) t, 8 v( x, y, t) = ( k k ) k tanh( kx k y) (tanh ( kx ky) )sech ( kx kyt ). 9 In this manner, the other components can be obtained. Substituting the above components into Eq. (6) and using the Taylor series, we obtain the closed form solutions as follows: k k b uxyt = k kx ky kt 0 (,, ) tanh( k vxyt (,, ) = b0 k tanh( kx ky kt (8) wxyt (,, ) = ( k k)sech ( kx ky kt where b0, k, k and k are arbitrary constants. With the initial conditions (7 the solitary wave solutions of Eq. () of kink-type for uxyt (,, ) and vxyt (,, ) and bell-type for wxyt (,, ). Table : The HPM results for uxyt (,, vxyt (,, ) and wxyt (,, ) for the first thirty approximation in comparison with the analytical solutions when b0 = 0., k = 0.05, k = 0., k = 0. and t =.5 for the solitary wave solutions with the initial conditions (7) of Eq. ( respectively. ( x, y ) u uhom i i exact otopy vexact vhomotopy wexact whomotopy (0.00,-40) (0.00,-0) (0.00,-0) (0.004,-0) (0.005,0) (0.006,0) (0.007,0) (0.008,0) 0.40E E E E E E E E E E E- 0.0E E E E E E E- 0.70E E E-0 0.4E E- 0.0E- 5. Numerical results of the HPM To demonstrate the convergence of the HPM, the results of the numerical example are presented and only few terms are required to obtain accurate solutions. The accuracy of the HPM for the Wu-Zhang equation is controllable, and absolute errors are very small with the present choice of x, y and t. The results are listed in Table. Moreover, as the decomposition method does not require discretization of the variables time and space, it is not affected by computation round off errors and one is not faced with the necessity of large computer memory and time. For an initial condition, we achieve a very good approximation to the partial exact solution by using only 0 terms of the decomposition series, 6

8 007 International Symposium on Nonlinear Dynamics (007 ISND) IOP Publishing Journal of Physics: Conference Series 96 (008) 08 doi:0.088/ /96//08 which shows that the speed of convergence of this method is very fast. It is evident that the overall errors can be made smaller by adding new terms of the decomposition series. 6. Summary and discussion In this paper, we considered a numerical treatment for the solutions of the WZ equation using the HPM. To the best of our knowledge, this is the first result on the application of the approach to this equation. This method transforms Eq. () into a recursive relation. The obtained numerical results compared with the analytical solution show that the method provides remarkable accuracy. Generally speaking, the HPM provides analytic, verifiable, rapidly convergent approximation that yields insight into the character and the behavior of the solution just as in the closed form solution. It solves nonlinear problems without requiring linearization, or unjustified assumption that may change the problem being solved. The method can also easily be extended to other similar physical equations, with the aid of Maple (or Matlab, Mathematica, etc. the course of solving nonlinear evaluation equations can be carried out in a computer. References [] Debtnath L 997 Nonlinear partial differential equations for scientists and engineers. Birkhäuser, Boston [] Wazwaz A M 00 Partial differential equations: Methods and applications. Balkema, Rotterdam [] Hirota R 97 Phys. Rev. Lett. 7 9 [4] Wang M L 996 Phys. Lett. A 79 [5] Yan Z Y and Zhang H Q 00 Phys. Lett. A [6] He J H 997 Commun Nonliear. Sci. 5 [7] He J H 998 Comput. Method. Appl. M [8] He J H 999 Comput. Method. Appl. M [9] He J H 998 Comput. Method. Appl. M [0] He J H 000 Int. J. Nonlinear. Mech.5 7 [] He J H 004 Appl. Math. Comput [] He J H 005 Chaos. Soliton. Fract [] He J H 004 Appl. Math. Comput [4] He J H 006 Phys. Lett. A [5] Abbasbandy S 006 Chaos. Soliton. Fract.0 06 [6] Wu T Y and Zhang J E 996 On modeling nonlinear long wave. PA: SIAM, Philadelphia [7] Chen C L, Tang X Y and Lou S Y 00 Phys. Rev. E