In this paper, we study the nonlinear dispersive K(m,n) equations which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the homotopy analysis method in K(m,n) equations. The nonlinear equations K(m,n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m,n) equations are established

Mohyud-Din, Syed T.

Yıldırım, Ahmet

Ünlü, Canan

English

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Applications and Applied Mathematics: An International Journal (AAM) Volume 5 Issue 3 Special Issue 1 – August 2010 Article 7 8-2010 Exact solitary-wave Special Solutions for the Nonlinear Dispersive K(m,n) Equations by Means of the Homotopy Analysis Method Ahmet Yıldırım Ege University Canan Ünlü İstanbul University Syed T. Mohyud-Din HITEC University Follow this and additional works at: https://digitalcommons.pvamu.edu/aam  Part of the Partial Differential Equations Commons Recommended Citation Yıldırım, Ahmet; Ünlü, Canan; and Mohyud-Din, Syed T. (2010). Exact solitary-wave Special Solutions for the Nonlinear Dispersive K(m,n) Equations by Means of the Homotopy Analysis Method, Applications and Applied Mathematics: An International Journal (AAM), Vol. 5, Iss. 3, Article 7. Available at: https://digitalcommons.pvamu.edu/aam/vol5/iss3/7 This Article is brought to you for free and open access by Digital Commons @PVAMU. It has been accepted for inclusion in Applications and Applied Mathematics: An International Journal (AAM) by an authorized editor of Digital Commons @PVAMU. For more information, please contact hvkoshy@pvamu.edu. 82  Available at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466  Special Issue No. 1 (August 2010)  pp. 82 – 93  Applications and Applied Mathematics:  An International Journal (AAM)     Exact solitary-wave Special Solutions for the Nonlinear Dispersive  K(m,n)  Equations by Means of the Homotopy Analysis Method    Ahmet Yıldırım* Ege University,  Department of Mathematics,  35100 Bornova –İzmir, Turkey ahmet.yildirim@ege.edu.tr , ahmetyildirim80@gmail.com  Canan Ünlü İstanbul University,  Department of Mathematics,  34134 Vezneciler –İstanbul, Turkey  Syed Tauseef Mohyud-Din HITEC University Taxila Cantt, Pakistan   Received: April 27, 2010; Accepted: May 10, 2010   Abstract  In this paper, we study the nonlinear dispersive K(m,n) equations which exhibit solutions with solitary patterns. New exact solitary solutions are found. The two special cases, K(2, 2) and K(3, 3), are chosen to illustrate the concrete features of the homotopy analysis method in K(m,n) equations. The nonlinear equations K(m,n) are studied for two different cases, namely when m = n being odd and even integers. General formulas for the solutions of K(m,n) equations are established.  Keywords: Homotopy analysis method, nonlinear dispersive K(m,n) equations  MSC (2010) #: 35C07, 35C09, 35C10  *Corresponding author: 1Y?ld?r?m et al.: Exact solitary-wave Special SolutionsPublished by Digital Commons @PVAMU, 2010AAM: Intern. J., Special Issue No. 1 (August 2010)  83  1. Introduction  Searching for solitary solutions for nonlinear equation in mathematical physics is attractive in the solitary theory. For example, Wadati (1972, 1973) developed the solitons for KdV and MKdV equations. In 1993, Rosenau and Hyman (1993) presented a family of fully nonlinear KdV equations K(m,n)   0, 0, 1 3m nt x xxxu u u m n  (1)  and introduced a class of solitary waves with compact support that are solutions of a two parameter family of fully nonlinear dispersive partial different equations such as K(2, 2) equation   2 2 0,t x xxxu u u  (2)  Recently, Wazwaz (2002a) gave exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations   0, , 1m nt x xxxu u u m n  (3)  The new solitary-wave special solutions with compact support for the nonlinear dispersive K(m,n)  equations   0, , 1m nt x xxxu u u m n  (4)  are presented by Wazwaz (2002b). Of course, other solitary-wave solutions of K(m,n) equations were also found by many authors [Rosenau (1994, 1997)].  In recent, Wazwaz (2001, 2002a, 2002b, 2004) have successfully used the Adomian decomposition method to construct solitary solutions for many nonlinear equations.  The main goal of this paper is to investigate the K(m,n) equations of the form   0, , 1m nt x xxxu u u m n  (5)  and we would like to extend the homotopy analysis method (HAM) to seek exact special solutions with solitary patterns for (5).  The HAM is developed in 1992 by Liao (1995, 1999, 2004). This method has been successfully applied to solve many types of nonlinear problems in science and engineering by many authors [Abbasbandy (2007), Ayub et al. (2003), Hayat et al. (2004)]. By the present method, numerical results can be obtained with using a few iterations. The HAM contains the auxiliary parameter  , which provides us with a simple way to adjust and control the convergence region of solution 2Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 7https://digitalcommons.pvamu.edu/aam/vol5/iss3/784  Yildirim et al. series for large values of  t. Other numerical methods are given low degree of accuracy for large values of t. Therefore, the HAM handles linear and nonlinear problems without any assumption and restriction.    2. The homotopy analysis method (HAM)  We apply the HAM to the nonlinear dispersive K(m,n) equations (5) . We consider the following differential equation   , 0,N u x t   (6)  where N  is a nonlinear operator for this problem, x  and  t denote independent variables, txu ,   is an unknown function.  In the frame of  HAM, we can construct the following zeroth-order deformation:   01 , ; , , , ; ,q L U x t q u x t q H x t N U x t q  (7)  where 1,0q  is the embedding parameter, 0  is an auxiliary parameter, 0,txH  is an auxiliary function, L  is an auxiliary linear operator,  txu ,0   is an initial guess of txu ,   and qtxU ;,  is an unknown function on the independent variables  x, t  and  q.  Obviously, when 0q   and  ,1q  it holds   ,,0;, 0 txutxU   .,1;, txutxU  (8)  Using the parameter q, we expand  qtxU ;,   in Taylor series as follows:   10 ,,,;,mmm qtxutxuqtxU  (9) where   , ;10!mm mU x t quqm q. (10)  Assume that the auxiliary linear operator, the initial guess, the auxiliary parameter    and the auxiliary function txH ,   are selected such that the series (9)  is convergent  at  1q , then due to (8) we have  3Y?ld?r?m et al.: Exact solitary-wave Special SolutionsPublished by Digital Commons @PVAMU, 2010AAM: Intern. J., Special Issue No. 1 (August 2010)  85  10 ,,,mm txutxutxu . (11)  Let us define the vector   txutxutxutxu nn ,,...,,,,, 10. (12)  Differentiating (7)  m  times with respect to the embedding parameter  q, then setting 0q  and finally dividing them by  !m , we have the so-called  mth-order deformation equation   ,,,, 11 mmmmm uRtxHtxutxuL  (13)  where    11 1, ;101 !mm m mN U x t qR uqm q, (14)  and   .11,10mmm  (15)  Finally, for the purpose of computation, we will approximate the HAM solution (11) by the following truncated series:   10, , .mm kkx t u x t  (16)   3. Applications  In this section, we would like to choose two special equations, namely K(2, 2) and K(3, 3) with specific initial conditions, to illustrate the above-mentioned scheme.  Example 1.  Consider the K(2, 2) equation with the initial condition   2 2 0,t x xxxu u u  (17a)  24,0 sinh ,3 4xu x v  (17b)  where v  is an arbitrary constant.  4Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 7https://digitalcommons.pvamu.edu/aam/vol5/iss3/786  Yildirim et al. According to (7), the zeroth-order deformation can be given by   2 201 , ; , , t x xxxq L U x t q u x t q H x t U U U . (18)  We can start with an initial approximation 204, sinh3 4xu x t v  and choose the auxiliary linear operator   , ;, ; ,U x t qL U x t qt  with the property   ,0CL   where  C  is an integral constant. We also choose the auxiliary function to be   .1,txH   Hence, the mth-order deformation can be given by   ,,,, 11 mmmmm uRtxHtxutxuL   where   31 111 1 130 0m mmm m i m i i m ii iuR u u u u ut x x. (19)  Now the solution of the mth-order deformation equations (19)  for 1m  becomes   .,, 111 mmmmm uRLtxutxu . (20)  Consequently, the first few terms of the HAM series solution are as follows:   204, sinh ,3 4xu x t v   21 , sinh ,3 2xu x t v t   2 22 2 3 22 , sinh sinh cosh ,3 2 3 2 12 2x x xu x t v t v t v t             . 5Y?ld?r?m et al.: Exact solitary-wave Special SolutionsPublished by Digital Commons @PVAMU, 2010AAM: Intern. J., Special Issue No. 1 (August 2010)  87 and so on. Hence, the HAM series solution (for )1  is   .....,,,,, 3210 txutxutxutxutxu             2 2 3 2 4 34 1 1 1sinh sinh cosh sinhh ......3 4 3 2 12 2 72 2x x x xv v t v t v t  (21)  Using Taylor series into (21), we find the closed form solution   24, sinh3 4x vtu x t v . (22)  In addition, we can develop another exact solution for the K(2, 2) equation. Now we consider another initial value problem of  K(2, 2) equation   2 2 0,t x xxxu u u  (23a)  24,0 cosh ,3 4xu x v  (23b)  Using the manner as discussed above, we obtain another exact solution given by   24, cosh3 4x vtu x t v . (24)  Example 2.   Consider the initial value problem K(3, 3)   3 3 0,t x xxxu u u  (25a)  6,0 sinh ,2 3v xu x  (25b)  where v  is an arbitrary constant.  According to (7), the zeroth-order deformation can be given by   3 301 , ; , , t x xxxq L U x t q u x t q H x t U U U  (26)  We can start with an initial approximation  06, sinh2 3v xu x t  and we choose the auxiliary linear operator  6Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 7https://digitalcommons.pvamu.edu/aam/vol5/iss3/788  Yildirim et al.  , ;, ; ,U x t qL U x t qt  with the property   ,0CL   where C is an integral constant. We also choose the auxiliary function to be   .1,txH   Hence, the mth-order deformation can be given by   ,,,, 11 mmmmm uRtxHtxutxuL   where   31 1 1 111 1 130 0 0 0m m i m m imm m i k m i k i k m i ki k i kuR u u u u u u ut x x (27)  Now the solution of the mth-order deformation equations (27) for 1m  become   .,, 111 mmmmm uRLtxutxu  (28)  Consequently, the first few terms of the HAM series solution are as follows:   06, sinh ,2 3v xu x t   3/ 216, cosh ,6 3xu x t v t   3/ 2 2 3/ 2 2 5/ 2 226 6 6, cosh cosh sinh ,6 3 6 3 36 3x x xu x t v t v t v t            .  and so on. Hence, the HAM series solution (for )1  is  .....,,,,, 3210 txutxutxutxutxu   3/ 2 5/ 2 2 7 / 2 36 6 6 6sinh cosh sinh cosh .....2 3 6 3 36 3 324 3v x x x xv t v t v t  (29)  7Y?ld?r?m et al.: Exact solitary-wave Special SolutionsPublished by Digital Commons @PVAMU, 2010AAM: Intern. J., Special Issue No. 1 (August 2010)  89 Using Taylor series into (29), we find the closed form solution   6, sinh2 3v x vtu x t  (30)  To obtain another exact solution for K(3, 3), we consider the initial value problem of K(3, 3) equation   3 3 0,t x xxxu u u  (31a)   6,0 sinh ,2 3v xu x  (31b)  According to the similar steps as discussed above, we have another exact solution given by   6, sinh2 3v x vtu x t   4. More exact solutions  4.1. The K(2,2) type  In Section 3, two exact solitary patterns solutions were developed in the form   24, sinh3 4x vtu x t v  (32) and   24, cosh3 4x vtu x t v  (33)  By combining the two results, we will find that   2 24 4, sinh cosh ,3 4 3 4x vt x vtu x t Mv Nv  (34)  satisfies the K(2, 2) equation, where  M  and  N  are constants if   M=N or M = 1-N (35)  (a)  When  M = N,  we can obtain the trivial solution 8Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 7https://digitalcommons.pvamu.edu/aam/vol5/iss3/790  Yildirim et al.  4,3u x t Nv  (36)  (b) When M = 1-N,   we can obtain  the new exact solution   2 24 4, 1 sinh cosh ,3 4 3 4x vt x vtu x t N v Nv  (37)  Moreover, adding a constant to the arguments in (32) and (33) will exhibit more exact solutions. In other words, we introduce the exact solutions   24, sinh3 4x vtu x t v c  (38)  and   24, cosh3 4x vtu x t v c  (39)  where  c  is a constant.  4.2. The K(3,3) type  As discussed before, the exact solutions for the K(3, 3) equation are given in the form   6, sinh2 3v x vtu x t  (40) and   6, sinh2 3v x vtu x t  (41)  We can obtain a new exact solution by combining the two results (40) and (41) and we find that   6 6, sinh sinh2 3 2 3v x vt v x vtu x t M N  (42)  satisfies the K(3, 3) equation if   M = N,  M = 1+N, M = -1+N, (43)  (a) When  M = N,  we can obtain the trivial solution  9Y?ld?r?m et al.: Exact solitary-wave Special SolutionsPublished by Digital Commons @PVAMU, 2010AAM: Intern. J., Special Issue No. 1 (August 2010)  91  , 0u x t  (44)  (b) When M = 1+N,   we can obtain the new exact solution   6 6, 1 sinh sinh2 3 2 3v x vt v x vtu x t N N  (45)  (c) When M = -1+N,   we can obtain the new exact solution   6 6, 1 sinh sinh2 3 2 3v x vt v x vtu x t N N  (46)  Moreover, adding a constant to the arguments in (40) and (41) will exhibit more exact solutions   6, sinh2 3v x vtu x t c  (47)  and   6, sinh2 3v x vtu x t c  (48)  where c  is a constant.   5. General solutions for K(m,n)   5.1. The K(m,n),  m= n Being Even Integer  When m = n being even integer, we can test several initial value problems by using homotopy analysis method. By properly observing several results, we introduce the general formulas   212 1, sinh1 2nvp nu x t x vt cp n (49)  and   212 1, cosh1 2nvp nu x t x vt cp n, (50)  10Applications and Applied Mathematics: An International Journal (AAM), Vol. 5 [2010], Iss. 3, Art. 7https://digitalcommons.pvamu.edu/aam/vol5/iss3/792  Yildirim et al. where  c  is a constant.   5.2. The  K(m,n),  m= n Being Odd Integer  For m = n being odd integer and n > 1, several problems were tested by using homotopy analysis method and by carefully observing several results, we introduce the general formulas   212 1, sinh1 2nvp nu x t x vt cp n (51)  and   212 1, sinh1 2nvp nu x t x vt cp n, (52)  where c  is a constant. And by combining the two results (51) and (52), we will find that   2 21 12 1 2 1, sinh sinh ,1 2 1 2n nvp n vp nu x t M x vt c N x vt cp n p n (53)  satisfies the K(n,n) equation with  n  being odd integer >1, where  M  and  N  are constants if   M = N,  M = 1+N, M = -1+N  (54)  Thus this completes our goal by establishing general formulas for solutions of the nonlinear dispersive K(n,n) that work for all values of  n > 1.   6.  Conclusion  In summary, we have presented the application of  homotopy analysis method to the nonlinear dispersive K(m,n) equations. We chose two special cases, K(2, 2) and K(3, 3) equations to illustrate the scheme such that new exact solutions with solitary patterns are of important significance. We developed the new exact solutions which are generated by combining two distinct solutions of the K(2, 2) and K(3, 3) equations. At last, we establish the general formulas for exact solutions of equations K(m,n) when m = n being even and odd integers for n > 1. It is worth noting that other solitary solutions of K(m,n) equation may be also constructed by using the homotopy analysis method.   11Y?ld?r?m et al.: Exact solitary-wave Special SolutionsPublished by Digital Commons @PVAMU, 2010AAM: Intern. J., Special Issue No. 1 (August 2010)  93      References  Abbasbandy, S. (2007). Homotopy analysis method for heat radiation equations. Int. Comm. Heat. Mass. Transfer Vol. 34, pp. 380–387. Ayub, M., Rasheed, A. and Hayat, T. (2003). Exact flow of a third grade fluid past a porous plate using homotopy analysis method. Int. J. Eng. Sci. Vol. 41, pp. 2091–2103. Hayat, T., Khan, M. and Asghar, S. (2004). Magnetohydrodynamic flow of an Oldroyd 6–constant fluid. Appl. Math. Comput.Vol. 155, pp. 417–425. Liao, S.J.(1995). An approximate solution technique which does not depend upon small parameters:   a special example. Int. J. Nonlinear. Mech Vol. 30, pp. 371–380. Liao, S.J.(1999). 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Exact solitary-wave Special Solutions for the Nonlinear Dispersive K(m,n) Equations by Means of the Homotopy Analysis Method

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Exact solitary-wave Special Solutions for the Nonlinear Dispersive K(m,n) Equations by Means of the Homotopy Analysis Method

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