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Exact and analytical solution for nonlinear dispersive K(m,p) equations using homotopy perturbation method
Physics Letters A 368 (2007) 266–270 www.elsevier.com/locate/pla
Exact and analytical solution for nonlinear dispersive K(m, p) equations using homotopy perturbation method Ganji Domairry ∗ , M. Ahangari, M. Jamshidi Noshiravani Technical University, PO Box 47415, Babol, Iran Received 1 March 2007; accepted 2 April 2007 Available online 5 April 2007 Communicated by A.R. Bishop
Abstract In the present Letter, we have implemented a homotopy perturbation method to approximately solve the nonlinear dispersive K(m, p) type equations. Using this scheme, the explicit exact solution is calculated in the form of a quickly convergent series with easily computable components. To illustrate the application of this method, numerical results are derived by using the calculated components of the variational series. The obtained results are found to be in good agreement with the exact solutions. © 2007 Elsevier B.V. All rights reserved. Keywords: Nonlinear dispersive K(m, p) equations; Homotopy perturbation method (HPM); Exact solution
1. Introduction There are many nonlinear equations, which are quite useful and applicable in engineering and physics such as the wellknown KdV equation [1], MKdV equation, BBM equation, Burgers equation, KdV–KSV equation, RLW equation [2] and so on. Since solving these equations needs some nonphysical assumptions, various approximate methods have recently been developed to solve linear and nonlinear differential equations [3–9], without the above mentioned shortcoming. In this Letter, we solve some evolution equations using homotopy perturbation method [10–12], which are widely applied to various engineering problems [13–22], and then compare the obtained results with the exact solutions. In the past decades, directly seeking for exact solutions of nonlinear partial differential equations has become one of the central themes of perpetual interest in mathematical physics. Nonlinear wave phenomena, which appears in many fields, such as fluid mechanics, plasma physics, biology, hydrodynamics, solid state physics, and optical fibers is often related to nonlinear wave equations. To solve these equations, many powerful * Corresponding author. Tel./fax: +98 1125234092.
E-mail address: [email protected] (G. Domairry). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.04.008
methods had been developed such as Backlunds transformation [23,24], Darboux transformation [25], the inverse scattering transformation [26], the bilinear method [27], the tanh method [28,29], the sine–cosine method [30,31], the homogeneous balance method [32], the Riccati method [33], the Jacobi elliptic function method [34], the extended Jacobi elliptic function method [35], etc. Here, we introduce a simple form of the well-known Korteweg–de Vries (KdV) equation: ut − auux + uxxx = 0.
(1)
The nonlinear term uux causes the sloping of the wave form. On the other hand, the dispersion term uxxx in this equation makes the wave form spread. Due to the balance between this weak nonlinearity and dispersion, solitons exist [36]. The genuinely nonlinear dispersive equation K(m, p) which generalizes the KdV Eq. (1), is given by: ut + a um x + up xxx = 0,
m, p 1,
(2)
see Rosenau and Hyman [37]. The aim of this Letter is to extend the He’s homotopy perturbation method [38–44] to derive the numerical and exact compacton solutions of the nonlinear dispersive K(m, p), Eq. (2), subject to the following initial con-
G. Domairry et al. / Physics Letters A 368 (2007) 266–270
dition: ut + um x + up xxx = 0,
3. Applications m > 1, 1 p 3,
u0 (x, t) = u(x, 0) = f (x).
(3)
2. Basic concepts of homotopy perturbation method (HPM)
r ∈ Ω.
(4)
Subject to following boundary condition: B(u, ∂u/∂n) = 0,
r ∈ Γ,
(5)
where A is a general differential operator, B a boundary operator, f (r) is a known analytical function, Γ is the boundary of domain Ω and ∂/∂n denotes differentiation along the normal drawn outwards from the domain Ω. The operator A can generally be divided into two parts, a linear part L and a nonlinear part N . Eq. (4) can therefore be rewritten as follows: L(ν) + N (ν) − f (r) = 0.
(6)
We construct a homotopy ν(r, p) : Ω × [0, 1] → which satisfies: H (ν, p) = (1 − p) L(ν) − L(u0 ) + p A(ν) − f (r) = 0, p ∈ [0, 1],
r ∈ Ω.
(7)
Which is equivalent to:
H (ν, p) = L(ν) − L(u0 ) + pL(u0 ) + p N (ν) − f (r) = 0,
(8)
where p ∈ [0, 1] is an embedding parameter, and u0 is an initial guess approximation of Eq. (4) which satisfies the boundary conditions. It follows from Eqs. (7) and (8) that: H (ν, 0) = L(ν) − L(u0 ) = 0, H (ν, 1) = A(ν) − f (r) = 0.
(9)
Thus, the changing process of p from zero to unity is just that of ν(r, p) from u0 (r) to u(r). In topology, this is called deformation and L(ν) − L(u0 ) and A(ν) − f (r) are called homotopy. Here the embedding parameter is introduced much more naturally, unaffected by artificial factors; besides, it can be considered as a small parameter for 0 p 1. So, it is very quite right to assume that the solutions of Eqs. (7) and (8) can be expressed as: ν = ν0 + pν1 + p ν2 + · · · . 2
(10)
The approximate solution of Eq. (4) can therefore be clearly obtained: u = lim ν = ν0 + ν1 + ν2 + · · · . p→1
We have chosen two examples to show the solution procedure of HPM and compare them with the exact solutions, namely K(2, 2) and K(3, 3). 3.1. Example
HPM is a combination of the classical perturbation method and the homotopy technique. To explain the basic idea of HPM, for solving nonlinear differential equations, we consider the following nonlinear differential equation: A(u) − f (r) = 0,
267
(11)
The convergence of series (11) has been proved by He in his paper [49].
4 1 u(x, 0) = c cos2 x , (12) 3 4 ν0 (x, t) = u0 (x, t) = u(x, 0), (13) 2 2 N(ν) = ν x + ν xxx . L(ν) = νt , (14) ut + u2 x + u2 xxx = 0,
One can now obtain a solution for equation system (14), in the form: 1 1 ν1 (x, t) = c2 t sin x , (15) 2 2 1 1 ν2 (x, t) = − c3 t 2 cos x , (16) 12 2 1 1 ν3 (x, t) = − c4 t 3 sin x , (17) 72 2 1 5 4 1 ν4 (x, t) = (18) c t cos x . 576 2 Having ui , i = 0, 1, . . . , 4, the solution, u(x, t), will be as follows: 4 4 1 1 2 2 1 u(x, t) = ui (x, t) = c cos x + c t sin x 3 4 2 2 i=0 1 1 1 1 − c3 t 2 cos x − c4 t 3 sin x 12 2 72 2 1 5 4 1 + (19) c t cos x . 576 2 This gives the solution in a close form: 4 c cos2 14 (x − ct) |x − ct| 2π, 3 u(x, t) = 0 otherwise,
(20)
which is exactly the same as that obtained by Adomian’s decomposition [45] and the finite difference methods [46]. The behavior of the solution (16) obtained by the homotopy perturbation method and the exact solution (17) are shown in Figs. 1 and 2. We have plotted these equations with some different values of c, t , versus distance x. 3.2. Example As in the solved previously example, we consider the following equation: ut + u3 x + u3 xxx = 0,
3c 1 u(x, 0) = (21) cos x , 2 3 ν0 (x, t) = u0 (x, t) = u(x, 0), N(ν) = ν 3 x + ν 3 xxx . L(ν) = νt ,
(22) (23)
268
G. Domairry et al. / Physics Letters A 368 (2007) 266–270
(a)
(b)
Fig. 1. The surfaces show the approximate solutions obtained by HPM and the exact solution of u(x, t), respectively (Example 1). (a) HPM plot (Eq. (19)); (b) Exact plot (Eq. (20)).
(a)
(b)
(c)
(d)
Fig. 2. The comparison of the results by HPM and the exact solutions for different values of c and t , versus distance x (Example 1). (a) C = 2, t = 1/2; (b) C = 2, t = (−1/2); (c) C = (−3/2), t = (3/2); (d) C = (−3/2), t = (−1/2).
G. Domairry et al. / Physics Letters A 368 (2007) 266–270
(a)
269
(b)
Fig. 3. The surfaces show the approximate solutions obtained by HPM and the exact solution of u(x, t), respectively (Example 2). (a) HPM plot; (b) exact plot.
(a)
(b)
(c)
(d)
Fig. 4. The comparison of the results by HPM and the exact solutions for different values of c and t , versus distance x (Example 2). (a) C = 3/2, t = 3/2; (b) C = 3/2, t = 1/10; (c) C = 5/2, t = 0; (d) C = 5/2, t = 7/2.
One can now obtain a solution for equation system above in the form: 1 √ 1 ν1 (x, t) = c 6c t sin x , (24a) 6 3 1 √ 1 ν2 (x, t) = − c2 6c t 2 cos x , (24b) 36 3 1 3√ 3 1 ν3 (x, t) = − (24c) c 6c t sin x , 324 3
ν4 (x, t) =
1 4√ 4 1 c 6c t cos x . 3888 3
(24d)
Having ui , i = 0, 1, . . . , 4, the solution u(x, t) is finally as:
4 1 1 √ 3c 1 cos x + c 6c t sin x ui (x, t) = u(x, t) = 2 3 6 3 i=0 1 1 1 2√ 2 1 3√ 3 − c 6c t cos x − c 6c t sin x 36 3 324 3
270
G. Domairry et al. / Physics Letters A 368 (2007) 266–270
1 4√ 4 1 + c 6c t cos x . 3888 3
(25)
This gives the solution in a close form: 1 3c 3π 2 cos 3 (x − ct) , |x − ct| 2 , u(x, t) = (26) 0 otherwise. This is exactly the same as that obtains by Adomian’s decomposition [45] and the finite difference methods [46]. The behavior of the solution (25) obtained by HPM and the exact solution (26) are shown in Figs. 3 and 4; with different values of c and t , versus distance x. We achieve a good agreement with the actual solution by using five terms only in homotopy perturbation method derived about. The general formula for the exact solution of Eq. (3) that is applicable for all values of m = p, p > 1 is given by: ⎧ 2/(m−1) 2cm ⎪ ⎨ m+1 cos m−1 , 2m (x − ct) u(x, t) = (27) mπ ⎪ |x − ct| m−1 , ⎩ 0, otherwise. See Wazwaz [45] and Rosenau [47]. 4. Conclusions In this Letter, we have presented a scheme used to obtain numerical and exact compacton solutions of the nonlinear dispersive K(2, 2) and K(3, 3) equations with initial conditions using homotopy perturbation method (HPM). The approximate solutions are compared with the exact ones in Figs. 1 to 4. The results show that the present method is a powerful mathematical tool for finding other compacton solutions of many nonlinear dispersive equations with initial conditions. Some difficulties in the Adomian’s decomposition methods and variational iteration methods disappear by this method. To point out an advantage of HPM over the decomposition procedure of Adomian’s, it is good to say that the present method provides a solution to the problem without calculating Adomian’s polynomials. In [48] and [50], examples for which the Adomian’s decomposition method and Variational iteration method terms are different from those of the homotopy perturbation method are given. In this work, Maple Package 9.5 has been used to calculate the series obtained from the homotopy perturbation method and also to plot the graphs. Acknowledgement The authors would really like to appreciate M. Rostamian for his help in the English and the electronic text.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]
D. Kaya, Appl. Math. Comput. 144 (2003) 353. K. Al-Khaled, Appl. Math. Comput. 171 (2005) 281. D.D. Ganji, Phys. Lett. A 355 (2006) 337. D.D. Ganji, M. Rafei, Phys. Lett. A 356 (2006) 131. D.D. Ganji, A. Rajabi, Int. Commun. Heat Mass Trans. 33 (3) (2006) 391. C. Jin, M. Liu, Appl. Math. Comput. 169 (2005) 953. G. Adomian, J. Math. Anal. Appl. 135 (1988) 501. J.H. He, Int. J. Mod. Phys. B 20 (2006) 1141. J.H. He, Non-Perturbative Methods for Strongly Nonlinear Problems, Dissertation, de-Verlag im Internet GmbH, Berlin, 2006. J.H. He, Comput. Methods Appl. Mech. Engrg. 178 (3–4) (1999) 257. J.H. He, Int. J. Non-Linear Mech. 35 (1) (2000) 37. J.H. He, Int. J. Mod. Phys. B 20 (2006) 2561. J.H. He, Appl. Math. Comput. 135 (1) (2003) 73. J.H. He, Appl. Math. Comput. 151 (1) (2004) 287. J.H. He, Phys. Lett. A 347 (4–6) (2005) 228. J.H. He, Chaos Solitons Fractals 26 (3) (2005) 695. J.H. He, Chaos Solitons Fractals 26 (3) (2005) 827. J.H. He, Int. J. Nonlinear Sci. Numer. Simul. 6 (2) (2005) 207. J.H. He, Phys. Lett. A 350 (1–2) (2006) 87. N. Bildik, A. Konuralp, Int. J. Nonlinear Sci. Numer. Simul. 7 (1) (2006) 65. J.H. He, X.H. Wu, Chaos Solitons Fractals 29 (1) (2006) 108. J.H. He, Appl. Math. Comput. 114 (2–3) (2000) 115. M.J. Ablowitz, P.A. Clarkson, Solitons: Nonlinear Evolution Equation and Inverse Scattering, Cambridge Univ. Press, Cambridge, 1991. M.R. Miura, Backlund Transformation, Springer, Berlin, 1978. C.H. Gu, et al., Darboux Transformation in Solitons Theory and Geometry Applications, Shangai Science Technology Press, Shanghai, 1999. C.S. Gardner, et al., Phys. Rev. Lett. 19 (1967) 1095. R. Hirota, Phys. Rev. Lett. 27 (1971) 1192. W. Malfliet, Am. J. Phys. 60 (1992) 659. M. Inc, E.G. Fan, Z. Naturforsch. 60 (2005) 7. Z. Yan, H.Q. Zhang, Phys. Lett. A 252 (1999) 291. M. Inc, D.J. Evans, Int. J. Comput. Math. 81 (2004) 473. M.L. Wang, Phys. Lett. A 215 (1996) 279. Z. Yan, H.Q. Zhang, Phys. Lett. A 285 (2001) 355. Z. Fu, et al., Phys. Lett. A 290 (2001) 72. Z. Yan, Chaos Solitons Fractals 15 (2003) 575. A.M. Wazwaz, M.A. Helal, Chaos Solitons Fractals 21 (2004) 579. P. Rosenau, J.M. Hyman, Phys. Rev. Lett. 70 (1993) 564. J.H. He, Commun. Nonlinear Sci. Numer. Simul. 2 (1997) 230. J.H. He, Commun. Nonlinear Sci. Numer. Simul. 2 (1997) 235. J.H. He, Int. J. Nonlinear Mech. 34 (1999) 699. J.H. He, Appl. Math. Comput. 114 (2000) 115. J.H. He, Chaos Solitons Fractals 19 (2004) 847. J.H. He, Appl. Math. Comput. 143 (2003) 539. J.H. He, Comput. Methods Appl. Mech. Eng. 167 (1998) 57. A.M. Wazwaz, Math. Comput. Simul. 56 (2001) 269. M.S. Ismail, T. Taha, Math. Comput. Simul. 47 (1998) 519. P. Rosenau, Physica D 123 (1998) 525. S. Momani, Z. Odibat, Phys. Lett. A 355 (2006) 271. J.H. He, Comput. Methods Appl. Mech. Eng. 178 (3–4) (1999) 257. M. Inc, Physica A 375 (2007) 447.
Exact and analytical solution for nonlinear dispersive K(m, p) equations using homotopy perturbation method Ganji Domairry ∗ , M. Ahangari, M. Jamshidi Noshiravani Technical University, PO Box 47415, Babol, Iran Received 1 March 2007; accepted 2 April 2007 Available online 5 April 2007 Communicated by A.R. Bishop
Abstract In the present Letter, we have implemented a homotopy perturbation method to approximately solve the nonlinear dispersive K(m, p) type equations. Using this scheme, the explicit exact solution is calculated in the form of a quickly convergent series with easily computable components. To illustrate the application of this method, numerical results are derived by using the calculated components of the variational series. The obtained results are found to be in good agreement with the exact solutions. © 2007 Elsevier B.V. All rights reserved. Keywords: Nonlinear dispersive K(m, p) equations; Homotopy perturbation method (HPM); Exact solution
1. Introduction There are many nonlinear equations, which are quite useful and applicable in engineering and physics such as the wellknown KdV equation [1], MKdV equation, BBM equation, Burgers equation, KdV–KSV equation, RLW equation [2] and so on. Since solving these equations needs some nonphysical assumptions, various approximate methods have recently been developed to solve linear and nonlinear differential equations [3–9], without the above mentioned shortcoming. In this Letter, we solve some evolution equations using homotopy perturbation method [10–12], which are widely applied to various engineering problems [13–22], and then compare the obtained results with the exact solutions. In the past decades, directly seeking for exact solutions of nonlinear partial differential equations has become one of the central themes of perpetual interest in mathematical physics. Nonlinear wave phenomena, which appears in many fields, such as fluid mechanics, plasma physics, biology, hydrodynamics, solid state physics, and optical fibers is often related to nonlinear wave equations. To solve these equations, many powerful * Corresponding author. Tel./fax: +98 1125234092.
E-mail address: [email protected] (G. Domairry). 0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.04.008
methods had been developed such as Backlunds transformation [23,24], Darboux transformation [25], the inverse scattering transformation [26], the bilinear method [27], the tanh method [28,29], the sine–cosine method [30,31], the homogeneous balance method [32], the Riccati method [33], the Jacobi elliptic function method [34], the extended Jacobi elliptic function method [35], etc. Here, we introduce a simple form of the well-known Korteweg–de Vries (KdV) equation: ut − auux + uxxx = 0.
(1)
The nonlinear term uux causes the sloping of the wave form. On the other hand, the dispersion term uxxx in this equation makes the wave form spread. Due to the balance between this weak nonlinearity and dispersion, solitons exist [36]. The genuinely nonlinear dispersive equation K(m, p) which generalizes the KdV Eq. (1), is given by: ut + a um x + up xxx = 0,
m, p 1,
(2)
see Rosenau and Hyman [37]. The aim of this Letter is to extend the He’s homotopy perturbation method [38–44] to derive the numerical and exact compacton solutions of the nonlinear dispersive K(m, p), Eq. (2), subject to the following initial con-
G. Domairry et al. / Physics Letters A 368 (2007) 266–270
dition: ut + um x + up xxx = 0,
3. Applications m > 1, 1 p 3,
u0 (x, t) = u(x, 0) = f (x).
(3)
2. Basic concepts of homotopy perturbation method (HPM)
r ∈ Ω.
(4)
Subject to following boundary condition: B(u, ∂u/∂n) = 0,
r ∈ Γ,
(5)
where A is a general differential operator, B a boundary operator, f (r) is a known analytical function, Γ is the boundary of domain Ω and ∂/∂n denotes differentiation along the normal drawn outwards from the domain Ω. The operator A can generally be divided into two parts, a linear part L and a nonlinear part N . Eq. (4) can therefore be rewritten as follows: L(ν) + N (ν) − f (r) = 0.
(6)
We construct a homotopy ν(r, p) : Ω × [0, 1] → which satisfies: H (ν, p) = (1 − p) L(ν) − L(u0 ) + p A(ν) − f (r) = 0, p ∈ [0, 1],
r ∈ Ω.
(7)
Which is equivalent to:
H (ν, p) = L(ν) − L(u0 ) + pL(u0 ) + p N (ν) − f (r) = 0,
(8)
where p ∈ [0, 1] is an embedding parameter, and u0 is an initial guess approximation of Eq. (4) which satisfies the boundary conditions. It follows from Eqs. (7) and (8) that: H (ν, 0) = L(ν) − L(u0 ) = 0, H (ν, 1) = A(ν) − f (r) = 0.
(9)
Thus, the changing process of p from zero to unity is just that of ν(r, p) from u0 (r) to u(r). In topology, this is called deformation and L(ν) − L(u0 ) and A(ν) − f (r) are called homotopy. Here the embedding parameter is introduced much more naturally, unaffected by artificial factors; besides, it can be considered as a small parameter for 0 p 1. So, it is very quite right to assume that the solutions of Eqs. (7) and (8) can be expressed as: ν = ν0 + pν1 + p ν2 + · · · . 2
(10)
The approximate solution of Eq. (4) can therefore be clearly obtained: u = lim ν = ν0 + ν1 + ν2 + · · · . p→1
We have chosen two examples to show the solution procedure of HPM and compare them with the exact solutions, namely K(2, 2) and K(3, 3). 3.1. Example
HPM is a combination of the classical perturbation method and the homotopy technique. To explain the basic idea of HPM, for solving nonlinear differential equations, we consider the following nonlinear differential equation: A(u) − f (r) = 0,
267
(11)
The convergence of series (11) has been proved by He in his paper [49].
4 1 u(x, 0) = c cos2 x , (12) 3 4 ν0 (x, t) = u0 (x, t) = u(x, 0), (13) 2 2 N(ν) = ν x + ν xxx . L(ν) = νt , (14) ut + u2 x + u2 xxx = 0,
One can now obtain a solution for equation system (14), in the form: 1 1 ν1 (x, t) = c2 t sin x , (15) 2 2 1 1 ν2 (x, t) = − c3 t 2 cos x , (16) 12 2 1 1 ν3 (x, t) = − c4 t 3 sin x , (17) 72 2 1 5 4 1 ν4 (x, t) = (18) c t cos x . 576 2 Having ui , i = 0, 1, . . . , 4, the solution, u(x, t), will be as follows: 4 4 1 1 2 2 1 u(x, t) = ui (x, t) = c cos x + c t sin x 3 4 2 2 i=0 1 1 1 1 − c3 t 2 cos x − c4 t 3 sin x 12 2 72 2 1 5 4 1 + (19) c t cos x . 576 2 This gives the solution in a close form: 4 c cos2 14 (x − ct) |x − ct| 2π, 3 u(x, t) = 0 otherwise,
(20)
which is exactly the same as that obtained by Adomian’s decomposition [45] and the finite difference methods [46]. The behavior of the solution (16) obtained by the homotopy perturbation method and the exact solution (17) are shown in Figs. 1 and 2. We have plotted these equations with some different values of c, t , versus distance x. 3.2. Example As in the solved previously example, we consider the following equation: ut + u3 x + u3 xxx = 0,
3c 1 u(x, 0) = (21) cos x , 2 3 ν0 (x, t) = u0 (x, t) = u(x, 0), N(ν) = ν 3 x + ν 3 xxx . L(ν) = νt ,
(22) (23)
268
G. Domairry et al. / Physics Letters A 368 (2007) 266–270
(a)
(b)
Fig. 1. The surfaces show the approximate solutions obtained by HPM and the exact solution of u(x, t), respectively (Example 1). (a) HPM plot (Eq. (19)); (b) Exact plot (Eq. (20)).
(a)
(b)
(c)
(d)
Fig. 2. The comparison of the results by HPM and the exact solutions for different values of c and t , versus distance x (Example 1). (a) C = 2, t = 1/2; (b) C = 2, t = (−1/2); (c) C = (−3/2), t = (3/2); (d) C = (−3/2), t = (−1/2).
G. Domairry et al. / Physics Letters A 368 (2007) 266–270
(a)
269
(b)
Fig. 3. The surfaces show the approximate solutions obtained by HPM and the exact solution of u(x, t), respectively (Example 2). (a) HPM plot; (b) exact plot.
(a)
(b)
(c)
(d)
Fig. 4. The comparison of the results by HPM and the exact solutions for different values of c and t , versus distance x (Example 2). (a) C = 3/2, t = 3/2; (b) C = 3/2, t = 1/10; (c) C = 5/2, t = 0; (d) C = 5/2, t = 7/2.
One can now obtain a solution for equation system above in the form: 1 √ 1 ν1 (x, t) = c 6c t sin x , (24a) 6 3 1 √ 1 ν2 (x, t) = − c2 6c t 2 cos x , (24b) 36 3 1 3√ 3 1 ν3 (x, t) = − (24c) c 6c t sin x , 324 3
ν4 (x, t) =
1 4√ 4 1 c 6c t cos x . 3888 3
(24d)
Having ui , i = 0, 1, . . . , 4, the solution u(x, t) is finally as:
4 1 1 √ 3c 1 cos x + c 6c t sin x ui (x, t) = u(x, t) = 2 3 6 3 i=0 1 1 1 2√ 2 1 3√ 3 − c 6c t cos x − c 6c t sin x 36 3 324 3
270
G. Domairry et al. / Physics Letters A 368 (2007) 266–270
1 4√ 4 1 + c 6c t cos x . 3888 3
(25)
This gives the solution in a close form: 1 3c 3π 2 cos 3 (x − ct) , |x − ct| 2 , u(x, t) = (26) 0 otherwise. This is exactly the same as that obtains by Adomian’s decomposition [45] and the finite difference methods [46]. The behavior of the solution (25) obtained by HPM and the exact solution (26) are shown in Figs. 3 and 4; with different values of c and t , versus distance x. We achieve a good agreement with the actual solution by using five terms only in homotopy perturbation method derived about. The general formula for the exact solution of Eq. (3) that is applicable for all values of m = p, p > 1 is given by: ⎧ 2/(m−1) 2cm ⎪ ⎨ m+1 cos m−1 , 2m (x − ct) u(x, t) = (27) mπ ⎪ |x − ct| m−1 , ⎩ 0, otherwise. See Wazwaz [45] and Rosenau [47]. 4. Conclusions In this Letter, we have presented a scheme used to obtain numerical and exact compacton solutions of the nonlinear dispersive K(2, 2) and K(3, 3) equations with initial conditions using homotopy perturbation method (HPM). The approximate solutions are compared with the exact ones in Figs. 1 to 4. The results show that the present method is a powerful mathematical tool for finding other compacton solutions of many nonlinear dispersive equations with initial conditions. Some difficulties in the Adomian’s decomposition methods and variational iteration methods disappear by this method. To point out an advantage of HPM over the decomposition procedure of Adomian’s, it is good to say that the present method provides a solution to the problem without calculating Adomian’s polynomials. In [48] and [50], examples for which the Adomian’s decomposition method and Variational iteration method terms are different from those of the homotopy perturbation method are given. In this work, Maple Package 9.5 has been used to calculate the series obtained from the homotopy perturbation method and also to plot the graphs. Acknowledgement The authors would really like to appreciate M. Rostamian for his help in the English and the electronic text.
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