Numerical and explicit solutions of the fifth-order Korteweg-de Vries equations

Chaos, Solitons and Fractals 39 (2009) 2484–2490 www.elsevier.com/locate/chaos

Numerical and explicit solutions of the fifth-order Korteweg-de Vries equations M.T. Darvishi b

a,*

, F. Khani

b

a Department of Mathematics, Razi University, Kermanshah 67149, Iran Department of Mathematics, Bakhtar Institute of Higher Education, P.O. Box. 696, Ilam, Iran

Accepted 10 July 2007

Communicated by Prof. L. Marek-Crnjac

Abstract In this paper, by means of variational iteration method numerical and explicit solutions are computed for some fifthorder Korteweg-de Vries equations, without any linearization or weak nonlinearity assumptions. These equations are the Kawahara equation, Lax’s fifth-order KdV equation and Sawada–Kotera equation. Comparison with Adomian decomposition method reveals that the variational iteration method is easier to be implemented. We conclude that the method is a promising method to various kinds of fifth-order Korteweg-de Vries equations. Ó 2007 Elsevier Ltd. All rights reserved.

1. Introduction The study of nonlinear problems is of crucial importance in all areas of mathematics and physics. Some of the most interesting features or physical systems are hidden in their nonlinear behavior, and can only be studied with appropriate method designed to tackle nonlinear problems. In this past several decades, many authors paid attention to study solitonic solutions of nonlinear equations by using various methods, among these are Backlund transformation [1], Darboux transformation [27], Inverse scattering method [4], Hirota’s bilinear method [14], the tanh-function method [19], the sine–cosine method [29], the homogeneous balance method [28]. The variational iteration method was first proposed by He [6,7], and was successfully applied to autonomous or ordinary differential equations by He [8], to nonlinear polycrystalline solids [20], nonlinear partial differential equations [2,11] and other fields. Variational iteration method has wide applications. For example, He and Wu [12] used this method to construct solitary solution and compacton-like solutions for nonlinear dispersive equations. Soliman [24] by using variational iteration method exactly obtained the solutions of Korteweg-de Vries Burgers and Lax’s seventh-order KdV equations. There are another applications of variational iteration method in Refs. [3,15,21,22,25,26]. The aim of this paper is to extend the variational iteration method to solve fifth-order Korteweg-de Vries equations, namely, Kawahara equation, Sawada–Kotera equation and Lax’s fifth-order KdV equation. *

Corresponding author. Tel.: +98 9123 58 2865; fax: +98 8314 274569. E-mail address: [email protected] (M.T. Darvishi).

0960-0779/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2007.07.034

M.T. Darvishi, F. Khani / Chaos, Solitons and Fractals 39 (2009) 2484–2490

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2. Variational iteration method Inokutti et al. [16] proposed a general Lagrange multiplier method to solve nonlinear problems. In this method the solution of a mathematical problem with linearization assumption is used as an initial approximation or trial function. Then a more highly precise approximation at some special point can be obtained. To illustrate this method, consider the following differential equation: LuðtÞ þ NuðtÞ ¼ gðtÞ;

ð1Þ

where L is a linear operator, N is a nonlinear operator and g(t) is an inhomogeneous term. He has modified the above method into an iteration method. According to the variational iteration method, or more precisely, He’s variational iteration method [5–8,10,13], we construct a correctional functional as follows: Z t kðLun ðsÞ þ N~un ðsÞ  gðsÞÞds; ð2Þ unþ1 ðtÞ ¼ un ðtÞ þ 0

where k is a general Lagrange multiplier [6,7,9], which can be identified optimally via the variational theory, the subscript n denotes the nth order approximation and ~un is consider as a restricted variation [6,7,9], i.e., d~ un ¼ 0. In the following section, we apply the variational iteration method to solve some test problems.

3. Applications 3.1. Kawahara equation We consider the Kawahara equation [23] ut þ uux þ uxxx  uxxxxx ¼ 0;

ð3Þ

with the following initial condition:   105 x  x0 uðx; 0Þ ¼ sech4 pffiffiffiffiffi ; 169 2 13

ð4Þ

where x0 is an arbitrary constant. Its correction variational functional in t-direction to obtain the solution of Kawahara equation (3) by variational iteration method can be expressed as follows:  Z t  oun o~un o3 ~un o5 ~ un ð5Þ k unþ1 ðx; tÞ ¼ un ðx; tÞ þ þ ~un þ 3  5 ds; os ox ox ox 0 un ¼ 0. Making the correction funcwhere k is general Lagrange multiplier, and u~n denote restricted variations, i.e., d~ tional equation (5), stationary  Z t  un oun o~un o3 ~un o5 ~ þ ~un þ 3  5 ds; ð6Þ k dunþ1 ðx; tÞ ¼ dun ðx; tÞ þ d os ox ox ox 0 Z t   oun ds; ð7Þ k dunþ1 ðx; tÞ ¼ dun ðx; tÞ þ d os 0 Z t k0 ðsÞdun ðx; sÞds ¼ 0; ð8Þ dunþ1 ðx; tÞ ¼ dun ðx; tÞ þ kðsÞdun ðx; tÞjs¼t  0

yields the following stationary conditions: dun : k0 ðsÞ ¼ 0; dun : 1 þ kðsÞjs¼t ¼ 0:

ð9Þ

The Lagrange multiplier, therefore, can be identified as k(s) =  1, and the following variational iteration formula can be obtained  Z t oun oun o3 un o5 un þ un þ 3  5 ds: ð10Þ unþ1 ðx; tÞ ¼ un ðx; tÞ  os ox ox ox 0

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  0ffi pffiffiffi , in t-direction. Thus, we can obtain By initial condition (4) we start with an initial approximation u0 ¼ 105 sech4 2xx 169 13 the following results: pffiffiffiffiffi       105 7560 13 x  x0 4 x  x0 4 x  x0 p ffiffiffiffiffi p ffiffiffiffiffi p ffiffiffiffiffi u1 ðx; tÞ ¼ þ tanh ; sech tsech 169 371293 2 13 2 13 2 13     105 2 x  x0 648 3240 x  x0  ; u2 ðx; tÞ ¼ u1 ðx; tÞ þ t sech4 pffiffiffiffiffi þ tanh2 pffiffiffiffiffi 169 371293 371293 2 13 2 13 or     68040 2 x  x0 x  x0 ; 3 þ 2 cosh pffiffiffiffiffi t sech6 pffiffiffiffiffi 62748517 13 2 13     105 3 x  x0 7776 x  x0 pffiffiffiffiffi tanh pffiffiffiffiffi u3 ðx; tÞ ¼ u2 ðx; tÞ þ  t sech4 pffiffiffiffiffi 169 2 13 2 13 62748517 13     144 x  x0 162 324 x  x0 2 pffiffiffiffiffi pffiffiffiffiffi tanh pffiffiffiffiffi þ  þ tanh 371293 371293 2 13 2 13 169 13     72 x  x0 324 972 x  x0 pffiffiffiffiffi tanh pffiffiffiffiffi  ; þ þ tanh2 pffiffiffiffiffi 371293 371293 169 13 2 13 2 13

u2 ðx; tÞ ¼ u1 ðx; tÞ þ

or u3 ðx; tÞ ¼ u2 ðx; tÞ þ

      x  x0 x  x0 x  x0 pffiffiffiffiffi t3 sech7 pffiffiffiffiffi 13 sinh pffiffiffiffiffi þ2 sinh pffiffiffiffiffi 2 13 2 13 2 13 10604499373 13 816480

and so on, in the same manner the rest of components of the iteration formula (10), can be obtained using Maple Package. The solution of u(x, t), in closed form, is readily found to be  36  x  x0  169 t 105 pffiffiffiffiffi ; ð11Þ uðx; tÞ ¼ sech4 169 2 13 which is exactly as same as the solution obtained by Adomian decomposition method [17]. 3.2. A generalized fifth-order KdV equation In this part we apply the variational iteration method to the generalized fifth-order KdV (gfKdV) equation. It has the following form: ut þ au2 ux þ bux uxx þ cuuxxx þ duxxxxx ¼ 0;

ð12Þ

where a, b, c and d are constants. This equation has known as the general form of the fifth-order KdV equation. Eq. (12) is the well-known Lax’s fifth-order KdV equation if we set a ¼ 30; b ¼ 30; c ¼ 10; d ¼ 1 and the Sawada–Kotera equation by setting a ¼ 45; b ¼ 15; c ¼ 15 and d = 1 [23]. To solve Eq. (12) by means of variational iteration method, we consider the correctional functional in t-direction as follows:  Z t  un un un oun o~un o~un o2 ~ o3 ~ o5 ~ unþ1 ðx; tÞ ¼ un ðx; tÞ þ þ d þ a~u2 þb ds; ð13Þ k þ c~ u n 3 os ox ox ox2 ox5 ox 0 where ~u is considered as restricted variations, i.e., d~u ¼ 0. Making the correction functional equation (13), stationary  Z t  un un un oun o~un o~ un o2 ~ o3 ~ o5 ~ ds; k þ c~ u þ d dunþ1 ðx; tÞ ¼ dun ðx; tÞ þ d þ a~u2 þb n os ox ox ox2 ox5 o3 x 0 Z t   oun ds; k dunþ1 ðx; tÞ ¼ dun ðx; tÞ þ d os 0 Z t k0 ðsÞdun ðx; sÞds ¼ 0; dunþ1 ðx; tÞ ¼ dun ðx; tÞ þ kðsÞdun ðx; tÞjs¼t  0

yields the following stationary conditions:

M.T. Darvishi, F. Khani / Chaos, Solitons and Fractals 39 (2009) 2484–2490

dun : k0 ðsÞ ¼ 0; dun : 1 þ kðsÞjs¼t ¼ 0:

2487

ð14Þ

The Lagrangian multiplier, therefore, can be identified as k(s) = 1, and the following variational iteration formula can be obtained:  Z t oun oun oun o2 un o3 un o5 un ds: ð15Þ þ cu þ d unþ1 ðx; tÞ ¼ un ðx; tÞ  þ au2 þb n 3 os ox ox ox2 ox5 ox 0 We obtain the solution of gfKdV in the following parts. 3.2.1. Lax’s fifth-order KdV equation For purpose of illustration of the variational iteration method to solve the generalized fifth-order KdV equation (12) in case of a ¼ 30; b ¼ 30; c ¼ 10 and d = 1, that is, the Lax’s fifth-order KdV equation, we start with an initial approximation u0 = u(x, 0) which is given by the following initial condition [23] uðx; 0Þ ¼ 2k 2 ð2  3tanh2 ðkðx  x0 ÞÞÞ;

ð16Þ

where k and x0 are arbitrary constants and k 5 0. By the iteration formula (15), we can obtain directly the other components as u1 ðx; tÞ ¼ u0 ðx; tÞ þ 672k 7 tsech2 ðkðx  x0 ÞÞ tanhðkðx  x0 ÞÞ; u2 ðx; tÞ ¼ u1 ðx; tÞ þ 18816k 12 t2 sech4 ðkðx  x0 ÞÞðcoshð2kðx  x0 ÞÞ  2Þ; u3 ðx; tÞ ¼ u2 ðx; tÞ þ 351232k 17 t3 sech5 ðkðx  x0 ÞÞðsinhð3kðx  x0 ÞÞ  11 sinhðkðx  x0 ÞÞÞ; and so on, in the same way the rest of components of the iteration formula (15), can be obtained using Maple Package. The solution of u(x, t), in closed form, is readily found to be uðx; tÞ ¼ 2k 2 ð2  3tanh2 ðkðx  56k 4 t  x0 ÞÞÞ;

ð17Þ

which is exactly the same as obtained by Adomian decomposition method [18]. 3.2.2. Sawada–Kotera equation A second case, considers the generalized fifth-order KdV equation (12), for a ¼ 45; b ¼ 15; c ¼ 15 and d = 1, in this case the initial condition has the following form [23]: uðx; 0Þ ¼ 2k 2 sech2 ðkðx  x0 ÞÞ;

ð18Þ

where k and x0 are arbitrary constants and k 5 0. By the iteration formula (15) and initial approximation u0 = u(x, 0) we can obtain directly the other components as u1 ðx; tÞ ¼ u0 ðx; tÞ þ 64k 7 tsech2 ðkðx  x0 ÞÞ tanhðkðx  x0 ÞÞ; u2 ðx; tÞ ¼ u1 ðx; tÞ þ 512k 12 t2 sech4 ðkðx  x0 ÞÞð2 þ coshð2kðx  x0 ÞÞÞ; 8192 17 3 k t sech5 ðkðx  x0 ÞÞð11 sinhðkðx  x0 ÞÞ þ sinhð3kðx  x0 ÞÞÞ; u3 ðx; tÞ ¼ u2 ðx; tÞ þ 3 and so on, in the same method the rest of components of the iteration formula (15), can be obtained using Maple Package. Hence the solution of u(x, t), in closed form, is readily found to be uðx; tÞ ¼ 2k 2 sech2 ðkðx  16k 4 t  x0 ÞÞ;

ð19Þ

which is exactly the same as obtained by Adomian decomposition method [18].

4. Numerical experiments In this section, we consider the Kawahara equation and two gfKdV equations for numerical comparisons. Based on the variational iteration method, we constructed the solution u(x, t) as iterative relations (10) and (15) for Kawahara equation and two gfKdV equations. In this section, we demonstrate how the approximate solutions of the Kawahara and gfKdV equations are close to their exact solutions.

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In order to, numerically, verify whether the proposed methodology lead to high accuracy, we evaluate the numerical solutions using the n-iterate approximations (10) and (15). Tables 1–3 show the absolute errors of approximate solutions of test problems. The graphs of the exact and approximate solutions are shown in Figs. 1–3. We consider, only six iterations to evaluate the approximate solutions. We achieved a very good approximations in comparison with the actual solutions of the equations by using six iterations for the variational iteration method. Numerical approximations show a high degree of accuracy and in most un, the n-iteration approximation, is accurate for quite low value of n. The numerical results which we obtained justify the advantage of this methodology, even in the few iterations approximations. Furthermore, as the variational iteration method dose not require discretization of the variables, i.e., time and

Table 1 juðx; tÞ  u6 ðx; tÞj for Kawahara equation with x0 = 4 ti/xi

0.5

1.0

1.5

2.0

0.5 1.0 1.5 2.0 2.5

1.12022E  17 1.86934E  17 2.36187E  17 3.04860E  17 3.79143E  17

4.36154E  17 5.01263E  17 5.89634E  17 6.11203E  17 7.10887E  17

3.21453E  17 3.68325E  17 4.01548E  17 4.35177E  17 5.10587E  17

1.15324E  17 1.89361E  17 2.54167E  17 2.98413E  17 3.15492E  17

Table 2 juðx; tÞ  u6 ðx; tÞj for Lax’s fifth-order KdV equation with k = 0.3 and x0 = 0.8 ti/xi

0.5

1.0

1.5

2.0

0.5 1.0 1.5 2.0 2.5

1.66682E  17 1.91996E  17 2.87980E  17 3.38959E  17 4.99713E  17

2.35687E  17 3.83986E  17 5.57556E  17 7.97008E  17 9.59653E  17

4.12563E  17 5.75977E  17 8.36390E  17 1.15191E  16 1.43256E  16

4.56348E  17 7.67962E  17 1.15191E  16 1.19375E  16 1.91637E  16

Table 3 juðx; tÞ  u6 ðx; tÞj for Sawada–Kotera equation with k = 0.4 and x0 = 0.6 ti/xi

0.5

1.0

1.5

2.0

0.5 1.0 1.5 2.0 2.5

4.80031E  18 9.59898E  18 1.32714E  17 2.39932E  17 3.78324E  17

9.53960E  18 1.39731E  17 2.00176E  17 3.26314E  17 4.53142E  17

1.34004E  17 2.87494E  17 3.63192E  17 4.23957E  17 5.36124E  17

1.91636E  17 3.83687E  17 5.32746E  17 6.12680E  17 7.14350E  16

a

b 0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0 5

4

3

t

2

1

0

-10

-5

0

5

x

10

0 5

4

3

t

2

1

0

-10

-5

0

5

10

x

Fig. 1. (a) The numerical results for u6(x, t) with fixed value of x0 = 5 for different time values and (b) the exact solution of Kawahara equation.

M.T. Darvishi, F. Khani / Chaos, Solitons and Fractals 39 (2009) 2484–2490

a

2489

b 0.15

0.15

0.1

0.1

0.05

0.05

0

0

-0.05 -6

-4

-2

x

0

2

4

6

1 0 0.5

2 1.5

3 2.5

-0.05 -6

-4

t

-2

x

0

2

4

6

0

2 1.5

3 2.5

t

1 0.5

Fig. 2. (a) The numerical results for u6(x, t) with fixed values of x0 = 0.5 and k = 0.2 for different time values and (b) the exact solution of Lax’s fifth-order KdV equation.

a 0.08

b

0.08

0.06

0.06

0.04

0.04

0.02

5

0.02

5

4 3

0 -20

2 -10

0

x

1 10

20

0

t

4 3

0 -20

2 -10

0

x

1 10

20

t

0

Fig. 3. (a) The numerical results for u6(x, t) with fixed values of x0 = 0.5 and k = 0.2 for different time values and (b) the exact solution of Sawada–Kotera equation.

space, it is not effected by computation a round off errors and one is not faced with necessity of large computer memory and time.

5. Conclusions In this paper we solved some problems by variational iteration method, this method dose not require small parameter in any equation as same as the perturbation approach. The results show that 1. A correction functional can be easily constructed by a general Lagrange multiplier, and this multiplier can be optimally identified by variational theory. The application of restricted variations in correction functional makes it much easier to determine the multiplier. 2. The initial approximation can be freely selected with unknown constants, which can be determined via various methods. 3. In comparison with results of Adomian’s decomposition method, we can see that the approximations that obtained by variational iteration method converges faster than Adomian’s decomposition method.

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