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Time-Fractal KdV Equation: Formulation and Solutionusing Variational Methods

S. A. El-Wakil, E. M. Abulw afa, M. A. Zahran and A. A. Mahmoud

Theoretical Physics Research Group, Physics Department, Faculty of Science, Mansoura University,

Mansoura 35516, Egypt

Abst ract: The Lagrangian of the time fractal KdV equation is used in similarform to the Lagrangian of the regular KdV equation. The variation of thefunctional of this Lagrangian leads to the Euler-Lagrange equation that leads tothe time fractal KdV equation. The Riemann-Liouvulle definition of thefractional derivative is used to describe the time fractal operator in the fractalKdV equation. The variational-iteration method given by He is used to solve thederived time fractal KdV equation. The calculations of the solution with initial

condition )(sec 2 cxh are carried out and represented in 3-dimensional and 2-

dimensional figures. The results are represented for different values of thenonlinear and dissipation coefficients, fractal order and constant (c).

1. Introduction

Because most classical processes observed in the physical world are nonconservative, it isimportant to be able to apply the power of variational methods to such cases. A method [1]used a Lagrangian that leads to an Euler-Lagrange equation that is, in some sense,equivalent to the desired equation of motion. Hamiltons equations are derived from theLagrangian and are equivalent to the Euler-Lagrange equation. If a Lagrangian is

constructed using noninteger-order derivatives, then the resulting equation of motion can benonconservative. It was shown that such fractional derivatives in the Lagrangian describenonconservative forces [2, 3]. Further study of the fractional Euler-Lagrange can be foundin the work of Agrawal [4-6], Baleanu and coworkers [7-9] and Tarasov and Zaslavsky [10,11].

During the last decades, Fractional Calculus has been applied to almost every field ofscience, engineering and mathematics. Some of the areas where Fractional Calculus has

been applied include viscoelasticity and rheology, electrical engineering, electrochemistry,biology, biophysics and bioengineering, signal and image processing, mechanics,mechatronics, physics, and control theory [12].

The Kortewegde Vries (KdV) equation has been found to be involved in a wide range ofphysics phenomena as a model for the evolution and interaction of nonlinear waves. It wasfirst derived as an evolution equation that governing a one dimensional, small amplitude,long surface gravity waves propagating in a shallow channel of water [13]. Subsequently theKdV equation has arisen in a number of other physical contexts as collision-freehydromagnetic waves, stratified internal waves, ion-acoustic waves, plasma physics, latticedynamics, etc [14]. Certain theoretical physics phenomena in the quantum mechanicsdomain are explained by means of a KdV model. It is used in fluid dynamics, aerodynamics,and continuum mechanics as a model for shock wave formation, solitons, turbulence,

boundary layer behavior, and mass transport. All of the physical phenomena may be


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considered as nonconservative, so they can be described using fractal differential equations.Therefore, in this paper, we will formulate a time-fractal KdV equation version using theEuler-Lagrange equation by variational method [4-6, 15].

Several methods have been used to solve fractional differential equations such as: theLaplace transform method [16], the Fourier transform method [17], the iteration method[18] and the operational method [19]. However, most of these methods are suitable for

special types of fractional differential equations, mainly the linear with constantcoefficients. Recently, there are some papers deal with the existence and multiplicity ofsolution of nonlinear fractional differential equation by the use of techniques of nonlinearanalysis (fixed-point theorems, LerayShauder theory, Adomian decomposition method,variational-iteration method, etc.), see [20-24]. In this paper, the resultant fractional KdVequation will be solved using a variational-iteration method (VIM) [25-27].

This paper is organized as follows: Section 2 is devoted to describe the formulation of thetime-fractal KdV (fKdV) equation using the variational Euler-Lagrange method. In section3, the resultant time-fKdV equation is solved approximately using VIM. Section 4 containsthe results of calculations and discussion of these results.

2. The time-fractal KdV equation

The regular KdV equation in (1+1) dimensions is given by [13]

0),(),(),(),(3

3

=

+

+

txu

xBtxu

xtxuAtxu

t, (1)

where ),( txu is a field variable, Rx is a space coordinate in the propagation direction of

the field and Tt (= ],0[ 0T ) is the time andAandBare known coefficients.

Using a potential function ),( txv where ),( txu = ),( txvx gives the potential equation of the

regular KdV equation (1) in the form

0),(),(),(),( =++ txvBtxvtxvAtxv xxxxxxxxt , (2)

where the subscripts denote the partial differentiation of the function with respect to theparameter. The Lagrangian of this regular KdV equation (1) can be defined using the semi-inverse method [28, 29] as follows.

The functional of the potential equation (2) can be represented by

)(vJ = TR dtdx {v(x, t)[ ),(),(),(),( 321 txBvctxvtxAvctxvc xxxxxxxxt ++ ]}, (3)

where c1, c2 and c3are constants to be determined. Integrating by parts and taking Rtv =

Rxv =

Txv = 0lead to

)(vJ = TR dtdx [ ),(),(21

),(),( 233

21 txBvctxAvctxvtxvc xxxxt + ]. (4)

The unknown constants (ci, i=1, 2, 3) can be determined by taking the variation of thefunctional (4) to make it optimal. Taking the variation of this functional, integrating eachterm by parts and make the variation optimum give the following relation

),(2),(),(3),(2 321 txBvctxvtxAvctxvc xxxxxxxxt ++ = 0. (5)


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As this equation must be equal to equation (2), the unknown constants are given as

c1= , c2= 1/3 and c3= . (6)

Therefore, the functional given by (4) gives the Lagrangian of the regular KdV equation as

),,( xxxt vvvL = ),(2

1),(

6

1),(),(

2

1 23 txBvtxAvtxvtxv xxxxt + . (7)

Similar to this form, the Lagrangian of the time-fractal version of the KdV equation can bewritten in the form

),,(0 xxxt vvvDF = ),(

2

1),(

6

1),()],([

2

1 230 txBvtxAvtxvtxvD xxxxt + , 10


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Substituting the Lagrangian of the time-fKdV equation (8) into this Euler-Lagrange formula(15) gives

0),(),(),(),(2

1),(

2

100

=+++ txvBtxvtxvAtxvDtxvD xxxxxxxxtxTt . (16)

Substituting for the potential function, ),( txvx = ),( txu , gives the time-fractal KdV equation

for the state function ),( txu in the form

0),(),(),(),(2

1),(

2

100

=++ txuBtxutxuAtxuDtxuD xxxxTtt , (17)

where the fractional derivatives ),(0 txuDt and ),(

0txuDTt

are, respectively the left and

right Riemann-Liouville fractional derivatives and are defined by equations (9) and (13).

The time-fKdV equation represented in (17) can be rewritten by the formula

0),(),(),(),( =++ txuBtxutxuAtxuD xxxxt , 10


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),(1 txun+ = ),( txun + )],(),(~),([)(

0 xfxuNxuLd nn

t+ , 0n , (21)

where )( is a Lagrangian multiplier and ),(~ xu is considered as a restricted variation

function [26, 27], i. e; 0),(~ = xu . Extreme the variation of the correction functional (21)

leads to the Lagrangian multiplier )( . The initial iteration ),(0 txu can be used as the

solution of the linear part of (20) or the initial value )0,(xu . As n tends to infinity, the

iteration leads to the exact solution of (20), i. e;

),(lim),( txutxu nn

= . (22)

For linear problems, the exact solution can be given using this method in only one stepwhere its Lagrangian multiplier can be exactly identified.

4. Time-fractal KdV equation Solution

The time-fKdV equation represented by (18) can be solved using the VIM by the iterationcorrection functional (21) as follows:

Affecting from left by the fractal operator 1tD on equation (20) leads to

),( txut

+ 1tD [ ),(),(),( 3

3

txux

Btxux

txuA nnn

+

] =0,

10


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which leads to the Lagrangian multiplier as

1)( = .

Therefore, the correction functional (25) is given in the form

),(1 txun+ = ),( txun

- td0 { ),(

xun

+ 1D [ ),(),(),( 3

3

xux

Bxux

xuA nnn+

]},

0n . (26)

The zero order correction of the solution can be taken as the initial value of the statevariable, which is taken in this case as

),(0 txu = )0,(xu = )(sec2

cxh . (27)

where cis a constant.

Substituting this zero order approximation into (26) and using the definition of the fractional

derivative definition of (19) lead to the first order approximation as),(1 txu = )(sec

2cxh

+ )](sec)12(4[)(sec)sinh(2 2223 cxhBcABccxhcxc +)1( +

t

. (28)

Substituting this equation into (26), using the definition (19) and the Maple package lead tothe second order approximation in the form

),(2 txu = )(sec2

cxh

+ )](sec)12(4[)(sec)sinh(22223

cxhBcABccxhcxc + )1( +

t

+ )(sec)635(1632)[(sec2 2222422 cxhBcABcBccxhc +

+ )(sec)16801763(2 42422 cxhBcABcA +

- )](sec)36042(7 62422 cxhBcABcA +)12(

2

+

t

+ )(sec)14(2432[)(sec)sinh(4 2222453 cxhBcABcBccxhcxAc +

+ )(sec)24032(442422

cxhBcABcA +

- )](sec)14424(5 62422 cxhBcABcA +)1(

)12(

+

+

)13(

3

+

t

. (29)

The higher order approximations can be calculated using the Maple or the Mathematicapackage to the appropriate order where the infinite approximation leads to the exactsolution.


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5. Results and calculations

The calculations are carried out for the solution of the fKdV equation using the IVM for twocases of the equation parameters of nonlinearity (A) and dissipation (B). The two cases areof (A= 1 andB= 1) and (A= 6 andB= 1). The initial value of the solution for all cases is

taken as )(sec 2 cxh where the constant (c) is used as B/A, B/4A, B/8A or B/12A. The

solution is calculated for different values of the fractal order (= 3/4, 1/2, 1/3 and 1/4).

The 3-dimensional representation of the solution of the fKdV equation with space and time

for constant c = B/4AandA= B= 1 for different values of the fractal order () is given inFig (1). While in Fig (2), the 3-dimensional representation of the solution of fKdV equation

withA=B= 1 is given for the fractal order = 1/4 for different values of the constant c=B/A, B/4A, B/8A and B/12A.

The same calculations are carried out and represented in 3-dimensional shape in Fig (3) andFig (4) for fKdV equation of the same parameters as in Fig (1) and Fig (2) except usingnonlinearity parameterA= 6 and dissipation parameterB= 1.

Figure (5) represents the solution of the fKdV equation for fractal parameter = 1/3 with

initial condition )(sec 2 cxh has constant c=B/4AusingA=B= 1 in case and in other caseA= 6,B=1.

In Fig (6), the solution of the fKdV equation u(x, t), forA=B= 1 with initial condition u(x,

0) = )(sec 2 cxh for constant c = B/4A, is calculated for two different values of = 3/4 and

1/4. These calculations are represented as 2-dimensional figures for different time values.

Figure (7) shows the calculations for = 1/2 for two different constant values c=B/4AandB/8Aas a function of space (x) at different time values.

The calculations of the solution of fKdV for fractal factor = 1/3 and constant c=B/4Aisrepresented as a 2-dimensional space graph for different time values. These calculations are

represented in Fig (8) for fKdV parametersA=B= 1 andA= 6,B= 1.


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(a) c=B/A (b) c=B/4A

(c) c=B/8A (d) c=B/12A

Fig. 2: Time Fractal KdV Equation with A=1 and B=1 and fractal order = 1/4 solvedusing VIM to the forth order approximation with zero-order solution

)(sec 20 cxhu = with different values of the constant (c).


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(a) = 3/4 (b) = 1/2

(c) = 1/3 (d) = 1/4

Fig. 3:Time Fractal KdV Equation withA=6 andB=1 solved using VIM to the forth order

approximation with zero-order solution )(sec 20 cxhu = with c = B/4Afor different

values of the fractal order ().


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(a) c=B/A (b) c=B/4A

(c) c=B/8A (d) c=B/12A

Fig. 4: Time Fractal KdV Equation with A=6 and B=1 and fractal order = 1/4 solvedusing VIM to the forth order approximation with zero-order solution

)(sec 20 cxhu = with different values of the constant (c).


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(a)A= 1,B= 1

(b)A= 6,B= 1

Fig. 5:Time Fractal KdV Equation with fractal order = 1/4 solved using VIM to the forth

order approximation with zero-order solution )(sec 20 cxhu = with constant c =

B/4Afor different values ofAandB.


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(a)

t=0 solid black, t=1 brown dot, t=3 blue dash, t=5 red dash-dot, t=7 green dash.

(b)t=0 black solid, t=5 brown dot, t=10 blue dash, t=15 red dash-dot and t=20 green dash.

Fig. 6: The distribution function u(x, t)as a function of space (x) for different time values

withA=B= 1 and initial condition )(sec2

0 cxhu = , c=B/4A:

(a) = 3/4 and (b) = 1/4.


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(a)

t=0 black solid, t=3 brown dot, t=5 blue dash, t=7 red dash-dot, t=10 green dash.

(b)

t=0 black solid, t=10 brown dot, t=20 blue dash, t=30 red dash-dot, t=40 green dash.

Fig. 7: The distribution function u(x, t) as a function of space (x) for different time values,

A=B=1 and initial condition )(sec 20 cxhu = , =1/2:

(a) c=B/4Aand (b) c=B/8A.


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(a)

t=0 solid black, t=5 brown dot, t=10 blue dash, t=15 red dash-dot and t=20 solid green

(b)

t=0 solid black, t=3 brown dot, t=5 blue dash, t=7 red dash-dot, t=10 green dash.

Fig. 8: The distribution function u(x, t) as a function of space (x) for different time values,

with initial condition )(sec 20 cxhu = , =1/3, c=B/4Aand forAandBas:

(a)A= 1,B= 1 and (b)A= 6,B=1.

time fractal

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