TimeFractional KdV Equation: Formulation and Solution using Variational Methods
Abstract
The Lagrangian of the time fractional KdV equation is derived in similar form to the Lagrangian of the regular KdV equation. The variation of the functional of this Lagrangian leads to the EulerLagrange equation that leads to the time fractional KdV equation. The RiemannLiouvulle definition of the fractional derivative is used to describe the time fractional operator in the functional of the EulerLagrange formula. The timefractional term of the derived KdV equation is represented as a Riesz fractional derivative. The variationaliteration method given by He [31] is used to solve the derived timefractional KdV equation. The calculations of the solution with initial condition h are carried out and demonstrated in 3dimensions and 2dimensions figures. We give the comparison and estimates of the role of fractional derivative to the nonlinear and dispersion terms in the fractional KdV equation for unit amplitude . It worth mentioned that, exploitation of the fractional calculus for fractional equation provides not only new types of mathematical construction, but also new physical features of the described phenomena.

EulerLagrange equation, RiemannLiouvulle definition of the fractional derivative, Riesz fractional derivative, fractional KdV equation, He’s variationaliteration method.

PACS: 05.45.Df, 05.30.Pr
1 Introduction
Because most classical processes observed in the physical world are nonconservative, it is important to be able to apply the power of variational methods to such cases. A method [1] used a Lagrangian that leads to an EulerLagrange equation that is, in some sense, equivalent to the desired equation of motion. Hamilton’s equations are derived from the Lagrangian and are equivalent to the EulerLagrange equation. If a Lagrangian is constructed using nonintegerorder derivatives, then the resulting equation of motion can be nonconservative. It was shown that such fractional derivatives in the Lagrangian describe nonconservative forces [2, 3]. Further study of the fractional EulerLagrange can be found in the work of Agrawal [46], Baleanu and coworkers [79] and Tarasov and Zaslavsky [10, 11].
Recently, fractional calculus has been applied to almost every field of science, engineering and mathematics. The awareness of the importance of this type of equation has grown continuously in last decade include for viscoelasticity and rheology, image processing, mechanics, mechatronics, physics, and control theory, see for instance [12].
On the other hand, the Korteweg–de Vries (KdV) equation has been found to be involved in a wide range of physics phenomena as a model for the evolution and interaction of nonlinear waves. It was first 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 the KdV equation has arisen in a number of other physical contexts as collisionfree hydromagnetic waves, stratified internal waves, ionacoustic waves, plasma physics, lattice dynamics, etc [14]. Certain theoretical physics phenomena in the quantum mechanics domain 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 considered as nonconservative, so they can be described using fractional differential equations. Therefore, in this paper, our motive is to formulate a timefractional KdV equation version using the EulerLagrange equation via what is called variational method [46, 15].
Several methods have been used to solve fractional differential equations such as: the Laplace transformation method [16, 17], the Fourier transformation method [16, 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 constant coefficients. Recently, there are some papers deal with the existence and multiplicity of solution of nonlinear fractional differential equation by the use of techniques of nonlinear analysis (fixedpoint theorems, Leray–Shauder theory, Adomian decomposition method, variationaliteration method, etc.), see [2024]. In this paper, the resultant fractional KdV equation will be solved using a variationaliteration method (VIM) [2527]. In addition, we give the comparison and estimates of the role of fractional derivative to the nonlinear and dispersion terms in the fractional KdV equation for unit amplitude .
This paper is organized as follows: Section 2 is devoted to describe the formulation of the timefractional KdV (FKdV) equation using the variational EulerLagrange method. In section 3, the resultant timeFKdV equation is solved approximately using VIM. Section 4 contains the results of calculations and discussion of these results.
2 The timefractional KdV equation
The regular KdV equation in (1+1) dimensions is given by [13]
(1) 
where is a field variable, is a space coordinate in the propagation direction of the field and () is the time variable and and are known coefficients.
Using a potential function , where , gives the potential equation of the regular KdV equation (1) in the form
(2) 
where the subscripts denote the partial differentiation of the function with respect to the parameter. The Lagrangian of this regular KdV equation (1) can be defined using the semiinverse method [28, 29] as follows.
The functional of the potential equation (2) can be represented by
(3) 
where , and are constants to be determined. Integrating by parts and taking lead to
(4) 
The unknown constants ( , , ) can be determined by taking the variation of the functional (4) to make it optimal. Taking the variation of this functional, integrating each term by parts and make the variation optimum give the following relation
(5) 
As this equation must be equal to equation (2), the unknown constants are given as
(6) 
Therefore, the functional given by (6) gives the Lagrangian of the regular KdV equation as
(7) 
Similar to this form, the Lagrangian of the timefractional version of the KdV equation can be written in the form
(1) 
where the fractional derivative is represented, using the left RiemannLiouville fractional derivative definition as [18, 19]
(9) 
The functional of the timeFKdV equation can be represented in the form
(10) 
where the timefractional Lagrangian is defined by (8).
Following Agrawal’s method [3, 4], the variation of functional (10) with respect to leads to
(11) 
The formula for fractional integration by parts reads [3, 18, 19]
(12) 
where , the right RiemannLiouville fractional derivative, is defined by [18, 19]
(13) 
Integrating the righthand side of (11) by parts using formula (12) leads to
(14) 
where it is assumed that .
Optimizing this variation of the functional , i. e; , gives the EulerLagrange equation for the timeFKdV equation in the form
(15) 
Substituting the Lagrangian of the timeFKdV equation (8) into this EulerLagrange formula (15) gives
(16) 
Substituting for the potential function, , gives the timeFKdV equation for the state function in the form
(17) 
where the fractional derivatives and are, respectively the left and right RiemannLiouville fractional derivatives and are defined by (9) and (13).
The timeFKdV equation represented in (17) can be rewritten by the formula
(18) 
where the fractional operator is called Riesz fractional derivative and can be represented by [4, 18, 19]
(2) 
The nonlinear fractional differential equations have been solved using different techniques [1823]. In this paper, a variationaliteration method (VIM) [24, 25] has been used to solve the timeFKdV equation that formulated using EulerLagrange variational technique.
3 VariationalIteration Method
Variationaliteration method (VIM) [2527] has been used successfully to solve different types of integer nonlinear differential equations [3135]. Also, VIM is used to solve linear and nonlinear fractional differential equations [25, 3638]. This VIM has been used in this paper to solve the formulated timeFKdV equation.
A general Lagrange multiplier method is constructed to solve nonlinear problems, which was first proposed to solve problems in quantum mechanics [25]. The VIM is a modification of this Lagrange multiplier method [26, 27]. The basic features of the VIM are as follows. The solution of a linear mathematical problem or the initial (boundary) condition of the nonlinear problem is used as initial approximation or trail function. A more highly precise approximation can be obtained using iteration correction functional. Considering a nonlinear partial differential equation consists of a linear part , nonlinear part and a free term represented as
(20) 
where is the linear operator and is the nonlinear operator. According to the VIM, the ()th approximation solution of (20) can be given by the iteration correction functional as [24, 25]
(21) 
where is a Lagrangian multiplier and is considered as a restricted variation function, i. e; . Extreme the variation of the correction functional (21) leads to the Lagrangian multiplier . The initial iteration can be used as the solution of the linear part of (20) or the initial value . As tends to infinity, the iteration leads to the exact solution of (20), i. e;
(22) 
For linear problems, the exact solution can be given using this method in only one step where its Lagrangian multiplier can be exactly identified.
4 Timefractional KdV equation Solution
The timeFKdV equation represented by (18) can be solved using the VIM by the iteration correction functional (21) as follows:
Affecting from left by the fractional operator on (18) leads to
(3) 
where the following fractional derivative property is used [18, 19]
(4) 
As , the Riesz fractional derivative is considered as Riesz fractional integral that is defined by [18, 19]
(25) 
where and are the left and right RiemannLiouvulle fractional integrals, respectively [18, 19].
The iterative correction functional of equation (23) is given as
(5)  
where and the function is considered as a restricted variation function, i. e; . The extreme of the variation of (26) using the restricted variation function leads to
This relation leads to the stationary conditions and , which leads to the Lagrangian multiplier as .
Therefore, the correction functional (32) is given by the form
(6)  
where .
In Physics, if denotes the timevariable, the right RiemannLiouville fractional derivative is interpreted as a future state of the process. For this reason, the rightderivative is usually neglected in applications, when the present state of the process does not depend on the results of the future development [3]. Therefore, the rightderivative is used equal to zero in the following calculations.
The zero order correction of the solution can be taken as the initial value of the state variable, which is taken in this case as
(28) 
where and are constants.
Substituting this zero order approximation into (27) and using the definition of the fractional derivative (19) lead to the first order approximation as
(7)  
Substituting this equation into (27), using the definition (19) and the Maple package lead to the second order approximation in the form
(8)  
The higher order approximations can be calculated using the Maple or the Mathematica package to the appropriate order where the infinite approximation leads to the exact solution.
5 Results and calculations
The calculations are carried out for the solution of the timeFKdV equation using the VIM for different values of the equation parameters of nonlinearity () and dispersion (). The initial value of the solution for all cases is taken as h where the amplitude is taken equal to unity and the constant () is used with different values as , , or . The solution is calculated for different values of the fractional order (, , and ).
The 3dimensional representation of the solution of the timeFKdV equation with space and time for constant and , for different values of the fractional order () is given in Fig (1). While in Fig (2), the 3dimensional representation of the solution of timeFKdV equation with , is given for the fractional order for different values of the constant , , and .
Figure (3) represents the solution of the timeFKdV equation for fractional order with using (, ), (, ), (, ) and (, ).
In Fig (4), the solution of the timeFKdV equation , for (, ) with , is calculated for . These calculations are represented as 2dimensional figure against the space for different values of time .
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Figure Captions
Fig. 1: The distribution function with , , initial condition h and for different values of the fractional order ().
Fig. 2: The distribution function with , , initial condition h and fractional order for different values of the constant ().
Fig. 3: The distribution function with fractional order , initial condition h and for different values of and .
Fig. 4: The distribution function as a function of space () for different values of time with , , initial condition h, and .