Formulation and solution to time-fractional generalized Korteweg-de Vries equation via variational methods
© Zhang; licensee Springer. 2014
Received: 7 November 2013
Accepted: 13 January 2014
Published: 14 February 2014
This paper presents the formulation of the time-fractional generalized Korteweg-de Vries equation (KdV) using the Euler-Lagrange variational technique in the Riemann-Liouville derivative sense and derives an approximate solitary wave solution. Our results show that He’s variational-iteration method is a very efficient technique for finding the solution of the proposed equation and extend the existing results.
The KdV equation has been used to describe a wide range of physics phenomena of the evolution and interaction of nonlinear waves. It was derived from the propagation of dispersive shallow water waves and is widely used in fluid dynamics, aerodynamics, continuum mechanics, as a model for shock wave formation, solitons, turbulence, boundary layer behavior, mass transport, and the solution representing the water’s free surface over a flat bed [1–3]. Camassa and Holm  put forward the derivation of solution as a model for dispersive shallow water waves and discovered that it is a formally integrable dimensional Hamiltonian system, and that its solitary waves are solitons. Most of the classical mechanics techniques have been used in studies of conservative systems, but most of the processes observed in the physical real world are nonconservative.
During the past three decades or so, fractional calculus has obtained considerable popularity and importance as generalization of integer-order evolution equations, and one has used it to model some meaningful things, such as fractional calculus in model price volatility in finance [5, 6], in hydrology to model fast spreading of pollutants ; the most common hydrologic application of fractional calculus is the generation of fractional Brownian motion as a representation of aquifer material with a long-range correlation structure [8, 9]. The fractional differential equation is used to model the particle motions in a heterogeneous environment and long particle jumps in anomalous diffusion in physics [10, 11]. For other exact descriptions of the applications of engineering, mechanics, mathematics, etc., we can refer to [12–16]. If the Lagrangian of conservative system is constructed using fractional derivatives, the resulting equations of motion can be nonconservative. Therefore, in many cases, real physical processes could be modeled in a reliable manner using fractional-order differential equations rather than integer-order equations . Based on the stochastic embedding theory, Cresson  defined the fractional embedding of differential operators and provided a fractional Euler-Lagrange equation for Lagrangian systems, then investigated a fractional Noether theorem and a fractional Hamiltonian formulation of fractional Lagrangian systems. Herzallah and Baleanu  presented the necessary and sufficient optimality conditions for the Euler-Lagrange fractional equations of fractional variational problems and made an effort in determining in which spaces the functional must exist. Malinowska  proposed the Euler-Lagrange equations for fractional variational problems with multiple integrals and proved the fractional Noether-type theorem for conservative and nonconservative generalized physical systems. Riewe [21, 22] formulated a version of the Euler-Lagrange equation for problems of calculus of variation with fractional derivatives. Wu and Baleanu  developed some new variational-iteration formulas to find approximate solutions of fractional differential equations and determined the Lagrange multiplier in a more accurate way. For generalized fractional Euler-Lagrange equations and a fractional-order Van der Pol-like oscillator, we can refer to the works by Odzijewicz [24, 25] and Attari et al. , respectively. For other the known results refer to Baleanu et al.  and Inokuti et al. . In view of the fact that most physical phenomena may be considered as nonconservative, they can be described using fractional-order differential equations. Recently, several methods have been used to solve nonlinear fractional evolution equations using techniques of nonlinear analysis, such as the Adomian decomposition method , the homotopy analysis method [30, 31], the homotopy perturbation method [32, 33] and the Laguerre spectral method [34–36]. It was mentioned that the variational-iteration method has been used successfully to solve different types of integer and fractional nonlinear evolution equations.
where a, b are constants, is a field variable, the subscripts denote the partial differentiation of the function with respect to the parameter x and t. () is a space coordinate in the propagation direction of the field and ( ()) is the time, which occurs in different contexts in mathematical physics. a, b are constant coefficients not equal to zero. The dissipative term provides damping at small scales, and the nonlinear term () (which has the same form as that in the KdV or 1-dimensional Navier-Stokes equations) stabilizes by transferring energy between large and small scales. For , we can refer to the known results of the time-fractional KdV equation: formulation and solution using variational methods [41, 42]. For , , there is little material on the formulation and solution to time-fractional KdV equation. Thus the present paper considers the formulation and solution to a time-fractional KdV equation as the index of the nonlinear term satisfies , . denotes the Riesz fractional derivative. Making use of the variational-iteration method, this work is devoted to a formulation of a time-fractional generalized KdV equation and derives an approximate solitary wave solution.
This paper is organized as follows: Section 2 states some background material from fractional calculus. Section 3 presents the principle of He’s variational-iteration method. Section 4 is devoted to a description of the formulation of the time-fractional generalized KdV equation using the Euler-Lagrange variational technique and to derive an approximate solitary wave solution. Section 5 presents an analysis for the obtained graphs and figures and discusses the present work.
Definition 1 A real multivariable function , is said to be in the space , with respect to t if there exists a real number r (>γ), such that , where . Obviously, if .
is valid under the assumptions that and that for arbitrary , , exist at every point and are continuous in t.
where and are, respectively, the left- and right-hand side Riemann-Liouville fractional integral operators.
where and are, respectively, the left- and right-hand side Riemann-Liouville fractional differential operators.
3 Variational-iteration method
The variational-iteration method provides an effective procedure for explicit and solitary wave solutions of a wide and general class of differential systems representing real physical problems. Moreover, the variational-iteration method can overcome the foregoing restrictions and limitations of approximate techniques so that it provides us with a possibility to analyze strongly nonlinear evolution equations. Therefore, we extend this method to solve the time-fractional KdV equation. The basic features of the variational-iteration method are outlined as follows.
4 Time-fractional generalized KdV equation
noting that .
and since , the fractional derivative operator reduces to the fractional integral operator by Remark 1.
where , is a constant.
The author wrote this paper carefully, gave a rigorous derivation process, and read and approved the final manuscript.
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