The functional variable method for solving the fractional Korteweg de Vries equations and the coupled Korteweg de Vries equations

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1 PRAMANA c Indian Academy of Sciences Vo. 85, No. 4 journa of October 015 physics pp The functiona variabe method for soving the fractiona Korteweg de Vries equations and the couped Korteweg de Vries equations M MATINFAR, M ESLAMI and M KORDY Department of Mathematics, Facuty of Mathematica Sciences, University of Mazandaran, Babosar, Iran Corresponding author. E-mai: m_kordy_1391@yahoo.com MS received 5 March 014; revised 3 May 014; accepted 1 August 014 DOI: /s ; epubication: 1 May 015 Abstract. This paper presents the exact soutions for the fractiona Korteweg de Vries equations and the couped Korteweg de Vries equations with time-fractiona derivatives using the functiona variabe method. The fractiona derivatives are described in the modified Riemann Liouvie derivative sense. It is demonstrated that the cacuations invoved in the functiona variabe method are extremey simpe and straightforward and this method is very effective for handing noninear fractiona equations. Keywords. Korteweg de Vries equation; couped Korteweg de Vries equation; functiona variabe method. PACS Nos 0.30.Jr; Yv 1. Introduction The physica and engineering processes have been modeed by means of fractiona cacuus, which are found to be best described by fractiona differentia equations. Unfortunatey, in many cases, the standard mathematica modes of integer-order derivatives, incuding noninear modes, do not work adequatey. Fractiona cacuus has payed a very important roe in various fieds such as economics, chemistry, notaby contro theory, eectricity, mechanics, ground water probems, bioogy and signa image processing. Earier, the investigation of traveing-wave soutions for noninear equations had a very effective roe in the study of noninear physica phenomena. A wide range of physics phenomena has been described by means of the Korteweg de Vries KdV) equation which has been used to mode the evoution and interaction of noninear waves. The evoution of the Korteweg de Vries equation, which has been a ong process, started about 80 years ago, began with the experiments of Scott Russe in 1834 [1], Pramana J. Phys., Vo. 85, No. 4, October

2 M Matinfar, M Esami and M Kordy the investigations of Boussinesq and Rayeigh around 1870 [ 5], and finay cuminated with the paper by Korteweg and De Vries in 1895 [6]. It was derived as an evoution equation governing a one-dimensiona, sma-ampitude, ong-surface gravity wave propagating in a shaow channe of water. Subsequenty, the KdV equation has thrived in other physica contexts as ion-acoustic waves, pasma physics, coision-free hydromagnetic waves, attice dynamics, stratified interna waves, etc. [7]. The KdV mode has been used to expain certain theoretica physics phenomena in the quantum mechanics domain. It is used as a mode for shock wave formation, soitons, turbuence, boundary ayer behaviour, and mass transport in fuid dynamics, aerodynamics, and continuum mechanics. Now consider the genera Korteweg de Vries equation of the form u t + p + 1)p + )u p u x + u xxx = gx,t), 1) where gx,t) is a given function and p = 1,,... with u,u x,u xx 0as x.if p = 0,p= 1, and p =, eq. 1) becomes inearized KdV, noninear KdV, and modified KdV equations, respectivey. The noninear KdV equation has been the focus of recent studies for finding exact soutions in [8 10] as we as numerica soutions in [11 13]. Recenty, there has been much interest in fractiona diffusion equations. These equations arise in continuous-time random waks, modeing of anomaous diffusive and sub-diffusive systems, unification of diffusion and wave propagation phenomenon, and simpification of the resuts [14]. The nature of the diffusion is characterized by a meansquare dispacement of the form r t) t α. ) For anomaous subdiffusion α<1 and for anomaous superdiffusion α>1, whereas in standard diffusion α = 1. For appications on both types of anomaous diffusions, one can refer to [15,16]. In this work, the exact soutions of the noninear KdV equation with time- and spacefractiona derivatives are considered which is of the foowing form: α u t + u α εu β x + u β v 3 = 0, t > 0, 3) x3 where ε and v are parameters and α and β are parameters describing the order of the fractiona time- and space-derivatives, respectivey. The function ux, t) is assumed to be a causa function of time and space, i.e., vanishing for t<0and x<0. When α = 1 and β = 1, the fractiona equation reduces to the cassica noninear KdV equation. Hirota and Satsuma proposed a couped KdV equation, which describes the interactions of two ong waves with different dispersion reations [17]. If one of the ong waves never affects the other, the atter obeys the ordinary KdV equation. However, the behaviour of the KdV soitons accepts the infuence of the existence of the former. Soutions show that the former determines the veocity of the KdV soiton. The couped KdV equations with time-fractiona derivatives are given as α u t + 6au u α x bv v x + a 3 u = 0, 0 <α 1, x3 β v t + 3bu v β x + v b 3 = 0, 0 <β 1, 4) x3 584 Pramana J. Phys., Vo. 85, No. 4, October 015

3 Soving the fractiona and couped Korteweg de Vries equations where a and b are constants, α and β are parameters describing the order of the timefractiona derivatives of ux;t)and vx;t), respectivey. The functions ux;t)and vx;t) are assumed to be causa functions of time and space, i.e., vanishing for t<0and x<0. When α = β = 1, the above system reduces to the cassica couped KdV equations. Many effective anaytic methods such as the Adomian decomposition method [18,19], homotopy perturbation method [0 ], variationa iteration method [3], fractiona subequation method [4], Lagrange characteristic method [5], first integra method [6 9], and so on [30 3], have been deveoped to derive approximate or exact soutions of fractiona ordinary differentia equations, integra equations, and fractiona partia differentia equations. A direct and effective method to sove noninear partia differentia equations was first proposed by Zerarka [33]. Soon after, this method became popuar among the other researchers and was deveoped by many in [34 36]. The aim of this paper is to show the abiity of this powerfu method for finding exact soutions of the fractiona KdV equations and couped KdV equations. The rest of this paper is organized as foows. In, the functiona variabe method is briefy described. In 3, the functiona variabe method is appied for finding exact soutions of the fractiona KdV equations and couped KdV equations.. The modified Riemann Liouvie derivative and the functiona variabe method The modified Riemann Liouvie derivative was defined by Jumarie [37,38] as 1 x x ξ) α 1 [ fξ) f0) ] dξ, α < 0, Ŵ α) 0 Dx a f x) = 1 d x x ξ) α [ fξ) f0) ] dξ, α < 0 < 1, Ŵ α) dx 0 f n) x) ) α n),n α n + 1, n 1, 5) Here Ŵ ) is the gamma function and f: R R,x fx) denote a continuous but not necessariy differentiabe) function. One of the properties of the fractiona-modified Riemann Liouvie derivative can be stated as foows: D α x xγ = Ŵ1 + γ) Ŵ1 + γ α) xγ α, γ > 0, 6) Dx α ux)vx)) = vx)dα x ux) + ux)dα x v x), 7) D α x [ ] fux)) = f u u)d x α ux) = Dα x fu) u ) α x. 8) For other properties, refer to [6,39]. Pramana J. Phys., Vo. 85, No. 4, October

4 M Matinfar, M Esami and M Kordy Let us present the features of the functiona variabe method. For a given time-fractiona differentia equation that it is written in severa independent variabes { t,x,y,z,...} and a dependent variabe u as D u,d α t u,u x,u y,u z,d α t u,u xy,u yz,u xz,... ) = 0, 9) where the subscripts denote partia derivatives. First, the variabe transformation is introduced ξ = 1 x + y + 3 z + λtα Ŵ1 + α), 10) where i and λ are constants to be determined ater to find the traveing wave soution of eq. 9), so that ut,x,y,z,...) = Uξ). 11) On using this transformation eq. 9) can be reduced to an ordinary differentia equation ODE) Q U,U ξ,u ξξ,u ξξξ,... ) = 0, 1) where Q is a poynomia in u and its tota derivatives. Then we make a transformation in which the unknown function Uξ) is considered as a functiona variabe in the form U ξ = Fu), 13) and some successive derivatives of U are U ξξ = 1 F ), U ξξξ = 1 F ) F, U ξξξξ = 1 [ F ) F + F ) F ) ], 14) where stands for d/du. Substituting 14) in 1), the ODE 1) can be reduced in terms of u, f, and its derivatives as R U,F,F,F,F,... ) = 0. 15) 586 Pramana J. Phys., Vo. 85, No. 4, October 015

5 Soving the fractiona and couped Korteweg de Vries equations Equation 15) is particuary important because it admits anaytica soutions for a arge cass of noninear wave-type equations. After integration, eq. 15) provides the expression for F, and this together with eq. 13) give reevant soutions to the origina probem. In order to iustrate how the method works, we examine some exampes treated by other approaches. This is discussed in the foowing section. 3. Appications In this section, two cases of fractiona KdV equations and couped-kdv equations are studied using of the functiona variabe method. Exampe 3.1. The KdV equation has an important roe to pay in noninear physics and it has been used in a number of other physica contexts. Now, consider the time-fractiona KdV equation D α t u + 6uu x + u xxx = 0, 0 <α 1, t > 0, 16) where α is a parameter describing the order of the fractiona time derivative, t is the time, and x is the space coordinate in the direction of propagation. Now, on appying the transformation ux,t) = Uξ), ξ = x λtα Ŵ1 + α), 17) to eq. 16) and integrating the resuting equation once, we get the ODE or 3 U λu + 3U = 0 U ξξ = λ 3 U 3 U. 18) Then we use the transformation U ξ = FU). 19) Using the above transformation in eq. 16) eads to 1 F U) ) = λ U 3 3 U 0) or F U) = λ 3 U U 3. 1) According to eq. 14), we get from eq. 1), the expression for the function FU) which reads as λ FU)=± U 1 U. ) 3 λ After appying the change of variabes Z = U, 3) λ Pramana J. Phys., Vo. 85, No. 4, October

6 M Matinfar, M Esami and M Kordy and using the reation 11), eq. ) gives the foowing soution: Uξ)= λ ) 1 λ sech ξ. 4) 3 Obviousy, by setting eq. 4) in eq. 17) the hyperboic soutions of the time-fractiona KdV equation is obtained as u 1 x,t) = λ ) 1 λ x ) sech λtα, 5) 3 Ŵ1 + α) u x,t) = λ 1 λ csch 3 ) x ) λtα. 6) Ŵ1 + α) For λ/ 3 < 0, the periodic soutions can be obtained as foows: u 3 x,t) = λ 1 λ ) ) sec x λtα, Ŵ1 + α) 7) u 4 x,t) = λ 1 λ ) ) csc x λtα. Ŵ1 + α) 8) Exampe 3.. Consider the foowing space and time-fractiona KdV equation: D α t u + udβ x + u xxx = 0. 9) For soving this equation first, we introduce the foowing transformation: ux,t) = Uξ), ξ = kxβ Ŵ1 + β) λtα Ŵ1 + α). 30) Substituting 30) in eq. 9) we get the ODE λu + kuu + k 3 U = 0. 31) After one integration, we get λu + ku + k3 U = 0. 3) We use the transformation U ξ = FU). 33) So eq. 3) can be converted to 1 F U) ) λ = k U 1 3 k U. 34) Then the expression for the function FU)is obtained as λ FU)=± k U 3 1 k U. 35) 3λ 588 Pramana J. Phys., Vo. 85, No. 4, October 015

7 Soving the fractiona and couped Korteweg de Vries equations For soving the above equation, the foowing change of variabe is appied: Z = k U. 36) 3λ Using the reation 13), eq. 35) has the foowing soution: Uξ)= 3λ ) 1 λ k sech k ξ. 37) 3 When λ/k 3 > 0, we can easiy get the hyperboic as foows: u 1 x,t) = 3λ 1 λ kx β ) ) k sech k 3 Ŵ1 + β) λtα, Ŵ1 + α) 38) u x,t) = 3λ 1 λ kx β ) ) k csch k 3 Ŵ1 + β) λtα. Ŵ1 + α) 39) If λ/k 3 < 0, it is evident that soutions 38) and 39) can reduce to periodic soutions as foows: u 3 x,t) = 3λ 1 k sec λ kx β ) ) k 3 Ŵ1 + β) λtα, 40) Ŵ1 + α) u 4 x,t) = 3λ 1 k csc λ kx β ) ) k 3 Ŵ1 + β) λtα. 41) Ŵ1 + α) Exampe 3.3. In this section, the couped KdV equations are soved with time-fractiona derivatives of the form α u t + 6au u α x bv v x + a 3 u = 0, 0 <α 1, x3 β v t + 3bu v β x + v b 3 = 0, 0 <β 1. 4) x3 In order to obtain soutions of eq. 4), the foowing transformations are introduced: vx,t) = 1 Uξ), ux,t) = Uξ), ξ = x λ 1t α Ŵ1 + α) λ t β Substituting 43) in eq. 4), the equation can be converted to the ODE Ŵ1 + β). 43) λ 1 λ ) U + 6a + 3b b ) UU + a + b ) U = 0. 44) After one integration, we get λ 1 λ ) U + 6a + 3b b ) U + a + b ) U = 0. 45) Pramana J. Phys., Vo. 85, No. 4, October

8 M Matinfar, M Esami and M Kordy Then we use the transformation U ξ = FU). 46) So, eq. 45) wi convert to 1 F U) ) λ 1 + λ = a + b U 6a + 3b b U. 47) a + b) Thus, we get from eq. 47) the expression for the function FU)which reads as λ1 + λ FU)=± a + b U The soutions of eq. 48) can be obtained as Uξ)= 3 λ 1 + λ ) 6a + 3b b sech 1 6a + 3b b U. 48) 3 λ 1 + λ ) 1 ) λ1 + λ a + b ξ. 49) For λ 1 + λ )/a + b) > 0, we get the foowing hyperboic soutions: u 1 x,t) = 3 λ 1 + λ ) 6a + 3b b sech 1 λ1 + λ a + b x λ 1t α Ŵ1 + α) λ t β ) ), 50) Ŵ 1 + β) v 1 x,t) = 1 ) 3 λ1 + λ ) 6a + 3b b sech 1 λ1 + λ a + b x λ 1t α Ŵ1 + α) λ t β ) ), 51) Ŵ 1 + β) and u x,t) = 3 λ 1 + λ ) 6a + 3b b csch 1 v x,t) = 1 csch 1 λ1 + λ a + b x λ 1t α Ŵ1 + α) λ t β ) ), 5) Ŵ 1 + β) ) 3 λ1 + λ ) 6a + 3b b λ1 + λ a + b x λ 1t α Ŵ1 + α) λ t β ) ). 53) Ŵ 1 + β) 590 Pramana J. Phys., Vo. 85, No. 4, October 015

9 Soving the fractiona and couped Korteweg de Vries equations For λ 1 + λ )/a + b) < 0, we get the foowing periodic soutions: u 3 x,t) = 3 λ 1 + λ ) 6a + 3b b sec 1 λ 1 + λ x λ 1t α a + b Ŵ1 + α) λ t β ) ), 54) Ŵ 1 + β) 3 λ1 + λ ) ) v 3 x,t) = 1 sec 1 λ 1 + λ a + b 6a + 3b b x λ 1t α Ŵ1 + α) λ t β Ŵ 1 + β) ) ), 55) and u 4 x,t) = 3 λ 1 + λ ) 6a + 3b b csc 1 λ 1 + λ x λ 1t α a + b Ŵ1 + α) λ t β ) ), 56) Ŵ 1 + β) 3 λ1 + λ ) ) 4. Concusions v 4 x,t) = 1 csc 1 λ 1 + λ a + b 6a + 3b b x λ 1t α Ŵ1 + α) λ t β Ŵ 1 + β) ) ). 57) In this paper, the functiona variabe method and the modified Riemann Liouvie derivative are presented for soving the fractiona KdV and the couped KdV equations. It is predicted that the obtained soutions in this paper wi be usefu for further investigating the compicated noninear physica phenomena. This method introduces a promising too for soving many fractiona partia differentia equations and it is aso a reiabe technique to hande noninear fractiona differentia equation. The cacuations of functiona variabe method are very simpe and straightforward. Thus, we deduce that this method can be appied to sove many systems of noninear fractiona partia differentia equations. References [1] J Scott Russe, J. Murray London) ) [] J Boussinesq, C. R. Acad. Sci. Paris) 7, ) [3] J Boussinesq, C. R. Acad. Sci. Paris) 73, ) [4] J Boussinesq, J. Math. Pure App. 17, ) Pramana J. Phys., Vo. 85, No. 4, October

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