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Travelling-Wave Solutions for Some Fractional Partial Differential International Journal of Applied Mathematical Research, 1 (2) (2012) 206-212 °c Science Publishing Corporation www.sciencepubco.com/index.php/IJAMR Travelling-wave solutions for some fractional partial di®erential equation by means of generalized trigonometry functions Zakia Hammouch and Tou¯k Mekkaoui D¶epartement de Math¶ematiquesFST Errachidia BP 509, Boutalamine 52000 Universit¶eMoulay Ismail Morocco [email protected] D¶epartement de Math¶ematiquesFST Errachidia BP 509, Boutalamine 52000 Universit¶eMoulay Ismail Morocco tou¯k¡[email protected] Abstract Making use of a Fan's sub-equation method and the Mittag-Le²er function with aid of the symbolic computation system Maple we con- struct a family of traveling wave solutions of the Burgers and the KdV equations. The obtained results include periodic, rational and soliton- like solutions. Keywords: Euler function, Mittag-Le²er function, TW-solution,Generalized trigonometry functions, Fractional derivative. 1 Introduction Recently it was found that many physical, chemical, biological and medical pro- cesses are governed by fractional di®erential equations (FDEs). As mathemat- ical models of the phenomena, the investigation of exact solutions of (FDEs) will help one to understand these phenomena more precisely. On the other hand the investigation of the exact travelling wave solutions for nonlinear par- tial di®erential equations plays a crucial role in the study of nonlinear physical phenomena. Such solutions when they exist can help us to understand the complicated physical phenomena modeled by these equations [2]. Many pow- erful methods for obtaining exact solutions of nonlinear partial di®erential Exact travelling wave solutions to some fractional di®erential equations 207 equations have been presented such as, Homotopy method [12], Hirota's bilin- ear method [7], Backlund transformation, homogeneous balance method [15], projective Riccati method [13], the sine-cosine method [14] etc. In the present paper we aim to ¯nd exact solitary wave solutions of some non- linear fractional partial di®erential equation with modi¯ed Riemann-Liouville- Jumarie derivatives. Using a sub-equation method [17] we obtain three types of solutions for the studied equations namely; two generalized hyperbolic function solutions, two generalized trigonometric function solutions and one rational so- lution. 2 Preliminaries There are di®erent de¯nitions for fractional derivatives for more details see [11] and the references therein. In this paper we use the modi¯ed Riemann- Liouville derivative de¯ned by Jumarie [8] 8 Z 1 x > ¡®¡1 > (x ¡ s) (f(s) ¡ f(0))ds; for ® < 0; > ¡(1 ¡ ®) 0 > > Z < 1 d x D®f(x) = (x ¡ s)¡®(f(s) ¡ f(0))ds; for 0 < ® < 1; x > ¡(1 ¡ ®) dx > 0 > > · ¸(n) > :> f (®¡n)(x) ; for n · ® < n + 1; n ¸ 1: In the following we shall outline the main idea of the fractional-Riccati equation method. For a given nonlinear fractional equation say ® ® F (u; ux; ut;Dx u; Dt u; :::) = 0; 0 < ® · 1; (1) ® ® where Dx u and Dt u are modi¯ed Riemann-Liouville derivatives of u with respect to x and t respectively. To determine the solution u = u(x; t) explicitly, we ¯rst introduce the following transformations u = U(») with » = kx + ct: (2) which yields 0 00 ® 2® F (U; U ;U ;D» U; D» U; :::) = 0; (3) Next we introduce a new variable ' = '(») which is a solution of the fractional Riccati equation ® 2 D» ' = σ + ' ; 0 < ® · 1: (4) 208 Zakia Hammouch, Tou¯k Mekkaoui Then we propose the following series expansion as a solution of (1) Xn i u(x; t) = U(») = ai' ; (5) i=0 where the positive integer n can be found via the balancing of the highest derivative term with the nonlinear term in equation (3). In a recent paper by Zhang et al.[17] a set of ¯ve di®erent solutions to equation (4) was introduced as follows 8 p p > ¡ ¡σ tanh ( ¡σ»); σ < 0 > ® > > p p > ¡ ¡σ coth ( ¡σ»); σ < 0 > ® > <> p p σ tan ( σ»); σ > 0 ® (6) > > p p > ¡ σ cot ( σ»); σ > 0: > ® > > > ¡(® + 1) :> ¡ ;! = const: σ = 0; »® + ! where the generalized hyperbolic and trigonometric functions [6] are expressed by the following ® ® E®(x ) ¡ E®(¡x ) tanh®(x) = ® ® ; E®(x ) + E®(¡x ) ® ® E®(x ) + E®(¡x ) coth®(x) = ® ® ; E®(x ) ¡ E®(¡x ) ® ® E®(ix ) ¡ E®(¡ix ) tan®(x) = ® ® ; i(E®(ix ) + E®(¡ix )) ® ® i(E®(ix ) + E®(¡ix )) cot®(x) = ® ® ; E®(ix ) + E®(¡ix ) where E® denotes the Mittag-Le²er function, given as X1 zk E (z) = : ® ¡(®k + 1) k=0 3 Applications In this section we give two illustrative examples, namely Burgers and the KdV equations. Exact travelling wave solutions to some fractional di®erential equations 209 3.1 Example.1 The Burgers equation [3] is considered as one of the fundamental model equa- tions in fluid mechanics. The equation demonstrates the coupling between di®usion and convection processes. Consider the following Burgers equation with fractional-order derivatives ® ® 2® Dt u + uDx u ¡ ºDx u = 0: (7) where º is a constant that de¯nes the kinematic viscosity. If the viscosity º = 0, the equation is called inviscid Burgers equation [16]. Our main task is to seek travelling wave solutions to equation (7). Using the wave transformation (2), we reduce equation (7) to an ODE ® ® ® ® 2® 2® c D» U + k UD» U ¡ ºk D» U = 0: (8) By (4) and balancing the terms D2®u with the term uD®u in (8), we obtain n = 1: Substituting equation (5) in (8) with n = 1 and collecting all coe±cients i of ' ; i = 0; :::; 3 we get the following system of algebraic equations for a0 and a1, 8 ® 2 2® > k a1 ¡ 2ºk a1 = 0; > > > ® ® <> c a1 + k a0a1 = 0; (9) > ® 2 2® > k a1σ ¡ 2ºk a1σ = 0; > > : ® ® c a1σ + k a1a0σ = 0; we ¯nd that c® a = ¡ a = 2ºk®: (10) 0 k® 1 Accordingly we obtain ² Two hyperbolic function solutions: c® p p u(x; t) = ¡ ¡ 2ºk® ¡σ tanh ( ¡σ(kx + ct)) σ < 0; k® ® c® p p u(x; t) = ¡ ¡ 2ºk® ¡σ coth ( ¡σ(kx + ct)) σ < 0; k® ® (11) ² Two trigonometric function solutions: c® p p u(x; t) = ¡ + 2ºk® σ tan ( σ(kx + ct)) σ > 0; k® ® c® p p u(x; t) = ¡ ¡ 2ºk® σ cot ( σ(kx + ct)) σ > 0; k® ® (12) 210 Zakia Hammouch, Tou¯k Mekkaoui ² One rational solution: c® ¡(1 + ®) u(x; t) = ¡ ¡ 2ºk® σ > 0 and ! = const: k® (kx + ct)® + ! (13) 3.2 Example.2 The Korteweg-deVries (KdV) equation arises in the study of shallow water waves [16]. In particular, the KdV equation is used to describe long waves traveling in canals. It is formally proved that this equation has solitary waves as solutions, hence it can have any number of solitons [5][16]. Consider the following fractional KdV equation with fractional-order derivatives ® ® 3® Dt u + 6uDx u + Dx u = 0; (14) Similarly, we use the wave transformation (2), we reduce equation (14) to an ODE ® ® ® ® 3® 3® c D» U + 6k UD» U + k D» U = 0; (15) 3® ® Thanks to (4) and the balancing of the terms D» U with the term UD» U in (15), we obtain n = 2: By substituting equation(5) in (15) with n = 2 and collecting all coe±cients of 'i; i = 0; :::; 5 we get the following system of algebraic equations for a0; a1; and a2, 8 ® 2 3® > 12k a2 + 24k a2 = 0; > > > ® 3® > 18k a1a2 + 6k a1 = 0; > > > ® ® ® 2 ® 2 3® < 2a2c + 12k a0a2 + 12k a2σ + 6k a1 + 40k a2σ = 0; (16) > 3® ® ® ® > 8k a1σ + 6k a0a1 + 18k a1a2σ + c a1 = 0; > > > ® 3® 2 ® ® 2 > 2c a2σ + 16k a2σ + 12k a0a2σ + 6k a1σ = 0; > > : ® 3® 2 ® c a1σ + 2k a1σ + 6k a0a1σ = 0: From the output of Maple, we get a solution namely 8k3®σ + c® a = ¡ a = 0; a = ¡2k2®: 0 6k® 1 2 In view of this we obtain Exact travelling wave solutions to some fractional di®erential equations 211 ² Two hyperbolic functions solution: 8k3®σ + c® p u(x; t) = ¡ + 2k2®σ tanh2 ( ¡σ(kx + ct)) σ < 0; 6k® ® 8k3®σ + c® p u(x; t) = ¡ + 2k2®σ coth2 ( ¡σ(kx + ct)) σ < 0; 6k® ® (17) ² Two trigonometric function solutions: 8k3®σ + c® p u(x; t) = ¡ ¡ 2k2®σ tan2 ( σ(kx + ct)) σ > 0; 6k® ® 8k3®σ + c® p u(x; t) = ¡ ¡ 2k2®σ cot2 ( σ(kx + ct)) σ > 0; 6k® ® (18) ² One rational solution: c® 2k2®¡2(® + 1) u(x; t) = ¡ ¡ ! = const: σ = 0: 6k® ((kx + ct)® + !)2 (19) 4 Conclusion In this paper we give some exact analytical solutions including the generalized hyperbolic function solutions, generalized trigonometric function solutions and rational solutions by using a sub-equation method. It can be concluded that the improved sub-equation method can be used for solving other nonlinear fractional partial di®erential equations with nonlinear terms of any order. All solutions obtained in this paper have been checked by Maple software. References [1] G. Adomian, Kluwer Academic Publishers, Boston, MA 1994. [2] N.Benhamidouche, Exact solutions to some nonlinear PDEs, travelling pro¯les method Electronic Journal of Qualitative Theory of Di®erential Equations 2008, No. 15, 1-7. [3] J.M. Burgers, Adv. Appl. Mech. 1, 171-199, (1948). [4] E.G.Fan, Phys. Lett. A 300 (2002) 243. 212 Zakia Hammouch, Tou¯k Mekkaoui [5] W.Hereman and A. Nuseir, Math. Comput. Simulation, 43, 1327, (1997). [6] http://mathworld.wolfram.com/GeneralizedHyperbolicFunctions.html [7] R.Hirota, Phys. Rev. Lett. 27 (1971) 1192. [8] G.Jumarie, Comput. Math. Appl. 51 (2006) 1367. [9] R.M. Miura, SIAM Rev., 18, 412459, (1976). [10] M.R.Miurs, Backlund Transformation, Springer, Berlin, 1978. [11] K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York 1974. [12] Foad Saadi, M.
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