Tema Tendências em Matemática Aplicada e Computacional, 18, N. 2 (2017),

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1 Tema Tendências em Matemática Aplicada e Computacional, 18, N ), Sociedade Brasileira de Matemática Aplicada e Computacional wwwscielobr/tema doi: /tema New Extension for Sub Equation Method and its Application to the Time-fractional Burgers Equation by using of Fractional Derivative A NEIRAMEH Received on August 20, 2016 / Accepted on February 17, 2017 ABSTRACT In this paper, we use the new fractional complex transform and the sub-equation method to study the nonlinear fractional differential equations and find the exact solutions These solitary wave solutions demonstrate the fact that solutions to the perturbed nonlinear Schrodinger equation with power law nonlinearity model can exhibit a variety of behaviors Keywords: Time-fractional Burgers equation, Fractional calculus, sub-equation method 1 INTRODUCTION With the availability of symbolic computation packages like Maple or Mathematica, direct searching for exact solutions of nonlinear systems of partial differential equations PDEs) has become more and more attractive Having exact solutions of nonlinear systems of PDEs makes it possible to study nonlinear physical phenomena thoroughly and facilitates testing the numerical solvers as well as aiding the stability analysis of solutions Wide classes of analytical methods have been proposed for solving the fractional differential equations, such as the fractional subequation method [1 3], the first integral method [4], and the G /G)-expansion method [5, 6], which can be used to construct the exact solutions for some time and space fractional differential equations Based on these methods, a variety of fractional differential equations have been investigated and solved In this present paper we applied the new extension of sub-equation method for finding new exact solitary wave solutions for time-fractional Burgers equation in the following form, u q εu t x v 2 u = 0, 0 < 1, t > 0 11) x2 Recently, a new modification of Riemann-Liouville derivative is proposed by Jumarie [7]: Dx a f x) = 1 d Ɣ1 ) dx x 0 x ε) f ε) f 0)) dε, 0 <<1 Department of Mathematics, Faculty of Sciences, Gonbad Kavous University, Gonbad, Iran s: aneirameh@gonbadacir; aneirameh@gmailcom

2 226 NEW EXTENSION FOR SUB EQUATION METHOD and gave some basic fractional calculus formulae, for example, formulae 4 12) and 4 13) in [7]: Dx u x)vx)) = vx) D x u x)) u x) D x v x)), Dx f u x))) = f u u) D x u x)) = D x u x) f u), 12) The last formula has been applied to solve the exact solutions to some nonlinear fractional order differential equations If this formula were true, then we could take the transformation kt ξ = x Ɣ1 ) and reduce the partial derivative U x, t) t to U ξ) Therefore the corresponding fractional differential equations become the ordinary differential equations which are easy to study But we must point out that Jumarie s basic formulae and are not correct, and therefore the corresponding results on differential equations are not true [8] Fractional derivative is as old as calculus The most popular definitions are [9 12]: i) Riemann-Liouville definition: If n is a positive integer and [n 1, n) the th derivative of f is given by d n Da f t) = 1 Ɣn ) dt n t a f x) t x) n1 dx ii) Caputo Definition For [n 1, n) the derivative of f is Da f t) = 1 t Ɣn ) a f n x) t x) n1 dx Now, all definitions are attempted to satisfy the usual properties of the standard derivative The only property inherited by all definitions of fractional derivative is the linearity property However, the following are the setbacks of one definition or another: i) The Riemann-Liouville derivative does not satisfy D a 1) = 0 D a 1)) = 0fortheCaputo derivative), if is not a natural number ii) All fractional derivatives do not satisfy the known product rule D a fg) = fd a g) gd a f ) iii) All fractional derivatives do not satisfy the known product rule ) f Da = fd a g) gd a f ) g g 2 Tend Mat Apl Comput, 18, N )

3 NEIRAMEH 227 iv) All fractional derivatives do not satisfy the known quotient rule: D a fog)t) = f g t)) g t) v) All fractional derivatives do not satisfy the chain rule: D D β f = D β f in general vi) Caputo definition assumes that the function f is differentiable Authors introduced a new definition of fractional derivative as follows [16]: For [0, 1), and f : [0, ) Rlet f t ξ t 1 ) f t) T f )t) = lim ξ 0 ξ For t > 0, 0, 1) T is called the conformable fractional derivative of f of order [17 18] Definition 11 Let f t) stands for T f )t) Hence f f t ξ t 1 ) f t) t) = lim ξ 0 ξ Iffis-differentiable in some 0, a),a> 0, and lim f t) exists, then by definition t 0 f 0) = lim f t) t 0 We should remark that T t μ ) = μt μ Further, this definition coincides with the classical definitions of R-L and of Caputo on polynomials up to a constant multiple) One can easily show that T satisfies all the properties in the theorem [15 16] Theorem 11 Let [0, 1) and f, gbe-differentiable at a point t Then: i) T af bg) = at f ) bt g),for all a, b R; ii) T t μ ) = μt μ, for all μ R; iii) T fg) = ft g) gt f ); ) f iv) T = ft g) gt f ) g g 2 If, in addition, f is differentiable, then T f )t) = t 1 df dt Theorem 12 Let f : [0, ) R be a function such that f is differentiable and also differentiable Let g be a function defined in the range of f and also differentiable; then, one has the following rule [17]: T fog)t) = t 1 g t) f g t)) The above rule is referred to as Atangana beta-rule We will present new derivative for some special functions Tend Mat Apl Comput, 18, N )

4 228 NEW EXTENSION FOR SUB EQUATION METHOD i) T e cx ) = cx 1 e cx, c R ii) T sin bx) = bx 1 cos bx, b R iii) T cos bx) =bx 1 sin bx, b R iv) T 1 x ) = 1 However, it is worth noting the following fractional derivatives of certain functions: i) T e 1 t ) = e 1 t ii) T sin 1 t ) = cos 1 t iii) T cos 1 t ) =sin 1 t Definition 12 Fractional Integral) Let a 0 and t a Also, let f be a function defined on a, t] and fthenthe-fractional integral of f is defined by, I a f )t) = t a f x) dx x1 if the Riemann improper integral exists It is interesting to observe that the -fractional derivative [15 16] Theorem 13 Inverse property) Let a 0, and 0, 1) Also, let f be a continuous function such that Ia f exists Then T I a f ) t) = f t), for t a In this paper, we obtain the exact solution of the fractional perturbed nonlinear Schrodinger equation with power law nonlinearity by means of the sub-equation method The sub-equation method is a powerful solution method for the computation of exact traveling wave solutions This method is one of the most direct and effective algebraic methods for finding exact solutions of nonlinear fractional partial differential equations FPDEs) The method is based on the homogeneous balance principle and the Jumarie s modified Riemann-Liouville derivative of fractional order 2 METHOD APPLIED Suppose that nonlinear fractional partial differential equations, say, in three independent variable x, y and t is given by ) G u, Dt u, D x u, D y u, D2 t u, Dx 2, D t D x u, = 0, 0 < 1 21) where Dx u, D y u and D t u are comformable fractional derivatives of u, u x, y, t) is an unknown function, G is a polynomial in u and its various partial derivatives, in which the highest order derivatives and nonlinear terms are involved This method consists of the following steps: Tend Mat Apl Comput, 18, N )

5 NEIRAMEH 229 Step 21 Using a wave transformation u = uξ ), ξ = k x l y c t, 22) Where k and c are real constants This enables us to use the following changes: D t ) = c d dξ, D x ) = k d dξ, D y ) = l d dξ, D2 x d2 ) = k2 dξ 2 Under the transformation 22), Eq21) becomes an ordinary differential equation where u = du dξ Nu, u, u, u,)= 0, 23) Step 22 We assume that the solution of Eq 23) is of the form uξ ) = m a i m Fξ )) i i=0 2m i=m1 a i m Fξ )) mi, 24) where a i i = 1, 2,,n) are real constants to be determined later Fξ ) expresses the solution of the auxiliary ordinary differential equation F ξ ) = γt) F 2 ξ ) βt) F ξ ) t), 25) Eq 25) admits the following solutions: { b tanh bξ), b < 0 9a) Fξ ) = b coth bξ), b < 0 9b) Fξ ) = { b tan bξ), b > 0 9c) b cot bξ), b > 0 9d) 26) Fξ ) = 1 ξξ 0, ξ 0 = const, b = 0 9e) Integer m in 24) can be determined by considering homogeneous balance between the nonlinear terms and the highest derivatives of uξ )in Eq 23) polynomial in Fξ ), equating each coefficient of the polynomial to zero yields a set of algebraic equations for a i, k, c Step 23 Solving the algebraic equations obtained in Step 3, and substituting the results into 23), then we obtain the exact traveling wave solutions for Eq 21) Tend Mat Apl Comput, 18, N )

6 230 NEW EXTENSION FOR SUB EQUATION METHOD 3 APPLICATION TO THE TIME-FRACTIONAL BURGERS EQUATAION Using a wave transformation ux, t) = Uξ ), ξ = kx ct, 31) by substituting Eq 31), into Eq 11) is reduced into an ODE by integrating once, we find Balancing U with U 2 in Eq 32) give cu UU k 2 vu = 0, ξ 0 cu 1 2 U 2 k 2 vu = 0 32) m 1 = 2m m = 1 We then assume that Eq 32) has the following formal solution: Uξ ) = a 0 a 1 h F) a 2 h F) 1, 33) by considering the F ξ) h = in Eq 33) we have and U ξ) = a 0 a 1 a 2 1, 34) = γ 2 β 2γ h) γ h 2 βh 35) Substituting Eqs 34) 35) into Eq 32) and collecting all terms with the same order of ψ j together, we convert the left-hand side of Eq 32) into a polynomial in F j Setting each coefficient of each polynomial to zero, we derive a set of algebraic equations for a 0, a 1, a 2 and h By solving these algebraic equations we have a 1 = 2kvγ ε a 0 = c k2 vβ 2k 2 vγ h a 2 = 2kv γ h2 βh) ε h = 1 β 2 γ, c = 2ξ 0 16k 4 v 2 γ 4k 4 v 2 β 2 So from 31) we have solitary wave solutions of Eq 11) as follows Tend Mat Apl Comput, 18, N )

7 NEIRAMEH 231 If b < 0 and If b > 0 and u 1 x, t) = ck2 vβ2k 2 vγ h 2kvγ 1 β ε 2 γ [ b b tanh kx 2kvγ h 2 βh) 1 β ε 2 γ [ b b tanh kx u 2 x, t) = ck2 vβ2k 2 vγ h 2kvγ 1 β ε 2 γ [ b b coth kx 2kvγ h 2 βh) 1 β ε 2 γ [ b b coth kx u 3 x, t) = ck2 vβ2k 2 vγ h [ 2kvγ 1 β b ε 2 γ b tan kx 2kvγ h 2 βh) 1 β ε 2 γ b tan 2ξ0 k16k 4 v 2 γ4k 4 v 2 β 2 t )]) 2ξ0 k16k 4 v 2 γ4k 4 v 2 β 2 t 2ξ0 k16k 4 v 2 γ4k 4 v 2 β 2 t 2ξ0 k16k 4 v 2 γ4k 4 v 2 β 2 t [ b kx u 4 x, t) = ck2 vβ2k 2 vγ h 2kvγ 1 β ε 2 γ [ b b cot kx 2kvγ h 2 βh) 1 β ε 2 γ [ b b cot kx If b = 0 we have solution of Eq 11) as follow u 5 x, t) = ck2 vβ2k 2 vγ h 2kvγ ε 4 CONCLUSION )]) 2ξ0 k16k 4 v 2 γ4k 4 v 2 β 2 t 2ξ0 k16k 4 v 2 γ4k 4 v 2 β 2 t 2ξ0 k16k 4 v 2 γ4k 4 v 2 β 2 t 1 β 2 γ kx 2ξ 0 16k 4 v 2 γ4k 4 v 2 β 2 t ξ 0 2kvγ h 2 βh) ε )]) )]) 2ξ0 k16k 4 v 2 γ4k 4 v 2 β 2 t ) 1 β 2 γ kx 2ξ 0 16k 4 v 2 γ4k 4 v 2 β 2 t ξ 0 ) 1 )]) 1, )]) 1 )]) 1, )]) 1 Now, we briefly summarize the results in this paper Firstly, the fractional complex transform is extremely simple but effective for solving nonlinear fractional differential equations Secondly, the sub-equation method for nonlinear fractional differential equations with fractional complex transform has its own advantages: direct, succinct, and basic; and it can be used for many other nonlinear equations Thirdly, to our knowledge, the solutions obtained in this paper have not been reported in the literature so far Tend Mat Apl Comput, 18, N )

8 232 NEW EXTENSION FOR SUB EQUATION METHOD 5 ACKNOWLEDGEMENTS I would like to express thanks to the editor and anonymous referees for their useful and valuable comments and suggestions RESUMO Neste artigo usamos uma nova transformação fracional complexa e o método da sub-equação para estudar equações diferenciais não lineares e encontrar soluções exatas As soluções de onda encontradas mostram que as soluções da equação não linear de Schrodinger perturbada com um modelo não linear de lei das potências pode apresentar diversos comportamentos diferentes Palavras-chave: Equação de Burgers, cálculo fracionário, método sub-equação REFERENCES [1] S Zhang & HQ Zhang Phys Lett A, ), 1069 [2] B Tong, Y He, L Wei & X Zhang Phys Lett A, ), 2588 [3] S Guo, L Mei, Y Li & Y Sun Phys Lett A, ), 407 [4] B Lu J Math Anal Appl, ), [5] B Zheng Commun Theor Phys, ), 623 [6] KA Gepreel & S Omran Chin Phys B, ), [7] G Jumarie Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results Computers and Mathematics with Applications, ), [8] C-s Liu Counter examples on Jumarie s two basic fractional calculus formulae Communications in Nonlinear Science and Numerical Simulation, 223) 2015), 9294 [9] I Podlubny Fractional Diferential Equations, Academic Press, 1999) [10] SG Samko, AA Kilbas & OI Marichev Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, 1993) [11] AA Kilbas, MH Srivastava & JJ Trujillo Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006) [12] AA Kilbas & M Saigo On solution of integral equation of Abel-Volterra type Diff Integr Equat, 85) 1995), [13] T Abdeljawad On conformable fractional calculus Journal of Computational and Applied Mathematics, ), [14] T Abdeljawad, M Al Horani & R Khalil Conformable fractional semigroup operators Journal of Semigroup Theory and Applications, 2015), Article 7, 1 9 [15] M Abu Hammad & R Khalil Conformable heat differential equation International Journal of Pure and Applied Mathematics, 942) 2014), [16] R Khalil, M Al Horani, A Yousef & M Sababheh A new definition of fractional derivative Journal of Computational and Applied Mathematics, ), Tend Mat Apl Comput, 18, N )