EXACT SOLUTIONS OF NON-LINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS BY FRACTIONAL SUB-EQUATION METHOD

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1 THERMAL SCIENCE, Year 15, Vol. 19, No. 4, pp EXACT SOLUTIONS OF NON-LINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS BY FRACTIONAL SUB-EQUATION METHOD by Hong-Cai MA a,b*, Dan-Dan YAO a, and Xiao-Fang PEN a a Department of Applied Mathematics, Donghua University, Shanghai, China b Department of Mathematics and Statistics, University of South Florida, Tampa, Florida, USA Original scientific paper DOI: 1.98/TSCI15439M This paper studies the space-time fractional Whitham-Broer-Kaup equations by the existed fractional sub-equation method, and exact solutions are obtained. Key words: fractional partial differential equations, exact solutions, the fractional sub-equation method Introduction Fractional partial differential equations are generalization of the classical differential equations of integer order. In recent decades, fractional differential equations have gained a lot of attention as they are widely used to describe a variety of complex phenomena in many fields [1-3]. In the past, many powerful methods were established and developed to obtain exact solutions and numerical solutions of the fractional differential equation (FDE), such as the finite difference method [4], the Adomian decomposition method [5], and so on. In this paper, we use the existed fractional sub-equation method to search for exact solutions for the space-time fractional Whitham-Broer-Kaup (WBK) equations in the sense of modified Riemann-Liouville derivative defined by Jumarie [6], which is a fractional version of the nown ('/) method [7]. This method is based on the following fractional ODE: t t D ( ) λd ( ) µ ( ) = (1) Jumarie's modified Riemann-Liouville derivative and existed fractional sub-equation method We list some important properties for the modified Riemann-Liouville derivative [6]: r t D t Γ (1 r) r = t () Γ (1 r ) t t t D [ f() tgt ()] = gtd () f() t f() td gt () (3) ' Dt f[ g( t)] fg[ g( t)] Dt g( t) Dt f[ g( t)][ g '( t)] * Corresponding author; hongcaima@dhu.edu.cn = = (4)

2 14 THERMAL SCIENCE, Year 15, Vol. 19, No. 4, pp In order to obtain the general solutions for eq. (1), we suppose () = H(η) and use the well-nown fractional complex transformation [8], η = /Γ(1 ). Then by using eq. () and the first equality in eq. (4) and eq. (1) can be turned into the following second ODE: H ''( η) λh '( η ) µ H( η) = since D ( ) = D H( η) = H'( η) D η = H'( η), we obtain: C1 sinh C cosh λ λ 4µ Γ (1 ) Γ (1 ), λ 4µ > C1 cosh C sinh Γ (1 ) Γ (1 ) D ( ) λ C Γ (1 ) =, λ 4µ = (5) ( ) C (1 ) C 1Γ 4µ λ 4µ λ C1 sin C cos λ 4µ λ Γ (1 ) Γ (1 ), λ 4µ < 4µ λ 4µ λ C1 cos C sin Γ (1 ) Γ (1 ) Description of the existed fractional sub-equation method method. In this section, we describe the main steps of the existed fractional sub-equation Step 1. Suppose that a non-linear FDE, say in two independent variables x and t: t x t x P( u, u, u, D u, D u, ) =, < 1 (6) where Dt u and Dx u are Jumarie's modified Riemann-Liouville derivatives of u, u = u(x, t) is an unnown function, P a polynomial in u, and its various partial derivatives, in which the highest order derivatives and non-linear terms are involved. Step. By using the traveling wave transformation: uxt (, ) = u( ), = x ct (7) then, by the second equality in eq. (4) and eq. (6) can be turned into the following fractional ODE with respect to the variable : P ( u, cu, u, c D u, D u, ) = (8) Step 3. Suppose that the solution of eq. (8) can be expressed by a polynomial in D / : i= i m D u( ) = ai (9)

3 THERMAL SCIENCE, Year 15, Vol. 19, No. 4, pp where = () satisfies eq. (1), and a i (i =, 1,, m) are constants to be determined later with a m. The positive integer m can be determined by considering the homogeneous balance between the highest order derivatives and non-linear terms appearing in eq. (8). Step 4. Substituting eq. (9) into eq. (8), using eq. (1) and collecting all terms with the same order of D / together, the left-hand side of eq. (8) is converted into another polynomial in D /. Equating each coefficient of this polynomial to zero yields a set of algebraic equations for a i (i =, 1,, m). Step 5. Solving the equation system in Step 4 and using eq. (5), we can construct a variety of exact solutions for eq. (6). Applications The space-time fractional WBK equations: Dt u udxu Dx v β Dx u = 3 Dt v Dx ( uv) βdx v γdx u = can be used to describe the dispersive long wave in shallow water. Here u = u(x, t) is the field of horizontal velocity, v = v(x, t) is the height deviating from equilibrium position of liquid, β and γ are real constants that represent different diffusion powers. When = 1, β =, and γ =, eq. (1) is the classical long-wave equations that describe the shallow water wave with diffusion. When = 1, β =, and γ = 1, eq. (1) reduces to the variant Boussinesq equations [9] which are very important in fluid mechanics. Suppose that u = u(x, t), v = v(x, t), where = x ct,, c, are all constants with, c. Then by use of the second equality in eq. (4) and eq. (1) can be turned into: (1) c D u ud u D v β D u = 3 3 c D v D ( uv) βd v γd u = (11) Assume that the solution of eq. (11) can be expressed by: m 1 D u( ) = ai i = i m D v( ) = bi i= i (1) 3 Balancing the order of D u, ud u, D u, and D ( uv) in eq. (11), we can obtain m 1 = 1, and m =. We have: D u( ) = a a1 D D v( ) = b b1 b (13)

4 14 THERMAL SCIENCE, Year 15, Vol. 19, No. 4, pp Substituting eq. (13) into eq. (11), using eq. (1) and collecting all the terms with the same power of D / together, equating each coefficient to zero, yields a set of algebraic equations. Solving these equations yields: a a 1 c = ± λ β γ =± β γ βµ β γ β µ µγ b = ± b1 = ± βλ β γ β λ λγ b = ± β β γ β γ (14) Substituting eq. (14) into eq. (13) and combining with eq. (5), we can obtain the exact solutions of eq. (1). When λ 4µ =, we have: c λ CΓ (1 ) u3( x, t) = ± λ β γ ± β γ C1Γ (1 ) C v3 ( x, t) βµ β γ β µ µγ = ± λ CΓ (1 ) ± βλ β γ β λ λγ C1Γ (1 ) C C (1 ) λ β β γ β γ Γ ± C1Γ (1 ) C (15) where = x ct. When λ 4µ > : c u1 ( x, t) = ± λ β γ ± C1sinh Ccosh λ λ 4µ Γ (1 ) Γ (1 ) ± β γ C1cosh Csinh Γ (1 ) Γ (1 ) (16)

5 THERMAL SCIENCE, Year 15, Vol. 19, No. 4, pp ( βµ β γ β µ µγ) ( βλ β γ β λ λγ) v ( x, t) = ± ± C1sinh Ccosh λ λ 4µ Γ (1 ) Γ (1 ) C1cosh Csinh Γ (1 ) Γ (1 ) ( β β γ β γ) ± C1sinh Ccosh λ λ 4µ Γ (1 ) Γ (1 ) (16) C1cosh Csinh Γ (1 ) Γ (1 ) where = x ct. When λ 4µ < : c u( x, t) = ± λ β γ ± 4µ λ 4µ λ C1 sin C cos λ 4µ λ Γ (1 ) Γ (1 ) ± β γ 4µ λ 4µ λ C1 cos C sin Γ (1 ) Γ (1 ) v ( x, t) = ± βµ β γ β µ µγ ± βλ β γ β λ λγ 4µ λ 4µ λ C1 sin C cos λ 4µ λ Γ (1 ) Γ (1 ) 4µ λ 4µ λ C1 cos C sin (1 ) (1 ) Γ Γ ± β β γ β γ 4µ λ 4µ λ C1 sin C cos λ 4µ λ Γ (1 ) Γ (1 ) C1 cos 4 µ λ C 4 µ λ sin Γ (1 ) Γ (1 ) where = x ct. (17)

6 144 THERMAL SCIENCE, Year 15, Vol. 19, No. 4, pp Conclusions In this paper, the existed fractional sub-equation method has been successfully obtained the exact solutions of the space-time fractional WBK equations. The above procedure shows that: the fractional sub-equation method is an efficient and powerful method in solving a wide class of equations, and the method is straightforward without any restrictive assumptions and special techniques. Whether we can introduce other new feasible algorithms to solve FDE, we hope this question will be further studied. Acnowledgments The wor is in part supported by the National Natural Science Foundation of China (project No ), the Fund of Science and Technology Commission of Shanghai Municipality (project No. ZX13714) and the Fundamental Research Funds for the Central Universities. References [1] Li, C., et al., Numerical Approaches to Fractional Calculus and Fractional Ordinary Differential Equation, J. Comput. Phys. 3 (11), 9, pp [] Odibat, Z., Momani, S., The Variational Iteration Method: An Efficient Scheme for Handling Fractional Partial Differential Equations in Fluid Mechanics, Comput. Math. with Appl. 58 (9), 11-1, pp [3] He, J.-H., et al., eometrical Explanation of the Fractional Complex Transform and Derivative Chain Rule for Fractional Calculus. Phys. Lett. A. 376 (1), 4, pp [4] Cui, M., Compact Finite Difference Method for the Fractional Diffusion Equation, J. Comput. Phys., 8 (9),, pp [5] El-Sayed, A. M. A., aber, M., The Adomian Decomposition Method for Solving Partial Differential Equations of Fractal Order in Finite Domains, Phys. Lett. A., 359 (6), 3, pp [6] Jumarie,., Modified Riemann-Liouville Derivative and Fractional Taylor Series of Nondifferentiable Functions Further Results, Comput. Math. Appl., 51 (6), 9-1, pp [7] Wang, M. L., et al., The ('/) Expansion Method and Travelling Wave Solutions of Non-Linear Evolution Equations in Mathematical Physics, Phys. Lett. A., 37 (8), 4, pp [8] He, J.-H. A Tutorial Review on Fractal Spacetime and Fractional Calculus, International Journal of Theoretical Physics, 53 (14), 11, pp [9] Ablowitz, M. J., Clarson, P. A., Solitons, Non-Linear Evolution Equations and Inverse Scattering, Cambridge Univ. Press., Cambridge, UK, 199 Paper submitted: January 5, 15 Paper revised: February 7, 15 Paper accepted: March 31, 15