Exp-function Method for Fractional Differential Equations

Transcription

1 From the SelectedWorks of Ji-Huan He 2013 Exp-function Method for Fractional Differential Equations Ji-Huan He Available at:

2 Citation Information: He JH. Exp-function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation. Volume 14, Issue 6, Pages , ISSN (Online) , ISSN (Print) , DOI: /ijnsns , August Exp-function Method for Fractional Differential Equations Ji-Huan He National Engineering Laboratory for Modern Silk, College of Textile and Engineering, Soochow University, 199 Ren-ai Road, Suzhou , China Tel: Abstract A fractional nonlinear wave equation is used as an example to elucidate how to solve fractional differential equations with local fractional derivatives via the fractional complex transform and the exp-function method. Keywords: Modified Riemann-Liouville Derivative, Fractional Complex Transform, Expfunction method, nonlinear dispersive equations, Phi-four equation, local fractional calculus

3 MSC A08, 74J35, 76B25, 35C08 1. Introduction The exp-function method[1-4] is routinely employed to search for solitary solutions for various nonlinear equations[5-16]. However, its application to fractional calculus is rare and primary, Zhang and his colleagues[14] suggested a fractional exp-function method with help of Mittag-Leffer function, which is, however, elusive to nonmathematicians. In this paper we will suggest a standard solution procedure for fractional differential equations by the expfunction method. Consider a fractional nonlinear wave equation in the form t u u x 2 2β 3 + a + bu+ cu = 0 2 2β, 0 <, β < 1, (1) where u/ t denotes the local fractional derivation of order with respect to t, which is defined as[17-20] ( (, ) ( 0, )) ( ) u Δ utx ut x = lim t t t 0 t t0 (2) The local fractional derivation is different from the modified Riemann-Liouville derivative defined as[21]. u 1 d t = ( t ξ ) ( u( ξ) u(0)) dξ t Γ(1 ) dt 0, (3) where u denotes a continuous (but not necessarily differentiable) function. The local fractional derivative is very similar to the fractal derivative defined as[22] Df ( x) = Dx lim Δ x= xa xb L0 f ( x ) f ( x ) A ( x x ) A B B (4)

4 where is the fractal dimensions, L 0 is a small value different from zero, L << 0 1. The fractal derivative is widely used to model the fractal heat transfer/water permeation in fractal media [23,24]. The chain rule for the local fractional derivative is given as follows[17-20] d f x dx ( ) () 1 ( ) = ( ) ( ) f g g x (5) where there exist ( 1 ) ( f ( g ) g ) and ( x). References [1] J.H.He, X.H. Wu, Exp-function method for nonlinear wave equations, Chaos Soliton. Fract., 30(2006) [2] X.H. Wu, J.H.He, Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method, Comput. Math. Applicat., 54(2007) [3] J.H. He, An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering, Int. J. Mod. Phys. B, 22(2008) [4] J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, 20(2006) [5] S. Zhang, H.Q. Zhang, An Exp-function method for new N-soliton solutions with arbitrary functions of a (2+1)-dimensional vcbk system, Comput. Math. Applicat., 61(2011) [6] H.M.Fu, Z.D. Dai, Double Exp-function Method and Application, Int. J. Nonlinear Sci. Num., 10(2009)

5 [7] A. Bekir, Ahmet Boz, Exact Solutions for a Class of Nonlinear Partial Differential Equations using Exp-Function Method, Int. J. Nonlinear Sci. Num., 8(2007), [8] G. Domairry, A. G. Davodi and A. G. Davodi, Solutions for the double Sine-Gordon equation by Exp-function, Tanh, and extended Tanh methods, Numer. Meth. Part. D. E., 26(2010) [9] H. Aminikhah, H. Moosaei and M. Hajipour, Exact solutions for nonlinear partial differential equations via Exp-function method, Numer. Meth. Part. D. E., 26(2010) , [10] M.M. Rashidi, D.D. Ganji and S. Dinarvand, Explicit analytical solutions of the generalized Burger and Burger Fisher equations by homotopy perturbation method, Numer. Meth. Part. D. E., 25(2009) [11] J.H. He, Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis, 2012, Article ID [12] S. Zhang, Exp-function Method: Solitary, Periodic and Rational Wave Solutions of Nonlinear Evolution Equations, Nonlinear Science Letters A, 1(2010) [13] E. Misirli, Y. Gurefe,The Exp-function Method to Solve the Generalized Burgers-Fisher Equation, Nonlinear Science Letters A, 1(2010) [14] S.Zhang, Q.A. Zong, D. Liu, Q. Gao, A Generalized Exp-Function Method for Fractional Riccati Differential Equations, Communications in Fractional Calculus, 1(2010) [15] D. D. Ganji, Seyed H. Hashemi Kachapi, Analysis of Nonlinear Equations in Fluids, Progress in Nonlinear Science, 2(2011) 1-293

6 [16] D. D. Ganji, Seyed H. Hashemi Kachapi, Analytical and Numerical Methods in Engineering and Applied Sciences, Progress in Nonlinear Science, 3(2011) [17] X. J. Yang, Local fractional integral transforms, Progress in Nonlinear Science, 4(2011),1-225 [18] X.J. Yang, Local Fractional Functional Analysis and Its Applications, Asian Academic publisher Limited, Hong Kong, China, [19] X.J. Yang, Advanced Local Fractional Calculus and Its Applications, World Science Publisher, New York, USA, [20] X.J. Yang, D. Baleanu, Fractal heat conduction problem solved by local fractional variation iteration method, Thermal Science, 2012, Doi: /TSCI Y. [21] G. Jumarie, Modified Riemann-Liouville Derivative and Fractional Taylor series of Nondifferentiable Functions Further Results, Computers and Mathematics with Applications, 51 (2006) [22] J.H. He, A new fractal derivation, Thermal Science, 15(2011), S145-S147 [23] J. Fan, J.H. He, Fractal derivative model for air permeability in hierarchic porous media, Abstract and Applied Analysis, 2012, [24] J. Fan, J.H. He, Biomimic design of multi-scale fabric with efficient heat transfer property, Thermal Science, 16(5)(2012) [25] Z. B. Li, J.H. He, Fractional Complex Transform for Fractional Differential Equations, Mathematical and Computational Applications, 15(2010) [26] Z. B.Li, An Extended Fractional Complex Transform, International Journal of Nonlinear Science and Numerical Simulation, 11(2010)

7 [27] J.H. He, Z.B. Li, Converting fractional differential equations into partial differential equations, Thermal Science, 16(2)(2012) [28] Z.B. Li, W.H. Zhu, J.H. He, Exact solutions of time-fractional heat conduction equation by the fractional complex transform, Thermal Science, 16(2)(2012) [29] Q.L. Wang, J.H. He, Z.B. Li, Fractional model for heat conduction in polar bear hairs, Thermal Science, 16(2)(2012) [30] He, J.H., Elagan, S. K.; Li, Z. B., Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376(4), [31] J.-H. He, F.J. Liu, Local fractional variational iteration method for fractal heat transfer in silk cocoon hierarchy, Nonlinear Science Letters A, 4(2013)15-20