From the SelectedWorks of Ji-Huan He 2013 Exp-function Method for Fractional Differential Equations Ji-Huan He Available at: https://works.bepress.com/ji_huan_he/73/
Citation Information: He JH. Exp-function Method for Fractional Differential Equations, International Journal of Nonlinear Sciences and Numerical Simulation. Volume 14, Issue 6, Pages 363 366, ISSN (Online) 2191-0294, ISSN (Print) 1565-1339, DOI: 10.1515/ijnsns-2011-0132, August 2013 http://www.degruyter.com/view/j/ijnsns.2013.14.issue-6/ijnsns-2011-0132/ijnsns-2011-0132.xml 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 215123, China Email: hejihuan@suda.edu.cn Tel: 86-512-6588-4633 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
MSC 2010 34A08, 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)
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) 700-708 [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) 966-986 [3] J.H. He, An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering, Int. J. Mod. Phys. B, 22(2008) 3487-3578 [4] J.H. He, Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, 20(2006) 1141-1199 [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) 1923-1930 [6] H.M.Fu, Z.D. Dai, Double Exp-function Method and Application, Int. J. Nonlinear Sci. Num., 10(2009) 927-933
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[27] J.H. He, Z.B. Li, Converting fractional differential equations into partial differential equations, Thermal Science, 16(2)(2012) 331-334 [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) 335-338 [29] Q.L. Wang, J.H. He, Z.B. Li, Fractional model for heat conduction in polar bear hairs, Thermal Science, 16(2)(2012) 339-342 [30] He, J.H., Elagan, S. K.; Li, Z. B., 2012. Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus, Phys. Lett. A, 376(4), 257-259 [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