The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics ✩

Abstract By introducing a new general ansatz, the improved fractional sub-equation method is proposed to construct analytical solutions of nonlinear evolution equations involving Jumarieʼs modified Riemann–Liouville derivative. By means of this method, the space–time fractional Whitham–Broer–Kaup and generalized Hirota–Satsuma coupled KdV equations are successfully solved. The obtained results show that the proposed method is quite effective, promising and convenient for solving nonlinear fractional differential equations.

[1]  M. A. Abdou.  The extended F-expansion method and its application for a class of nonlinear evolution equations , 2007 .

[2]  Ahmed M. A. El-Sayed,et al.  Adomian's decomposition method for solving an intermediate fractional advection-dispersion equation , 2010, Comput. Math. Appl..

[3]  Mohd. Salmi Md. Noorani,et al.  Variational iteration method for fractional heat- and wave-like equations , 2009 .

[4]  Wenchang Tan,et al.  An exact solution of start-up flow for the fractional generalized Burgers' fluid in a porous half-space , 2008 .

[5]  Mingliang Wang.  SOLITARY WAVE SOLUTIONS FOR VARIANT BOUSSINESQ EQUATIONS , 1995 .

[6]  Shaher Momani,et al.  The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics , 2009, Comput. Math. Appl..

[7]  Mingrong Cui,et al.  Compact finite difference method for the fractional diffusion equation , 2009, J. Comput. Phys..

[8]  Hong-qing Zhang,et al.  Fractional sub-equation method and its applications to nonlinear fractional PDEs , 2011 .

[9]  Changpin Li,et al.  Numerical approaches to fractional calculus and fractional ordinary differential equation , 2011, J. Comput. Phys..

[10]  Xianguo Geng,et al.  A generalized Hirota–Satsuma coupled Korteweg–de Vries equation and Miura transformations , 1999 .

[11]  Sheng Zhang,et al.  Application of Exp-function method to Riccati equation and new exact solutions with three arbitrary functions of Broer–Kaup–Kupershmidt equations , 2008 .

[12]  Ji-Huan He,et al.  Variational iteration method for delay differential equations , 1997 .

[13]  Bo Tian,et al.  New extension of the tanh-function method and application to the Whitham-Broer-Kaup shallow water model with symbolic computation , 2007 .

[14]  M. A. Abdou,et al.  The extended mapping method and its applications for nonlinear evolution equations , 2006 .

[15]  Davood Domiri Ganji,et al.  Solitary wave solutions for a time-fraction generalized Hirota–Satsuma coupled KdV equation by an analytical technique , 2009 .

[16]  Sheng Zhang,et al.  Exp-function method for solving Maccari's system , 2007 .

[17]  F. Khani,et al.  Some new exact solutions of the (2+1)-dimensional variable coefficient Broer-Kaup system using the Exp-function method , 2009, Comput. Math. Appl..

[18]  Guanhua Huang,et al.  A finite element solution for the fractional advection–dispersion equation , 2008 .

[19]  Liquan Mei,et al.  The fractional variational iteration method using He's polynomials , 2011 .

[20]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[21]  G. Jumarie,et al.  Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results , 2006, Comput. Math. Appl..

[22]  B. M. Fulk.  MATH , 1992 .

[23]  B. Kupershmidt,et al.  Mathematics of dispersive water waves , 1985 .

[24]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[25]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[26]  Ji-Huan He.  A coupling method of a homotopy technique and a perturbation technique for non-linear problems , 2000 .

[27]  Zhang Ping.  New exact solutions to breaking soliton equations and Whitham–Broer–Kaup equations , 2010 .

[28]  Guy Jumarie,et al.  Cauchy's integral formula via the modified Riemann-Liouville derivative for analytic functions of fractional order , 2010, Appl. Math. Lett..

[29]  Ahmed M. A. El-Sayed,et al.  The Adomian decomposition method for solving partial differential equations of fractal order in finite domains , 2006 .

[30]  Shaher Momani,et al.  A generalized differential transform method for linear partial differential equations of fractional order , 2008, Appl. Math. Lett..

[31]  Sheng Zhang,et al.  Exp-function method for constructing explicit and exact solutions of a lattice equation , 2008, Appl. Math. Comput..

[32]  Wen-Xiu Ma,et al.  Complexiton solutions to the Korteweg–de Vries equation , 2002 .

[33]  Z. Odibat.  Compact and noncompact structures for nonlinear fractional evolution equations , 2008 .

[34]  Ji-Huan He,et al.  Solitary solutions, periodic solutions and compacton-like solutions using the Exp-function method , 2007, Comput. Math. Appl..

[35]  Ji-Huan He,et al.  A SHORT REMARK ON FRACTIONAL VARIATIONAL ITERATION METHOD , 2011 .

[36]  Eric Wai Ming Lee,et al.  FRACTIONAL VARIATIONAL ITERATION METHOD AND ITS APPLICATION , 2010 .

[37]  O. P. Singh,et al.  An analytic algorithm for the space-time fractional advection-dispersion equation , 2011, Comput. Phys. Commun..

[38]  Ji-Huan He,et al.  Exp-function method for nonlinear wave equations , 2006 .

[39]  Guy Jumarie,et al.  Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution , 2007 .

[40]  A. Bekir,et al.  Exact solutions for nonlinear evolution equations using Exp-function method , 2008 .

[41]  Gisèle M. Mophou,et al.  Existence and uniqueness of mild solutions to impulsive fractional differential equations , 2010 .

[42]  Wei Jiang,et al.  Approximate solution of the fractional advection-dispersion equation , 2010, Comput. Phys. Commun..

[43]  Hyman,et al.  Compactons: Solitons with finite wavelength. , 1993, Physical review letters.

[44]  Andrew G. Glen,et al.  APPL , 2001 .

[45]  Mingliang Wang,et al.  Periodic wave solutions to a coupled KdV equations with variable coefficients , 2003 .

[46]  Reza Abazari,et al.  Numerical simulation of generalized Hirota-Satsuma coupled KdV equation by RDTM and comparison with DTM , 2012 .

[47]  M. A. Abdou,et al.  New exact travelling wave solutions using modified extended tanh-function method , 2007 .