Improved ()-Expansion Method for the Space and Time Fractional Foam Drainage and KdV Equations

The fractional complex transformation is used to transform nonlinear partial differential equations to nonlinear ordinary differential equations. The improved ()-expansion method is suggested to solve the space and time fractional foam drainage and KdV equations. The obtained results show that the presented method is effective and appropriate for solving nonlinear fractional differential equations.

[1]  Guy Jumarie,et al.  New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations , 2006, Math. Comput. Model..

[2]  Ji-Huan He,et al.  Homotopy Perturbation Method with an Auxiliary Term , 2012 .

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

[4]  Mohammad Taghi Darvishi,et al.  New solitary wave and periodic solutions of the foam drainage equation using the Exp-function method , 2009 .

[5]  Guy Jumarie,et al.  Lagrange characteristic method for solving a class of nonlinear partial differential equations of fractional order , 2006, Appl. Math. Lett..

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

[7]  A. Golbabai,et al.  Analytical modelling of fractional advection-dispersion equation defined in a bounded space domain , 2011, Math. Comput. Model..

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

[9]  M. Inc..  The approximate and exact solutions of the space- and time-fractional Burgers equations with initial conditions by variational iteration method , 2008 .

[10]  Khaled A. Gepreel,et al.  Exact solutions for nonlinear partial fractional differential equations , 2012 .

[11]  Nikolai A. Kudryashov,et al.  Seven common errors in finding exact solutions of nonlinear differential equations , 2009, 1011.4268.

[12]  Howard A. Stone,et al.  Dynamics of foam drainage , 1998 .

[13]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[14]  Hessameddin Yaghoobi,et al.  Application of the Homotopy Perturbation Method for Solving the Foam Drainage Equation , 2011 .

[15]  M. A. Helal,et al.  The tanh method and Adomian decomposition method for solving the foam drainage equation , 2007, Appl. Math. Comput..

[16]  Zheng Bin,et al.  （G＇/G）-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics , 2012 .

[17]  郑滨.  (G'/G)-Expansion Method for Solving Fractional Partial Differential Equations in the Theory of Mathematical Physics , 2012 .

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

[19]  N. Kudryashov,et al.  Popular ansatz methods and solitary wave solutions of the Kuramoto-Sivashinsky equation , 2009 .

[20]  Ji-Huan He,et al.  CONVERTING FRACTIONAL DIFFERENTIAL EQUATIONS INTO PARTIAL DIFFERENTIAL EQUATIONS , 2012 .

[21]  Xiao-Jun Yang,et al.  Local Fractional Integral Transforms , 2011 .

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

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

[24]  Jincun Liu,et al.  Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method , 2011, Appl. Math. Comput..

[25]  Hossein Jafari,et al.  Solving a multi-order fractional differential equation using adomian decomposition , 2007, Appl. Math. Comput..

[26]  Yinnian He,et al.  A generalized fractional sub-equation method for fractional differential equations with variable coefficients☆ , 2012 .

[27]  Alain Oustaloup,et al.  A Lyapunov approach to the stability of fractional differential equations , 2009, Signal Process..

[28]  M. Ganjiani,et al.  Solution of nonlinear fractional differential equations using homotopy analysis method , 2010 .

[29]  B. Lu.  The first integral method for some time fractional differential equations , 2012 .

[30]  Guy Jumarie,et al.  Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions , 2009, Appl. Math. Lett..

[31]  Sachin Bhalekar,et al.  Solving multi-term linear and non-linear diffusion-wave equations of fractional order by Adomian decomposition method , 2008, Appl. Math. Comput..

[32]  Nikolai A. Kudryashov,et al.  A note on the G'/G-expansion method , 2010, Appl. Math. Comput..

[33]  Weiguo Zhang,et al.  Application of Improved (G′/G)–Expansion Method to Traveling Wave Solutions of Two Nonlinear Evolution Equations , 2012 .

[34]  S. Momani,et al.  A novel method for nonlinear fractional partial differential equations: Combination of DTM and generalized Taylor's formula , 2008 .

[35]  A. Golbabai,et al.  Solitary pattern solutions for fractional Zakharov-Kuznetsov equations with fully nonlinear dispersion , 2012, Appl. Math. Lett..

[36]  Khosro Sayevand,et al.  Analysis of differential equations of fractional order , 2012 .

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

[38]  Nikolai A. Kudryashov,et al.  Be careful with the Exp-function method , 2009, 1011.4265.

[39]  A. Golbabai,et al.  Analytical treatment of differential equations with fractional coordinate derivatives , 2011, Comput. Math. Appl..

[40]  Liquan Mei,et al.  Compacton and solitary pattern solutions for nonlinear dispersive KdV-type equations involving Jumarieʼs fractional derivative , 2012 .