Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method

Abstract In this paper, by introducing the fractional derivative in the sense of Caputo, the generalized two-dimensional differential transform method (DTM) is directly applied to solve the coupled Burgers equations with space- and time-fractional derivatives. The presented method is a numerical method based on the generalized Taylor series formula which constructs an analytical solution in the form of a polynomial. Several illustrative examples are given to demonstrate the effectiveness of the generalized two-dimensional DTM for the equations.

[1]  Cha'o-Kuang Chen,et al.  Solving partial differential equations by two-dimensional differential transform method , 1999, Appl. Math. Comput..

[2]  Fatma Ayaz,et al.  Solutions of the system of differential equations by differential transform method , 2004, Appl. Math. Comput..

[3]  M. Bayram,et al.  Approximate analytical solution for the fractional modified KdV by differential transform method , 2010 .

[4]  A. S. V. Ravi Kanth,et al.  Differential transform method for solving the linear and nonlinear Klein-Gordon equation , 2009, Comput. Phys. Commun..

[5]  Changpin Li,et al.  Numerical algorithm based on Adomian decomposition for fractional differential equations , 2009, Comput. Math. Appl..

[6]  Hossein Jafari,et al.  SOLVING A SYSTEM OF NONLINEAR FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS USING HOMOTOPY ANALYSIS METHOD , 2009 .

[7]  Yang Zhang,et al.  A finite difference method for fractional partial differential equation , 2009, Appl. Math. Comput..

[8]  Shih-Hsiang Chang,et al.  A new algorithm for calculating two-dimensional differential transform of nonlinear functions , 2009, Appl. Math. Comput..

[9]  Shijun Liao,et al.  Analysis of nonlinear fractional partial differential equations with the homotopy analysis method , 2009 .

[10]  A. S. V. Ravi Kanth,et al.  Two-dimensional differential transform method for solving linear and non-linear Schrödinger equations , 2009 .

[11]  Shaher Momani,et al.  Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation , 2007 .

[12]  Fawang Liu,et al.  Finite difference approximations for the fractional Fokker–Planck equation , 2009 .

[13]  M. Caputo.  Linear Models of Dissipation whose Q is almost Frequency Independent-II , 1967 .

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

[15]  Cha'o-Kuang Chen,et al.  Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform , 1999 .

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

[17]  Ming-Jyi Jang,et al.  On solving the initial-value problems using the differential transformation method , 2000, Appl. Math. Comput..

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

[19]  S. Ray,et al.  Analytical solution for the space fractional diffusion equation by two-step Adomian Decomposition Method , 2009 .

[20]  I. Podlubny.  Fractional differential equations , 1998 .

[21]  Yong Chen,et al.  Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives , 2008, Appl. Math. Comput..

[22]  A. Kanth,et al.  Solution of singular two-point boundary value problems using differential transformation method , 2008 .