Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part

Convergence and stability are main issues when an asymptotical method like the Homotopy Perturbation Method (HPM) has been used to solve differential equations. In this paper, convergence of the solution of fractional differential equations is maintained. Meanwhile, an effective method is suggested to select the linear part in the HPM to keep the inherent stability of fractional equations. Riccati fractional differential equations as a case study are then solved, using the Enhanced Homotopy Perturbation Method (EHPM). Current results are compared with those derived from the established Adams-Bashforth-Moulton method, in order to verify the accuracy of the EHPM. It is shown that there is excellent agreement between the two sets of results. This finding confirms that the EHPM is powerful and efficient tool for solving nonlinear fractional differential equations.

[1]  Changpin Li,et al.  Chaos in Chen's system with a fractional order , 2004 .

[2]  A. Ranjbar,et al.  Maintaining the stability of nonlinear differential equations by the enhancement of HPM , 2008 .

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

[4]  Shijun Liao,et al.  SERIES SOLUTIONS OF NON-LINEAR RICCATI DIFFERENTIAL EQUATIONS WITH FRACTIONAL ORDER , 2009 .

[5]  S. Momani,et al.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 2008 .

[6]  Shaher Momani,et al.  Solving systems of fractional differential equations by homotopy-perturbation method , 2008 .

[7]  Zheng-Ming Ge,et al.  Chaos synchronization of fractional order modified duffing systems with parameters excited by a chaotic signal , 2008 .

[8]  K. Diethelm.  AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER , 1997 .

[9]  D. Matignon.  Stability results for fractional differential equations with applications to control processing , 1996 .

[10]  S. Momani,et al.  Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order , 2008 .

[11]  E. Ahmed,et al.  Equilibrium points, stability and numerical solutions of fractional-order predator–prey and rabies models , 2007 .

[12]  Hossein Jafari,et al.  SOLVING FRACTIONAL DIFFUSION AND WAVE EQUATIONS BY MODIFIED HOMOTOPY PERTURBATION METHOD , 2007 .

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

[14]  Ali Akbar Ranjbar,et al.  A Solution of Riccati Nonlinear Differential Equation using Enhanced Homotopy Perturbation Method (EHPM) , 2008 .

[15]  Prince Abdullah bin Ghazi,et al.  Modified homotopy perturbation method : Application to quadratic Riccati differential equation of fractional order , 2007 .

[16]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[17]  Davood Domiri Ganji,et al.  Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations , 2007 .

[18]  Alan D. Freed,et al.  Detailed Error Analysis for a Fractional Adams Method , 2004, Numerical Algorithms.

[19]  D. Ganji.  The application of He's homotopy perturbation method to nonlinear equations arising in heat transfer , 2006 .

[20]  Ji-Huan He.  SOME ASYMPTOTIC METHODS FOR STRONGLY NONLINEAR EQUATIONS , 2006 .

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

[22]  Mohammad Saleh Tavazoei,et al.  A necessary condition for double scroll attractor existence in fractional-order systems , 2007 .