Homotopy perturbation method for nonlinear partial differential equations of fractional order

The aim of this Letter is to present an efficient and reliable treatment of the homotopy perturbation method (HPM) for nonlinear partial differential equations with fractional time derivative. The fractional derivative is described in the Caputo sense. The modified algorithm provides approximate solutions in the form of convergent series with easily computable components. The obtained results are in good agreement with the existing ones in open literature and it is shown that the technique introduced here is robust, efficient and easy to implement.

[1]  Ji-Huan He.  Homotopy Perturbation Method for Bifurcation of Nonlinear Problems , 2005 .

[2]  Saeid Abbasbandy,et al.  Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method , 2006, Appl. Math. Comput..

[3]  K. Miller,et al.  An Introduction to the Fractional Calculus and Fractional Differential Equations , 1993 .

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

[5]  Shaher Momani,et al.  An explicit and numerical solutions of the fractional KdV equation , 2005, Math. Comput. Simul..

[6]  Saeid Abbasbandy,et al.  Numerical solutions of the integral equations: Homotopy perturbation method and Adomian's decomposition method , 2006, Appl. Math. Comput..

[7]  A. Siddiqui,et al.  Couette and Poiseuille Flows for Non-Newtonian Fluids , 2006 .

[8]  Shaher Momani,et al.  Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method , 2006, Appl. Math. Comput..

[9]  Ji-Huan He.  Limit cycle and bifurcation of nonlinear problems , 2005 .

[10]  Ji-Huan He,et al.  The homotopy perturbation method for nonlinear oscillators with discontinuities , 2004, Appl. Math. Comput..

[11]  Ji-Huan He,et al.  Addendum:. New Interpretation of Homotopy Perturbation Method , 2006 .

[12]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

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

[14]  Ji-Huan He.  HOMOTOPY PERTURBATION METHOD FOR SOLVING BOUNDARY VALUE PROBLEMS , 2006 .

[15]  S. Momani,et al.  Analytical approach to linear fractional partial differential equations arising in fluid mechanics , 2006 .

[16]  Ji-Huan He.  Approximate analytical solution for seepage flow with fractional derivatives in porous media , 1998 .

[17]  Shaher Momani,et al.  Non-perturbative analytical solutions of the space- and time-fractional Burgers equations , 2006 .

[18]  Ji-Huan He.  Application of homotopy perturbation method to nonlinear wave equations , 2005 .

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

[20]  Shaher Momani,et al.  Approximate solutions for boundary value problems of time-fractional wave equation , 2006, Appl. Math. Comput..

[21]  Ji-Huan He,et al.  Comparison of homotopy perturbation method and homotopy analysis method , 2004, Appl. Math. Comput..

[22]  S. Momani,et al.  Numerical methods for nonlinear partial differential equations of fractional order , 2008 .

[23]  M. El-shahed.  Application of He's Homotopy Perturbation Method to Volterra's Integro-differential Equation , 2005 .

[24]  Ji-Huan He,et al.  Asymptotology by homotopy perturbation method , 2004, Appl. Math. Comput..

[25]  R. Mahmood,et al.  Thin Film Flow of a Third Grade Fluid on a Moving Belt by He's Homotopy Perturbation Method , 2006 .

[26]  S. Momani,et al.  Numerical comparison of methods for solving linear differential equations of fractional order , 2007 .

[27]  Ji-Huan He.  Periodic solutions and bifurcations of delay-differential equations , 2005 .

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

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