Analytical approach to fractional partial differential equations in fluid mechanics by means of the homotopy perturbation method

Purpose – This paper aims to present a general framework of the homotopy perturbation method (HPM) for analytic treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense.Design/methodology/approach – Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation and fractional Klein‐Gordon equation are investigated to show the pertinent features of the technique.Findings – HPM is a powerful and efficient technique in finding exact and approximate solutions for fractional partial differential equations in fluid mechanics. The implementation of the noise terms, if they exist, is a powerful tool to accelerate the convergence of the solution. The results so obtained reinforce the conclusions made by many researchers that the efficiency of the HPM and related phenomena gives it much wider applicability.Originality/value – The essential idea of this method is to introduce a homo...

[1]  Qi Wang.  Homotopy perturbation method for fractional KdV-Burgers equation , 2008 .

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

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

[4]  Mehdi Dehghan,et al.  Solution of a partial differential equation subject to temperature overspecification by He's homotopy perturbation method , 2007 .

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

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

[7]  F. Mainardi,et al.  The fundamental solution of the space-time fractional diffusion equation , 2007, cond-mat/0702419.

[8]  Ji-Huan He.  AN ELEMENTARY INTRODUCTION TO RECENTLY DEVELOPED ASYMPTOTIC METHODS AND NANOMECHANICS IN TEXTILE ENGINEERING , 2008 .

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

[10]  Qi Wang,et al.  Homotopy perturbation method for fractional KdV equation , 2007, Appl. Math. Comput..

[11]  Mehdi Dehghan,et al.  The numerical solution of the second Painlevé equation , 2009 .

[12]  Md. Sazzad Hossien Chowdhury,et al.  Solutions of Emden-Fowler equations by homotopy-perturbation method , 2009 .

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

[14]  M. Dehghan,et al.  Application of He’s homotopy perturbation method for non-linear system of second-order boundary value problems , 2009 .

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

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

[17]  M. Dehghan,et al.  The use of the decomposition procedure of Adomian for solving a delay differential equation arising in electrodynamics , 2008 .

[18]  Mehdi Dehghan,et al.  SOLUTION OF AN INTEGRO-DIFFERENTIAL EQUATION ARISING IN OSCILLATING MAGNETIC FIELDS USING HE'S HOMOTOPY PERTURBATION METHOD , 2008 .

[19]  Asghar Ghorbani,et al.  Beyond Adomian polynomials: He polynomials , 2009 .

[20]  Ahmet Yildirim,et al.  A note on He’s homotopy perturbation method for van der Pol oscillator with very strong nonlinearity , 2007 .

[21]  G. Drăgănescu,et al.  Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives , 2006 .

[22]  S. Momani,et al.  Application of Variational Iteration Method to Nonlinear Differential Equations of Fractional Order , 2006 .

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

[24]  Ji-Huan He,et al.  Recent development of the homotopy perturbation method , 2008 .

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

[26]  E. Yusufoglu,et al.  Homotopy Perturbation Method for Solving a Nonlinear System of Second Order Boundary Value Problems , 2007 .

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

[28]  Mehdi Dehghan,et al.  Solution of delay differential equations via a homotopy perturbation method , 2008, Math. Comput. Model..

[29]  E. Yusufoglu,et al.  Variational Iteration Method for Construction of Some Compact and Noncompact Structures of Klein-Gordon Equations , 2007 .

[30]  Shaher Momani,et al.  Analytic and approximate solutions of the space- and time-fractional telegraph equations , 2005, Appl. Math. Comput..

[31]  Mehdi Dehghan,et al.  USE OF HES HOMOTOPY PERTURBATION METHOD FOR SOLVING A PARTIAL DIFFERENTIAL EQUATION ARISING IN MODELING OF FLOW IN POROUS MEDIA , 2008 .

[32]  Shaher Momani,et al.  Analytical approximate solutions of the fractional convection–diffusion equation with nonlinear source term by He's homotopy perturbation method , 2010, Int. J. Comput. Math..

[33]  Mehdi Dehghan,et al.  Solution of the differential algebraic equations via homotopy perturbation method and their engineering applications , 2010, Int. J. Comput. Math..

[34]  Ahmet Yildirim,et al.  Determination of periodic solution for a u1/3 force by He's modified Lindstedt–Poincaré method , 2007 .

[35]  Fawang Liu,et al.  The fundamental solution of the space-time fractional advection-dispersion equation , 2005 .

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

[37]  Ahmet Yildirim,et al.  Solutions of singular IVPs of Lane–Emden type by homotopy perturbation method , 2007 .

[38]  Subir Das,et al.  Solution of Fractional Vibration Equation by the Variational Iteration Method and Modified Decomposition Method , 2008 .

[39]  Mehdi Dehghan,et al.  Inverse problem of diffusion equation by He's homotopy perturbation method , 2007 .

[40]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[41]  Kamel Al-khaled,et al.  An approximate solution for a fractional diffusion-wave equation using the decomposition method , 2005, Appl. Math. Comput..

[42]  E. S. Ismail,et al.  Approximate analytical solution to fractional modified KdV equations , 2009, Math. Comput. Model..

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

[44]  Ji-Huan He.  New interpretation of homotopy perturbation method , 2006 .

[45]  Asghar Ghorbani,et al.  He's Homotopy Perturbation Method for Calculating Adomian Polynomials , 2007 .

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

[47]  Alan L. Andrew,et al.  Asymptotic correction and inverse eigenvalue problems: an overview , 2005 .

[48]  Shaher Momani,et al.  Homotopy perturbation method for nonlinear partial differential equations of fractional order , 2007 .

[49]  I. Hashim,et al.  Application of multistage homotopy-perturbation method for the solutions of the Chen system , 2009 .

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

[51]  Ahmet Yildirim,et al.  Traveling Wave Solution of Korteweg-de Vries Equation using He's Homotopy Perturbation Method , 2007 .

[52]  Ahmet Yildirim,et al.  A Comparative Study of He's Homotopy Perturbation Method for Determining Frequency-amplitude Relation of a Nonlinear Oscillator with Discontinuities , 2007 .

[53]  Elçin Yusufoglu,et al.  An efficient algorithm for solving integro-differential equations system , 2007, Appl. Math. Comput..

[54]  Andrzej Hanygad,et al.  Multidimensional solutions of time-fractional diffusion-wave equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.