Solving nonlinear fractional partial differential equations using the homotopy analysis method

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM‐Burgers, cubic Boussinesq, coupled KdV, and Boussinesq‐like B(m,n) equations with initial conditions, which are introduced by replacing some integer‐order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer‐order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

[1]  W. Marsden.  I and J , 2012 .

[2]  Mehdi Dehghan,et al.  Solution of parabolic integro‐differential equations arising in heat conduction in materials with memory via He's variational iteration technique , 2010 .

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

[4]  Mehdi Dehghan,et al.  Variational iteration method for solving the wave equation subject to an integral conservation condition , 2009 .

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

[6]  Lina Song,et al.  Solving the fractional BBM–Burgers equation using the homotopy analysis method , 2009 .

[7]  Hossein Jafari,et al.  Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation , 2009 .

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

[9]  M. A. Abdou,et al.  An improved variational iteration method for solving coupled KdV and Boussinesq-like B(m, n) equations , 2009 .

[10]  Md. Sazzad Hossien Chowdhury,et al.  Comparison of homotopy analysis method and homotopy-perturbation method for purely nonlinear fin-type problems , 2009 .

[11]  T. Hayat,et al.  Comparison of HAM and HPM methods in nonlinear heat conduction and convection equations , 2008 .

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

[13]  Mehdi Dehghan,et al.  Solution of a model describing biological species living together using the variational iteration method , 2008, Math. Comput. Model..

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

[15]  M. Dehghan,et al.  Application of He's variational iteration method for solving the Cauchy reaction-diffusion problem , 2008 .

[16]  Mehdi Dehghan,et al.  Identifying an unknown function in a parabolic equation with overspecified data via He's variational iteration method , 2008 .

[17]  S. Abbasbandy.  Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method , 2008 .

[18]  Hang Xu,et al.  Analysis of a time fractional wave-like equation with the homotopy analysis method , 2008 .

[19]  M. Dehghan,et al.  Approximate solution of a differential equation arising in astrophysics using the variational iteration method , 2008 .

[20]  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 .

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

[22]  A. Wazwaz.  The variational iteration method for rational solutions for KdV, K(2,2), Burgers, and cubic Boussinesq equations , 2007 .

[23]  Mehdi Dehghan,et al.  On the convergence of He's variational iteration method , 2007 .

[24]  Mehdi Dehghan,et al.  Numerical solution of a biological population model using He's variational iteration method , 2007, Comput. Math. Appl..

[25]  Lina Song,et al.  Application of homotopy analysis method to fractional KdV–Burgers–Kuramoto equation , 2007 .

[26]  Mehdi Dehghan,et al.  He’s variational iteration method for computing a control parameter in a semi-linear inverse parabolic equation , 2007 .

[27]  Mehdi Dehghan,et al.  The solution of coupled Burgers' equations using Adomian-Pade technique , 2007, Appl. Math. Comput..

[28]  M. Dehghan,et al.  Solution of a semilinear parabolic equation with an unknown control function using the decomposition procedure of Adomian , 2007 .

[29]  M. Dehghan,et al.  Solution of problems in calculus of variations via He's variational iteration method , 2007 .

[30]  Dong-Hong Ji,et al.  Document reranking by term distribution and maximal marginal relevance for chinese information retrieval , 2007, Inf. Process. Manag..

[31]  T. Hayat,et al.  Homotopy analysis of MHD boundary layer flow of an upper-convected Maxwell fluid , 2007 .

[32]  Mehdi Dehghan,et al.  Numerical solution of the Klein–Gordon equation via He’s variational iteration method , 2007 .

[33]  Shaher Momani,et al.  Decomposition method for solving fractional Riccati differential equations , 2006, Appl. Math. Comput..

[34]  S. Liao.  Series Solutions of Unsteady Boundary‐Layer Flows over a Stretching Flat Plate , 2006 .

[35]  Mehdi Dehghan,et al.  The use of He's variational iteration method for solving a Fokker–Planck equation , 2006 .

[36]  Q. Chang,et al.  Exact solitary solutions with compact support for the nonlinear dispersive Boussinesq-like B(m,n) equations , 2005 .

[37]  Santanu Saha Ray,et al.  An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method , 2005, Appl. Math. Comput..

[38]  Shaher Momani,et al.  Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method , 2005, Appl. Math. Comput..

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

[40]  M. Dehghan,et al.  Numerical Solution of Laplace Equation in a Disk using the Adomian Decomposition Method , 2005 .

[41]  Dogan Kaya,et al.  Exact and numerical traveling wave solutions for nonlinear coupled equations using symbolic computation , 2004, Appl. Math. Comput..

[42]  S. Liao.  An analytic approximate approach for free oscillations of self-excited systems , 2004 .

[43]  Tasawar Hayat,et al.  Periodic unidirectional flows of a viscoelastic fluid with the fractional Maxwell model , 2004, Appl. Math. Comput..

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

[45]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[46]  S. Liao.  On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet , 2003, Journal of Fluid Mechanics.

[47]  Igor M. Sokolov,et al.  Physics of Fractal Operators , 2003 .

[48]  R. Feynman,et al.  RECENT APPLICATIONS OF FRACTIONAL CALCULUS TO SCIENCE AND ENGINEERING , 2003 .

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

[50]  Zhenya Yan,et al.  New families of solitons with compact support for Boussinesq-like B(m,n) equations with fully nonlinear dispersion , 2002 .

[51]  Nabil T. Shawagfeh,et al.  Analytical approximate solutions for nonlinear fractional differential equations , 2002, Appl. Math. Comput..

[52]  Abdul-Majid Wazwaz,et al.  A new method for solving singular initial value problems in the second-order ordinary differential equations , 2002, Appl. Math. Comput..

[53]  Shijun Liao,et al.  Analytic solutions of the temperature distribution in Blasius viscous flow problems , 2002, Journal of Fluid Mechanics.

[54]  B. Tian,et al.  Ion-acoustic shocks in space and laboratory dusty plasmas: Two-dimensional and non-traveling-wave observable effects , 2001 .

[55]  J. Trujillo,et al.  Differential equations of fractional order:methods results and problem —I , 2001 .

[56]  D. Kaya.  Explicit Solutions of Generalized Nonlinear Boussinesq Equations , 2001 .

[57]  Virginia Kiryakova,et al.  Multiple (multiindex) Mittag-Leffler functions and relations to generalized fractional calculus , 2000 .

[58]  R. Schreiber.  Numerical Methods for Partial Differential Equations , 1999 .

[59]  S. Liao.  An explicit, totally analytic approximate solution for Blasius’ viscous flow problems , 1999 .

[60]  Ming-xin Wang,et al.  Exact soliton solutions of some nonlinear physical models 1 This work was supported by PRC grants NS , 1999 .

[61]  I. Podlubny.  Fractional differential equations : an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications , 1999 .

[62]  S. Liao.  AN EXPLICIT TOTALLY ANALYTIC APPROXIMATION OF BLASIUS VISCOUS FLOW PROBLEMS , 1999 .

[63]  M. Gandarias,et al.  Classical and Nonclassical Symmetries of a Generalized Boussinesq Equation , 1998, math-ph/9801206.

[64]  S. Liao.  A kind of approximate solution technique which does not depend upon small parameters — II. An application in fluid mechanics , 1997 .

[65]  Existence and convergence of solutions for the generalized BBM-Burgers equations with dissipative term , 1997 .

[66]  A. Wazwaz.  Necessary conditions for the appearance of noise terms in decomposition solutions series , 1997 .

[67]  Shijun Liao,et al.  AN APPROXIMATE SOLUTION TECHNIQUE WHICH DOES NOT DEPEND UPON SMALL PARAMETERS (PART 2): AN APPLICATION IN FLUID MECHANICS , 1997 .

[68]  A. Osborne,et al.  The inverse scattering transform: Tools for the nonlinear fourier analysis and filtering of ocean surface waves , 1995 .

[69]  S. Liao.  An approximate solution technique not depending on small parameters: A special example , 1995 .

[70]  Shijun Liao,et al.  AN APPROXIMATE SOLUTION TECHNIQUE WHICH DOES NOT DEPEND UPON SMALL PARAMETERS: A SPECIAL EXAMPLE , 1995 .

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

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

[73]  Hyman,et al.  Compactons: Solitons with finite wavelength. , 1993, Physical review letters.

[74]  P. Christiansen,et al.  Solitary wave solutions to a system of Boussinesq-like equations , 1992 .

[75]  M. Ablowitz,et al.  Solitons, Nonlinear Evolution Equations and Inverse Scattering , 1992 .

[76]  G. Adomian.  A review of the decomposition method in applied mathematics , 1988 .

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

[78]  M. Caputo.  Linear models of dissipation whose Q is almost frequency independent , 1966 .