The variational iteration method: An efficient scheme for handling fractional partial differential equations in fluid mechanics

Variational iteration method has been used to handle linear and nonlinear differential equations. The main property of the method lies in its flexibility and ability to solve nonlinear equations accurately and conveniently. In this work, a general framework of the variational iteration method is presented for analytical treatment of fractional partial differential equations in fluid mechanics. The fractional derivatives are described in the Caputo sense. Numerical illustrations that include the fractional wave equation, fractional Burgers equation, fractional KdV equation, fractional Klein-Gordon equation and fractional Boussinesq-like equation are investigated to show the pertinent features of the technique. Comparison of the results obtained by the variational iteration method with those obtained by Adomian decomposition method reveals that the first method is very effective and convenient. The basic idea described in this paper is expected to be further employed to solve other similar linear and nonlinear problems in fractional calculus.

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

[2]  Zaid M. Odibat,et al.  Reliable approaches of variational iteration method for nonlinear operators , 2008, Math. Comput. Model..

[3]  Ji-Huan He.  Approximate solution of nonlinear differential equations with convolution product nonlinearities , 1998 .

[4]  Shaher Momani,et al.  Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations , 2007, Comput. Math. Appl..

[5]  Ahmet Yildirim,et al.  Comparison between Adomian's method and He's homotopy perturbation method , 2008, Comput. Math. Appl..

[6]  Igor Podlubny,et al.  Geometric and Physical Interpretation of Fractional Integration and Fractional Differentiation , 2001, math/0110241.

[7]  George Adomian,et al.  Solving Frontier Problems of Physics: The Decomposition Method , 1993 .

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

[9]  Y. Cherruault.  Convergence of Adomian's method , 1990 .

[10]  Ji-Huan He,et al.  Variational Theory for Linear Magneto-Electro-Elasticity , 2001 .

[11]  Abdul-Majid Wazwaz,et al.  The variational iteration method: A reliable analytic tool for solving linear and nonlinear wave equations , 2007, Comput. Math. Appl..

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

[13]  H. Sekine,et al.  General Use of the Lagrange Multiplier in Nonlinear Mathematical Physics1 , 1980 .

[14]  Abdul-Majid Wazwaz,et al.  The variational iteration method: A powerful scheme for handling linear and nonlinear diffusion equations , 2007, Comput. Math. Appl..

[15]  Ahmed Alawneh,et al.  Variational iteration method for solving the space‐ and time‐fractional KdV equation , 2008 .

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

[17]  Shaher Momani,et al.  Application of He’s variational iteration method to Helmholtz equation , 2006 .

[18]  Ji-Huan He,et al.  Variational iteration method: New development and applications , 2007, Comput. Math. Appl..

[19]  Reza Mokhtari,et al.  Variational Iteration Method for Solving Nonlinear Differential- Difference Equations , 2008 .

[20]  Ji-Huan He.  Variational iteration method—Some recent results and new interpretations , 2007 .

[21]  Ji-Huan He,et al.  Semi-Inverse Method of Establishing Generalized Variational Principles for Fluid Mechanics With Emphasis on Turbomachinery Aerodynamics , 1997 .

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

[23]  Jianhong Wu,et al.  Numerical steady state and Hopf bifurcation analysis on the diffusive Nicholson's blowflies equation , 2000, Appl. Math. Comput..

[24]  Y. Cherruault,et al.  Decomposition methods: A new proof of convergence , 1993 .

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

[26]  Alan Solomon,et al.  The initial-value problem for the equation (_{})_{}=ₓₓ , 1970 .

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

[28]  A. Yildirim,et al.  Solutions of singular IVPs of Lane–Emden type by the variational iteration method , 2009 .

[29]  Abdul-Majid Wazwaz,et al.  A new modification of the Adomian decomposition method for linear and nonlinear operators , 2001, Appl. Math. Comput..

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

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

[32]  Fawang Liu,et al.  The time fractional diffusion equation and the advection-dispersion equation , 2005, The ANZIAM Journal.

[33]  Ji-Huan He.  Variational iteration method – a kind of non-linear analytical technique: some examples , 1999 .

[34]  Ji-Huan He,et al.  Variational Principle for Nano Thin Film Lubrication , 2003 .

[35]  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.

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

[37]  Shaher Momani,et al.  Numerical approach to differential equations of fractional order , 2007 .

[38]  K. B. Oldham,et al.  The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order , 1974 .

[39]  Abdul-Majid Wazwaz,et al.  A new algorithm for calculating adomian polynomials for nonlinear operators , 2000, Appl. Math. Comput..

[40]  Ji-Huan He,et al.  Variational iteration method for autonomous ordinary differential systems , 2000, Appl. Math. Comput..

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

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

[43]  Ji-Huan He,et al.  Variational iteration method for delay differential equations , 1997 .

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

[45]  Ji-Huan He,et al.  Variational principles for some nonlinear partial differential equations with variable coefficients , 2004 .

[46]  Abdul-Majid Wazwaz,et al.  The variational iteration method for solving linear and nonlinear systems of PDEs , 2007, Comput. Math. Appl..

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

[48]  Jafar Biazar,et al.  He's Variational Iteration Method for Solving Hyperbolic Differential Equations , 2007 .

[49]  Huang Feng-hui.  The Fundamental Solution of the Time-Space Fractional Advection-Dispersion Equation , 2009 .

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

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

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

[53]  H. Ozer,,et al.  Application of the Variational Iteration Method to the Boundary Value Problems with Jump Discontinuities arising in Solid Mechanics , 2007 .