Applied Mathematics, 2011, 2, 10911095 doi:10.4236/am.2011.29150 Published Online September 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Application of He’s Variational Iteration Method for the Analytical Solution of Space Fractional Diffusion Equation Mehdi Safari Department of Mechanical Engineering, Islamic Azad University, Aligoodarz Branch, Aligoodarz, Iran Email: ms_safari2005@yahoo.com Received July 1, 2011; revised August 3, 2011; accepted August 11, 2011 Abstract Spatially fractional order diffusion equations are generalizations of classical diffusion equations which are increasingly used in modeling practical super diffusive problems in fluid flow, finance and others areas of application. This paper presents the analytical solutions of the space fractional diffusion equations by varia tional iteration method (VIM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. Two examples, the first one is onedimensional and the second one is twodimensional fractional diffusion equation, are presented to show the application of the present tech niques. The present method performs extremely well in terms of efficiency and simplicity. Keywords: He’s Variational Iteration Method, Fractional Derivative, Fractional Diffusion Equation 1. Introduction Fractional diffusion equations are used to model prob lems in Physics [13], Finance [47], and Hydrology [812]. Fractional space derivatives may be used to for mulate anomalous dispersion models, where a particle plume spreads at a rate that is different than the classical Brownian motion model. When a fractional derivative of order 1 < < 2 replaces the second derivative in a diffusion or dispersion model, it leads to a super diffu sive flow model. Nowadays, fractional diffusion equa tion plays important roles in modeling anomalous diffu sion and subdiffusion systems, description of fractional random walk , unification of diffu sion and w ave prop aga tion phenomenon, see, e.g. the reviews in [116], and references therein. Consider a onedimensional fractional diffusion equation considered in [17] (,) (,) () (,) uxt uxt dx qxt tx (1) on a finite domain R xx with 12 . We assume that the diffusion coefficient (or diffusivity) . We also assume an initial condition for 0dx (, 0uxt) ()sx R xx (, L ux tand Dirichlet bound ary conditions of the form and R . Equation (1) uses a Riemann fractional derivative of order )0 (,) R uxt ()b t . Consider a twodimensional fractional diffusion equa tion considered in [18] (,,) (,,) (,) (,,) (,)(,,) uxytuxyt dxy tx ux yt exy qxyt x (2) on a finite rectangular domain H xx and R yyy , with fractional orders 1 < 2 and 1 < 2 , where the diffusion coefficients , 0dxy and exy, 0. The “forcing” function t, , qxy can be used to represent sources and sinks. We will as sume that this fractional diffusion equation has a unique and sufficiently smooth solution under the following initial and boundary conditions. Assume the initial con dition u(x, y, t = 0) = f(x, y) for H xx and R yyy , and Dirichlet boundary condition , , yt, , uxytBx on the boundary (perimeter) of the rectangular region H xx , R y (, ,) LL Bxyt yy (, ,)Bx yt, with the additional restriction that . In physical applications, this means that the left/lower boundary is set far away enough from an evolving plume that no significant concentrations reach that boundary. The classical dispersion equation in twodimensions is given by 0 2 . The values of 1 < 2 and 1 < 2 model a super diffusive process in that coor dinate. Equation (2) also uses Riemann fractional deriva tives of order and . In this paper, we use the v ar ia tional iteration method (VIM) to obtain the solutions of
M. SAFARI 1092 the fractional diffusion Equations (1) and (2). The varia tional iteration method (VIM) established in (1999) by He in [1922] is thoroughly used by many researchers to handle linear and non linear models. The reliability of the method and the reduction in the size of computational domain gave this method a wider applicability. The method has been proved by many authors [2326], and the references therein, to be reliable and efficient for a wide variety of scientific applications, linear and nonlin ear as well. The method gives rapidly convergent suc cessive approximations of the exact solution if such a solution exists. For co ncrete problems, a few numbers of approximations can be used for numerical purposes with high degree of accuracy. The VIM does not require spe cific transformations or nonlinear terms as required by some existing techniques. However, we use the VIM to solve fractional diffusion Equations (1) and (2) and fi nally the results are illustrated in graphical figures. 2. Mathematical Aspects The mathematical definition of fractional calculus has been the subject of several different approaches [27,28]. The most frequently enco untered defin ition of an integral of fractional order is the RiemannLiouville integral, in which the fractional order integral is defined as 1 0 d() 1()d () () d( qt q tqq ) tf Dft q tt tx x (3) while the definition of fractional order derivative is () () 1 0 d d()1d()d () () dd d() nnq n t q tnnqn nq t Dft nq tt ttx ftx n (6) where ( and ) is the order of the opera tion and n is an integer that satisfies . q0qqR1 nq 3. Basic Idea of He’s Variational Iteration Method To clarify the basic ideas of VIM, we consider the fol lowing differential equation: LuNugt (5) where is a linear operator, a nonlinear operator and L N t an inhomogeneous term. According to VIM, we can write down a correction functional as follows: 10d t nnn n ututLu Nu g (6) where is a general Lagrangian multiplier which can be identified optimally via the variational theory. The subscript indicates the nth approximation and is considered as a restricted variation nn u 0 n u . 4. The Fractional Diffusion Equation Model and Its Solution by VIM Now we adopt variational iteration method for solving Equation (1). In the light of this method we assume that () 10() (,)d tx nn nn ututudxu qxt (7) where () indicates a differential with respect to x and dot denotes a differential with respect to t, is general Lagrangian multiplier.Similarly, fo r Equation (2) using variational iteration method, we can obtain 1 () () 0(, )(, )(, ,)d nn txy nn n utut u dxyuexyuqxyt (8) 5. Numerical Illustrations 5.1. Example 1 Let us consider a onedimensional fractional diffusion equation for the Equation (1), as taken in [17] 1.8 1.8 (,) (,) () (,) uxt uxt dx qxt tx (9) on a finite domain 01 , with the diffusion coeffi cient 2.8 2.8 ( )(2.2)60.183634dx xx (10) the source/sink function 3 (,)(1) t qxtxex (11) the initial condition 3 (,0) ,ux x (12) for 01x and the boundary conditions (0,)0,(1, ),0 t utut e fort (13) Implementation of Variational Iteration Method for Example 1 Now we consider the application of VIM to one dimensional fractional diffusion equation with the initial condition of: 3 (,0) ,ux x (14) for 01x Its correction variational functional in x and t can be expressed, respectively, as follows: (1.8 ) 10 (,) (,)()(,)d tx nn nn uxtu xtudxuqxt (15) where (1.8 ) indicates a differential with respect to x and dot denotes a differential with respect to t, is general Lagrangian multiplier. After some calculations, we obtain the following stationary conditions: 0 (16) 1 t 0 (17) Copyright © 2011 SciRes. AM
M. SAFARI1093 Equation (16) is called LagrangeEuler equation and Equation (17) is natural boundary condition. The La grange multiplier can therefore, be identified as 1 and the variational iteration formula is obtained in the form of: (1.8 ) 10 (,) (,)()(,)d tx nn nn uxtuxtudxuqxt (18) We start with the initial approximation of given by Equation (14). Using the above iteration for mula (18), we can directly obtain the other components as follows: (,0)ux 3 0(,) ,uxtx (19) 44 1(, )1.000001369(), t uxtxxt exx 34 (20) (1.8 ) 21 11 0 (,) (,)()(,)d tx uxtuxtudxuqxt (21) In Figure 1 we can see the 3D result of approximate solution of the onedimensional fractional diffusion equation by VIM. 5.2. Example 2 Let us consider a twodimensional fractional diffusion equation for the Equation (2), considered in [18]. 1.8 1.8 1.6 1.6 (,,) (,,) (, ) (,,) +(,)(,,), uxytuxyt dxy tx ux yt exy qxyt y (22) on a finite rectangular domain 01 , , for with the diffusion coefficients 01y end 0Tt Figure 1. For the onedimensional fractional diffusion equa tion with the initial condition (12) of Equation (9), VIM result for . (,)uxt 2.8 (,)(2.2)/ 6dxyx y (23) and 2.6 ( ,)2/(4.6)exyx y (24) and the forcing function 33.6 (,,)(1 2) t qxytxyexy (25) with the initial cond ition 33.6 (,,0)uxy xy (26) and Dirichlet bo undary cond itions on the r ectangle in the form and , for all . 3 (,0,)(0,,)0,(,1, ), t uxtuytuxte x 3.6t ey 0t(1,, )uyt Implementation of Variational Iteration Method for Example 2 Again we consider the application of VIM fractional diffusion equation with the initial condition of: 33.6 (,0) ,ux xy (27) for 01, 0y1x Its correction variational functional in x and t can be expressed, respectively, as follows: 1 (1.8 )(1.6) 0 (,,) (,,) (, )(, )(, ,)d nn txy nn n uxytuxyt u dxyuexyuqxyt (28) where (1.8 ) indicates a differential with respect to x, indicates a differential with respect to y and dot denotes a differential with respect to t, also (1.6 )y is gen eral Lagrangian multiplier. After some calculations, we obtain the following stationary conditions: 0 (29) 1 t 0 (30) Equation (29) is called LagrangeEuler equation and Equation (30) is natural boundary condition. The La grange multiplier can therefore, be identified as 1 and the variational iteration formula is obtained in the form of: 1 (1.8 )(1.6) 0 (,,) (,,) (, )(, )(,,)d nn txy nn n uxytuxyt u dxyuexyuqxyt (31) We start with the initial approximation of given by Equation (27). Using the above iteration for mula (31), we can directly obtain the other components as follows: (,0)ux 33.6 0(,,) ,uxyt xy (32) 23 23 44 55 1 18 23 34 55 (,,)22 2, tt u xytxyxyt exy exy (33) Copyright © 2011 SciRes. AM
M. SAFARI 1094 Figure 2. For the twodimensional fractional diffusion equa tion with the initial condition (26) of Equation (22), VIM result for with . , ()xtu1y 21 (1.8 )(1.6) 11 1 0 (,,) (,,) (, )(, )(, ,)d txy uxyt uxyt u dxyuexyuqxyt (34) In Figure 2 we can see the 3D result of approximate solution of the onedimensional fractional diffusion equa tion by VIM. 6. Conclusions In this paper, He’s variational iteration method has been successfully applied to find the solution of space frac tional diffusion equation. All cases show that the results of the VIM method are very good and the obtained solu tions are shown graphically. In our work, we use the Maple Package to calculate the functions obtained from the He’s variational iteration method. 7. References [1] R. Metzler, E. Barkai and J. Klafter, “Anomalous Diffu sion and Relaxation Close to Thermal Equilibrium: A Fractional FokkerPlanck Equation Approach,” Physics Review Letters, Vol. 82, No. 18, 1999, pp. 35633567. doi:10.1103/PhysRevLett.82.3563 [2] R. Metzler and J. Klafter, “The Random Walk’s Guide to Anomalous Diffusion: A Fractional Dynamics Approach,” Physics Reports, Vol. 339, No. 1, 2000, pp. 177. doi:10.1016/S03701573(00)000703 [3] R. Metzler and J. Klafter, “The Restaurant at the End of the Random Walk: Recent Developments in the Descrip tion of Anomalous Transport by Fractional Dynamics,” Journal of Physics A, Vol. 37, No. 31, 2004, pp. 161208. [4] R. Gorenflo, F. Mainardi, E. Scalas and M. Raberto, “Fractional Calculus and ContinuousTime Finance. III. The Diffusion Limit,” Mathematical Finance, Konstanz, 2000, pp. 171180. [5] F. Mainardi, M. Raberto, R. Gorenflo and E. Scalas, “Fractional Calculus and ContinuousTime Finance II: The WaitingTime Distribution,” Physica A, Vol. 287, no. 34, 2000, pp. 468481. doi:10.1016/S03784371(00)003861 [6] E. Scalas, R. Gorenflo and F. Mainardi, “Fractional Calcu lus and ContinuousTime Finance,” Physica A, Vol. 284, No. 14, 2000, pp. 376384. doi:10.1016/S03784371(00)002557 [7] M. Raberto, E. Scalas and F. Mainardi, “WaitingTimes and Returns in High Frequency Financial Data: An Em pirical Study,” Physica A, Vol. 314, No. 14, 2002, pp. 749755. doi:10.1016/S03784371(02)010488 [8] D. A. Benson, S. Wheatcraft and M. M. Meerschaert, “Application of a Fractional Advection Dispersion Equa tion,” Wate r Resource Research, Vol. 36, No. 6, 2000, pp. 14031412. doi:10.1029/2000WR900031 [9] B. Baeumer, M. M. Meerschaert, D. A. Benson and S. W. Wheatcraft, “Subordinated AdvectionDispersion Equa tion for Contaminant Transport,” Water Resource Re search, Vol. 37, No. 6, 2001, pp. 15431550. [10] D. A. Benson, R. Schumer, M. M. Meerschaert and S. W. Wheatcraft, “Fractional Dispersion, Lévy Motions, and the MADE Tracer Tests,” Transport in Porous Media, Vol. 42, No. 12, 2001, pp. 211240. doi:10.1023/A:1006733002131 [11] R. Schumer, D. A. Benson, M. M. Meerschaert and S. W. Wheatcraft, “Eulerian Derivation of the Fractional Advec tionDispersion Equation,” Journal of Contaminant Hy drology, Vol. 48, No. 12, 2001, pp. 6988. doi:10.1016/S01697722(00)001704 [12] R. Schumer, D. A. Benson, M. M. Meerschaert and B. Baeumer, “Multiscaling Fractional AdvectionDispersion Equations and Their Solutions,” Water Resource Re search, Vol. 39, No. 1, 2003, pp. 10221032. [13] A. Carpinteri and F. Mainardi, “Fractals and Fractional Calculus in Continuum Mechanics,” SpringerVerlag, Wien, New York, 1997, pp. 291348. [14] F. Mainardi and G. Pagnini, “The Wright Functions as Solutions of the Time Fractional Diffusion Equations,” Applied Mathematics and Computation, Vol. 141, No. 1, 2003, pp. 5162. doi:10.1016/S00963003(02)00320X [15] O. P. Agrawal, “Solution for a Fractional DiffusionWave Equation Defined in a Bounded Domain,” Nonlinear Dy namics, Vol. 29, No. 14, 2002, pp. 145155. doi:10.1023/A:1016539022492 [16] W. R. Schneider and W. Wyss, “Fractional Diffusion and Wave Equations,” Journal of Mathematical Physics, Vol. 30, No. 1, 1989, pp. 134144. doi:10.1063/1.528578 [17] M. M. Meerschaert, H. Scheffler and C. Tadjeran, “Finite Difference Methods for TwoDimensional Fractional Dis persion Equation,” Journal of Computational Physics, Copyright © 2011 SciRes. AM
M. SAFARI Copyright © 2011 SciRes. AM 1095 Vol. 211, No. 1, 2006, pp. 249261. doi:10.1016/j.jcp.2005.05.017 [18] C. Tadjeran, M. M. Meerschaert and H. Scheffler, “Finite Difference Methods for TwoDimensional Fractional Dispersion Equation,” Journal of Computational Physics, Vol. 213, No. 1, 2006, pp. 205213. doi:10.1016/j.jcp.2005.08.008 [19] J. H. He, “Some Asymptotic Methods for Strongly Nonlinear Equations,” International Journal of Modern Physics B, Vol. 20, No. 10, 2006, pp. 11411199. doi:10.1142/S0217979206033796 [20] J. H. He, “Approximate Analytical Solution for Seepage Flow with Fractional Derivatives in Porous Media,” Computer Methods in Applied Mechanics and Engineer ing, Vol. 167, No. 12, 1998, pp. 5768. doi:10.1016/S00457825(98)00108X [21] J. H. He, “Variational Iteration Method for Autonomous Ordinary Differential Systems,” Applied Mathematics and Computation, Vol. 114, No. 23, 2000, pp. 115123. doi:10.1016/S00963003(99)001046 [22] J. H. He and X. H. Wu, “Construction of Solitary Solution and CompactonLike Solution by Variational Iteration Method,” Chaos, Solitons & Fractals, Vol. 29, No. 1, 2006, pp. 108113. [23] D. D. Ganji, E. M. M. Sadeghi and M. Safari, “App lica tion of He’s Variational Iteration Method and Adomian’s Decomposition Method Method to Prochhammer Chree Equation,” International Journal of Modern Physics B, Vol. 23, No. 3, 2009, pp. 435446. doi:10.1142/S0217979209049656 [24] M. Safari, D. D. Ganji and M. Moslemi, “Application of He’s Variational Iteration Method and Adomian’s De composition Method to the Fractional KdvBurgers Kuramoto Equation,” Computers and Mathematics with Applications, Vol. 58, 2009, pp. 20912097. doi:10.1016/j.camwa.2009.03.043 [25] D. D. Ganji, M. Safari and R. Ghayor, “Application of He’s Variational Iteration Method and Adomian’s De composition Method to SawadaKoteraIto SeventhOrder Equation,” Numerical Methods for Partial Differential Equations, Vol. 27, No. 4, 2011, pp. 887897. doi:10.1002/num.20559 [26] M. Safari, D. D. Ganji and E. M. M. Sadeghi, “ Ap pl i ca ti on of He’s Homotopy Perturbation and He’s Variational Iteration Methods for Solution of BenneyLin Equation,” International Journal of Computer Mathe matics, Vol. 87, No. 8, 2010, pp. 18721884. doi:10.1080/00207160802524770 [27] I. Podlubny, “Fractional Differential Equations,” Aca demic Press, San Diego, 1999. [28] K. B. Oldham and J. Spanier, “The Fractional Calculus,” Academic P r ess, New Y ork and London, 1974.
