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 Applied Mathematics, 2011, 2, 1091-1095 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 E-mail: 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 one-dimensional and the second one is two-dimensional 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 [1-3], Finance [4-7], and Hydrology [8-12]. 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 [1-16], and references therein. Consider a one-dimensional fractional diffusion equation considered in [17] (,) (,)() (,)uxt uxtdx qxttx (1) on a finite domain LRxxx with 12. We assume that the diffusion coefficient (or diffusivity) . We also assume an initial condition for 0dx(, 0uxt) ()sxLRxxx(,Lux tand Dirichlet bound-ary conditions of the form and R. Equation (1) uses a Riemann fractional derivative of order)0(,)Ruxt ()b t. Consider a two-dimensional fractional diffusion equa-tion considered in [18] (,,) (,,)(,)(,,) (,)(,,)uxytuxytdxytxux ytexy qxytx (2) on a finite rectangular domain LHxxx and LRyyy, 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 LHxxx and LRyyy, and Dirichlet boundary condition , , yt, , uxytBx on the boundary (perimeter) of the rectangular region LHxxx ,LRy(, ,)LLBxyt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 two-dimensions is given by 02. 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 [19-22] 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 [23-26], 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 Riemann-Liouville integral, in which the fractional order integral is defined as 10d() 1()d() ()d(qtqtqq)ftfDft qtttxx (3) while the definition of fractional order derivative is ()() 10d d()1d()d() ()dd d()nnq ntqtnnqn nqftDft nqtt ttx   ftxn (6) where ( and ) is the order of the opera-tion and n is an integer that satisfies . q0qqR1nq  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Ngt an inhomogeneous term. According to VIM, we can write down a correction functional as follows:  10dtnnn nututLu 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 nnu0nu. 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() (,)dtxnn nnututudxu qxt  (7) where ()x 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(, )(, )(, ,)dnntxynn nututu dxyuexyuqxyt  (8) 5. Numerical Illustrations 5.1. Example 1 Let us consider a one-dimensional fractional diffusion equation for the Equation (1), as taken in [17] 1.81.8(,) (,)() (,)uxt uxtdx qxttx (9) on a finite domain 01x, with the diffusion coeffi-cient 2.8 2.8( )(2.2)60.183634dx xx  (10) the source/sink function 3(,)(1) tqxtxex  (11) the initial condition 3(,0) ,ux x (12) for 01xand the boundary conditions (0,)0,(1, ),0tutut 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(,) (,)()(,)dtxnn nnuxtu xtudxuqxt (15) where (1.8 )xindicates 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) 1t0 (17) Copyright © 2011 SciRes. AM M. SAFARI1093 Equation (16) is called Lagrange-Euler 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(,) (,)()(,)dtxnn nnuxtuxtudxuqxt(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)ux30(,) ,uxtx (19) 441(, )1.000001369(),tuxtxxt exx 34 (20) (1.8 )21 110(,) (,)()(,)dtxuxtuxtudxuqxt (21) In Figure 1 we can see the 3-D result of approximate solution of the one-dimensional fractional diffusion equation by VIM. 5.2. Example 2 Let us consider a two-dimensional fractional diffusion equation for the Equation (2), considered in [18]. 1.81.81.61.6(,,) (,,)(, )(,,) +(,)(,,),uxytuxytdxytxux ytexy qxyty (22) on a finite rectangular domain 01x, , for with the diffusion coefficients 01yend0Tt Figure 1. For the one-dimensional fractional diffusion equa- tion with the initial condition (12) of Equation (9), VIM result for . (,)uxt2.8(,)(2.2)/ 6dxyx y (23) and 2.6( ,)2/(4.6)exyx y (24) and the forcing function 33.6(,,)(1 2) tqxytxyexy  (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, ),tuxtuytuxte x 3.6tey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(,,) (,,)(, )(, )(, ,)dnntxynn nuxytuxytu dxyuexyuqxyt (28) where (1.8 )x 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) 1t0 (30) Equation (29) is called Lagrange-Euler 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(,,) (,,)(, )(, )(,,)dnntxynn nuxytuxytu 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)ux33.60(,,) ,uxyt xy (32) 23 234455118 233455(,,)22 2,ttu xytxyxytexy exy  (33) Copyright © 2011 SciRes. AM M. SAFARI 1094 Figure 2. For the two-dimensional fractional diffusion equa-tion with the initial condition (26) of Equation (22), VIM result for with . ,()xtu1y 21(1.8 )(1.6)11 10(,,) (,,)(, )(, )(, ,)dtxyuxyt uxytu dxyuexyuqxyt (34) In Figure 2 we can see the 3-D result of approximate solution of the one-dimensional 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. 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