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 Advances in Pure Mathematics, 2011, 1, 345-350 doi:10.4236/apm.2011.16062 Published Online November 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Application of Adomian’s Decomposition Method for the Analytical Solution of Space Fractional Diffusion Equation Mehdi Safari1, Mohammad Danesh2 1Department of Mechanical Engineering, Islamic Azad University, Aligoodarz Branch, Aligoodarz, Iran 2Department of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran E-mail: ms_safari2005@yahoo.com Received July 1, 2011; revised September 26, 2011; accepted October 5, 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 Adomian’s decomposition method (ADM). 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: Adomian’s Decomposition 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 formu-late 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 < α < 2 replaces the second derivative in a diffu-sion or dispersion model, it leads to a super diffusive flow model. Nowadays, fractional diffusion equation plays important roles in modeling anomalous diffusion and subdiffusion systems, description of fractional ran-dom walk, unification of diffusion and wave propagation phenomenon, see, e.g. the reviews in [1-16], and refer-ences 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) d(x) > 0. We also assume an initial condition u(x, t = 0) = s(x) for LRxxxLux and Dirichlet boundary conditions of the form and ,t0,RtRb tux. Equation (1) uses a Riemann fractio nal derivative of or der . Consider a two-dimensional fractional diffusion equa-tion considered in [18] ,,,, ,,xyxx,, ,,uxytuxytu tdxy exytqxyt , (2) on a finite rectangular domain LHxxx and LRyyy, with fractional orders 1 < 2 and 1 < 2, where the diffusion coefficients d(x, y) > 0 and e(x, y) > 0. The ‘forcing’ function q(x, y, t) can be used to represen t sources and sinks. We will assume that this fractional diffusion equation has a unique and suffi-ciently smooth solution under the following initial and boundary conditio ns. Assume the initial condition u(x, y, t = 0) = f(x, y) for LHxxx and LR, and Dirichlet boundary condition u(x, y, t) = B(x, y, t) on the boundary (perimeter) of the rectangular region yyyLHxxx, LRyyy, with the additional restric-tion that ,t,,yt ,Bxy 0LLBx. In physical appli-cations, this means that the left/lower bou ndary is set far away enough from an evolving plume that no significant concentrations reach that boundary. The classical disper-sion equation in two-dimensions is given by 2. M. SAFARI ET AL 346 The values of 1 < 2 and 1 < 2 model a super diffusive process in that coordinate. Equation (2) also uses Riemann fractional derivatives of order  and . In this paper, we use the Adomian’s decomposi-tion method (ADM) to obtain the solutions of the frac-tional diffusion Equations (1) and (2). Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the ADM. The de-composition method provides an effective procedure for analytical solution of a wide and general class of dy-namical systems representing real physical problems [19-25]. This method efficiently works for initial-value or boundary-value problems and for linear or nonlinear, ordinary or partial differential equations and even for stochastic systems. Moreover, we have the advantage of a single global method for solving ordinary or partial differential equations as well as many types of other equations. Recently, the so lution of fractional differential equation has been obtained through ADM by the re-searchers [26-28]. The application of ADM for the solu-tion of nonlinear fractional differential equations has also been established by Shawagfeh, Saha Ray and Bera [27, 28]. However, we use the ADM to solve fractional diffu-sion Equations (1) and (2) and finally the results are il-lustrated in graphi cal fi g ures . 2. Mathematical Aspects The mathematical definition of fractional calculus has been the subject of several different approaches [29,30]. 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  10d1tqqqddqtftftxDft qtxt, (3) while the definition of fractional order derivative is 10ddddd1d dnqnqtnnqntnnftDftttqftxnq ttx  , (4) where ( and q) is the order of the opera-tion and n is an integ er that satisfies . q0qR1nq n 3. Basic Idea of Adomian’s Decomposition Method We begin with the equati o n LuR uFugt , (5) where L is the operator of the highest-ordered derivatives with respect to t and R is the remainder of the linear op-erator. The nonlinear term is represented by F(u). Thus we get Lug tR uFu , (6) The inverse 1L is assumed an integral operator given by 10dttLt, (7) The operating with the operator on both sides of Equation (6) we have 1L10uf LgtRuFu , (8) where 0f is the solution of homogeneous equation 0Lu , (9) involving the constants of integration. The integration constants involved in the solution of homogeneous Equation (9) are to be determined by the initial or boundary condition according as the problem is initial- value problem or boundary-value problem. The ADM assumes that the unknown function ,uxt can be expressed by an infinite series of the form  0,nnuxtu xt,, (10) and the nonlinear operator Fu can be decomposed by an infinite series of polynomials given by 0nnFuA, (11) where ,nuxtwill be determined recurrently, and nA are the so-called polynomials of defined by 01,,,nuu u001d , 0,1,2,!dniinnAnF unn, (12) 4. The Fractional Diffusion Equation Model and Its Solution by ADM We adopt Adomian decomposition method for solving Equation (1). In the light of this method we assume that 0nnuu, (13) to be the solutio n of Equation (1 ). Now, Equation (1) can be rewritten as ,,txLuxtdx Dxtqxt,, (14) where tLtwhich is an easily invertible linear opera- tor, xD is the Riemann-Liouville derivative of order Copyright © 2011 SciRes. APM M. SAFARI ET AL347 .Therefore, by Adomian decomposition method, we can write,  110,,0 ,txntnuxtuxLdxDuL qxt  , (15) Each term of series (13) is given by Adomian decom-position method recurrence relation 0,uf (16) 11,0nt xnuLdxDun,, (17) where  1,0, .tfuxL qxt It is worth noting that once the zeroth component 0 is defined, then the remaining components n can be completely determined; each term is computed by using the previous term. As a result, the components 01 are identified and the series solutions thus en-tirely determined. However, in many cases the exact so-lution in a closed form may be obtained. u1,un,,uuSimilarly, for Equation (2) using Adomian decompo-sition method, we can obtain   10110,, ,,0, ,,,,txnntyntnuxytuxyL dxyDuLexyD u Lqxyt, (18) The Adomian decomposition method recurrence scheme is 0,uf (19) 111,,0,ntxntynuL dxyDuLexyDun (20) where ,,0,, .t1fuxyL qxyt 5. Numerical Illustration 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, (21) on a finite domain 0 < x < 1, with the diffusion coeffi-cient  2.8 2.82.26 0.183634dx xx , (22) the source/sink function 3,1etqxtx x , (23) the initial condition 3,0, for 01ux xx, (24) and the boundary conditions 0,0, 1,e, for 0tut utt, (25) Implementation of Adomian’s Decomposition Method Equation (21) can be rewritten in operator form as 1.8,,txLuxtdxDux tqxt,, (26) where tLt symbolizes the easily invertible linear differential operator, 1.8xD is the Riemann-Liouville derivative of order 1.8. I f the i nvertible oper a tor 10dttLt is applied to Equation (26), then 111.8,,tttxLLuxtLdxD uxtqxt,, (27) is obtained. By this 11.8,,0 ,,txuxtuxLdxD uxtqxt , (28) is found. Here the main point is that the solution of the decomposition method is in the form of  0,nnuxtu xt,, (29) Substituting from Equation (29) into Equation (28), we find    011.80,,0 ,,nntxnnuxt uxLdxDuxtqxt, (30) is found. Thus according to Equation (7) approximate solution can be obtained as follows: 30,uxt x, (31) 344 341,1.000001369 e,tuxtxxxtxx (32)  1.8210,,txuxtdxDuxt qxtt,d, (33) The approximate solution of the one-dimensional frac-tional diffusion equation is obtained as 012,,,uxtuxt uxt uxt,, (34) In Figure 1 we can see the 3-D result of approximate solution of the one-dimensional fractional diffusion equation by ADM. Copyright © 2011 SciRes. APM M. SAFARI ET AL 348 Figure 1. For the one-dimensional fractional diffusion equation with the initial condition (24) of Equation (21), ADM result for u(x, t). 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,, ,,,,, ,,,,uxyt uxytdxytxuxytexy qxyty (35) on a finite rectangular domain 0 < x < 1, 0 < y < 1, for 0 < t < Tend with the diffusion coefficients  2.8,2.2dxyx y 6, (36) and  2.6,2 4.6exyxy, (37) and the forcing function  33.6,,12 etqxytxyxy , (38) with the initial condition 33.6,,0uxy xy, (39) and Dirichlet bound ary conditio ns on the re ctangle in th e form and , for all .  3,0,0,,0, ,1,e,tuxtuytux tx 3.6,e tt y0t1,uy Implementation of Adomian’s Decomposition Method Now, Equation (35) can be rewritten in operator form as 1.81.6,, ,,, ,,,,,,,txyLuxytdxyDxy texyt Du xytq xyt (40) where tLt symbolizes the easily invertible linear differential operator, 1.8xD and and are the Riemann–Liouville derivatives of order 1.8 and 1.6, re-spectively. 1.6yDIf the invertible operator is applied to 10dttLtEquation (40) , then  111.81.6,,, ,, ,,,,,,,ttt xyLLuxytLdxyD uxytexyt Du xytq xyt (41) is obtained. By this  11.81.6(,,) ,,0,,, ,,,,,,,txyuxytuxyLdxyD uxytexyt Du xytqxyt (42) is found. Here the main point is that the solution of the decomposition method is in the form of  0,, ,,nnuxytu xyt, (43) Substituting from Equation (43) into Equation (42), we find   01 1.801.60,, ,,0 ,,, ,,,,,,,nntxnnynnuxyt uxyLdxyD uxytexytDuxytqxyt (44) is found. Thus according to Equation (7) approximate solution can be obtained as follows: 33.60,, ,uxyt xy (45) 18 23 2334455118 233455,,2 2 e2e,ttuxytxyxyxy txy xy 5 (46)   1.82101.6 1,,,,, ,,,,,,d,txyux ytdx yDux ytexyt Duxytqxytt(47) The approximate solution of the two-dimensional frac-tional diffusion equation is obtained as 012,,,uxtuxt uxtuxt,In Figure 2 we can see the 3-D result of approximate , (48) Copyright © 2011 SciRes. APM M. SAFARI ET AL349 Figure 2. For the two-dimensional fractional diffusion lution of the two-dimensional fractional diff. 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