Advances in Pure Mathematics, 2011, 1, 345350 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 Email: 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 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: Adomian’s Decomposition Method, Fractional Derivative, Fractional Diffusion Equation 1. Introduction Fractional diffusion equations are used to model prob lems in Physics [13], Finance [47], 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 [116], and refer ences 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) d(x) > 0. We also assume an initial condition u(x, t = 0) = s(x) for R xx L ux and Dirichlet boundary conditions of the form and ,t0 , R t R b tux. Equation (1) uses a Riemann fractio nal derivative of or der . Consider a twodimensional fractional diffusion equa tion considered in [18] ,,,, ,,xy xx ,, ,, uxytuxytu t dxy exy t qxyt , (2) on a finite rectangular domain H xx and R yyy , 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 H xx and R, and Dirichlet boundary condition u(x, y, t) = B(x, y, t) on the boundary (perimeter) of the rectangular region yyy H xx , R yyy , with the additional restric tion that ,t,,yt ,Bxy 0 LL Bx . 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 twodimensions 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 [1925]. This method efficiently works for initialvalue or boundaryvalue 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 [2628]. 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 RiemannLiouville integral, in which the fractional order integral is defined as 1 0 d 1t q qq d d q t tftx Dft qtx t , (3) while the definition of fractional order derivative is 1 0 d d dd d 1d d nq n q tnnq nt nn ft Dfttt q tx nq ttx , (4) where ( and q) is the order of the opera tion and n is an integ er that satisfies . q0qR1nq 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 highestordered 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 1 L is assumed an integral operator given by 1 0 d t t L t , (7) The operating with the operator on both sides of Equation (6) we have 1 L 1 0 uf LgtRuFu , (8) where 0 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 boundaryvalue problem. The ADM assumes that the unknown function ,uxt can be expressed by an infinite series of the form 0 , n n uxtu xt , , (10) and the nonlinear operator u can be decomposed by an infinite series of polynomials given by 0n n u A, (11) where , n uxtwill be determined recurrently, and n are the socalled polynomials of defined by 01 ,,, n uu u 00 1d , 0,1,2, !d n i i n n AnF un n , (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 0n n uu , (13) to be the solutio n of Equation (1 ). Now, Equation (1) can be rewritten as ,, tx Luxtdx Dxtqxt ,, (14) where t Lt which is an easily invertible linear opera tor, x D is the RiemannLiouville derivative of order Copyright © 2011 SciRes. APM
M. SAFARI ET AL347 .Therefore, by Adomian decomposition method, we can write, 11 0 ,,0 , txnt n uxtuxLdxDuL qxt , (15) Each term of series (13) is given by Adomian decom position method recurrence relation 0,uf (16) 1 1,0 nt xn uLdxDun , , (17) where 1 ,0, . t uxL 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. u 1,un ,,uu Similarly, for Equation (2) using Adomian decompo sition method, we can obtain 1 0 11 0 ,, ,,0, ,,,, txn n tynt n uxytuxyL dxyDu LexyD u Lqxyt , (18) The Adomian decomposition method recurrence scheme is 0,uf (19) 11 1,, 0, ntxntyn uL dxyDuLexyDu n (20) where ,,0,, . t 1 uxyL qxyt 5. Numerical Illustration 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 , (21) on a finite domain 0 < x < 1, with the diffusion coeffi cient 2.8 2.8 2.26 0.183634dx xx , (22) the source/sink function 3 ,1e t qxtx x , (23) the initial condition 3 ,0, for 01ux xx , (24) and the boundary conditions 0,0, 1,e, for 0 t ut utt , (25) Implementation of Adomian’s Decomposition Method Equation (21) can be rewritten in operator form as 1.8 ,, tx LuxtdxDux tqxt ,, (26) where t Lt symbolizes the easily invertible linear differential operator, 1.8 x D is the RiemannLiouville derivative of order 1.8. I f the i nvertible oper a tor 1 0 d t t L t is applied to Equation (26), then 111.8 ,, tttx LLuxtLdxD uxtqxt , , (27) is obtained. By this 11.8 ,,0 ,, tx uxtuxLdxD uxtqxt , (28) is found. Here the main point is that the solution of the decomposition method is in the form of 0 , n n uxtu xt ,, (29) Substituting from Equation (29) into Equation (28), we find 0 11.8 0 ,,0 ,, n n txn n uxt ux LdxDuxtqxt , (30) is found. Thus according to Equation (7) approximate solution can be obtained as follows: 3 0,uxt x, (31) 344 34 1,1.000001369 e, t uxtxxxtxx (32) 1.8 21 0 ,, t x uxtdxDuxt qxtt ,d, (33) The approximate solution of the onedimensional frac tional diffusion equation is obtained as 012 ,,,uxtuxt uxt uxt , , (34) In Figure 1 we can see the 3D result of approximate solution of the onedimensional fractional diffusion equation by ADM. Copyright © 2011 SciRes. APM
M. SAFARI ET AL 348 Figure 1. For the onedimensional 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 twodimensional fractional diffusion equation for the Equation (2), considered in [18]. 1.8 1.8 1.6 1.6 ,, ,, , ,, ,,,, uxyt uxyt dxy tx uxyt exy qxyt y (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 e t qxytxyxy , (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, t uxtuytux tx 3.6 ,e t t y 0t1,uy Implementation of Adomian’s Decomposition Method Now, Equation (35) can be rewritten in operator form as 1.8 1.6 ,, ,,, ,,,,,,, tx y LuxytdxyDxy t exyt Du xytq xyt (40) where t Lt symbolizes the easily invertible linear differential operator, 1.8 x D and and are the Riemann–Liouville derivatives of order 1.8 and 1.6, re spectively. 1.6 y D If the invertible operator is applied to 1 0 d t t L t Equation (40) , then 111.8 1.6 ,,, ,, ,,,,,,, ttt x y LLuxytLdxyD uxyt exyt Du xytq xyt (41) is obtained. By this 11.8 1.6 (,,) ,,0,,, ,,,,,,, tx y uxytuxyLdxyD uxyt exyt Du xytqxyt (42) is found. Here the main point is that the solution of the decomposition method is in the form of 0 ,, ,, n n uxytu xyt , (43) Substituting from Equation (43) into Equation (42), we find 0 1 1.8 0 1.6 0 ,, ,,0 ,,, ,,,,,,, n n txn n yn n uxyt uxy LdxyD uxyt exytDuxytqxyt (44) is found. Thus according to Equation (7) approximate solution can be obtained as follows: 33.6 0,, ,uxyt xy (45) 18 23 23 344 55 1 18 23 34 55 ,,2 2 e2e, tt uxytxyxyxy t xy xy 5 (46) 1.8 21 0 1.6 1 ,,,,, ,,,,,,d, t x y ux ytdx yDux yt exyt Duxytqxytt (47) The approximate solution of the twodimensional frac tional diffusion equation is obtained as 012 ,,,uxtuxt uxtuxt , In Figure 2 we can see the 3D result of approximate , (48) Copyright © 2011 SciRes. APM
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