- Email: [email protected]
Application of modified decomposition method for the analytical solution of space fractional diffusion equation
Available online at www.sciencedirect.com
Applied Mathematics and Computation 196 (2008) 294–302 www.elsevier.com/locate/amc
Application of modified decomposition method for the analytical solution of space fractional diffusion equation S. Saha Ray a
a,*
, K.S. Chaudhuri b, R.K. Bera
a
Department of Mathematics, Heritage Institute of Technology, Anandapur, P.O.-East Kolkata Township, Kolkata 700107, India b Department of Mathematics, Jadavpur University, Kolkata 700032, India
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 Modified decomposition method (MDM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. The decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self-cancelling ‘‘noise’’ phenomenon. 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 technique. The present method performs extremely well in terms of efficiency and simplicity. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Fractional derivative; Fractional diffusion equation; Adomian decomposition method; Modified decomposition method; The self-cancelling noise terms
1. Introduction Fractional diffusion equations are used to model problems in Physics [1–3], Finance [4–7], and Hydrology [8–12]. Fractional space derivatives may be used to formulate 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 < a < 2 replaces the second derivative in a diffusion 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 random walk, unification of diffusion and wave propagation phenomenon, see, e.g. the reviews in [1–16], and references therein. *
Corresponding author. E-mail addresses: [email protected] (S.S. Ray), [email protected] (K.S. Chaudhuri), [email protected] (R.K. Bera). 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.05.048
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
295
Consider a one-dimensional fractional diffusion equation considered in [17] ouðx; tÞ oa uðx; tÞ þ qðx; tÞ; ¼ dðxÞ ot oxa
ð1:1Þ
on a finite domain xL < x < xR with 1 < a 6 2. We assume that the diffusion coefficient (or diffusivity) d(x) > 0. We also assume an initial condition u(x,t = 0) = s(x) for xL < x < xR and Dirichlet boundary conditions of the form u(xL, t) = 0 and u(xR, t) = bR(t). Eq. (1.1) uses a Riemann fractional derivative of order a. Consider a two-dimensional fractional diffusion equation considered in [18] ouðx; y; tÞ oa uðx; y; tÞ ob uðx; y; tÞ ¼ dðx; yÞ þ eðx; yÞ þ qðx; y; tÞ; ot oxa oy b
ð1:2Þ
on a finite rectangular domain xL < x < xH and yL < y < yH, with fractional orders 1 < a 6 2 and 1 < b 6 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 represent sources and sinks. We will assume that this fractional diffusion equation has a unique and sufficiently smooth solution under the following initial and boundary conditions. Assume the initial condition u(x, y, t = 0) = f(x, y) for xL < x < xH, yL < y < yH, and Dirichlet boundary condition u(x, y, t) = B(x, y, t) on the boundary (perimeter) of the rectangular region xL 6 x 6 xH, yL 6 y 6 yH, with the additional restriction that B(xL, y, t) = B(x, yL, t) = 0. 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 a = b = 2. The values of 1 < a < 2, or 1 < b < 2 model a super diffusive process in that coordinate. Eq. (1.2) also uses Riemann fractional derivatives of order a and b. In this paper, we use the Adomian decomposition method (ADM) [19,20] to obtain the solutions of the fractional diffusion Eqs. (1.1) and (1.2). Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the ADM [19–26]. A reliable modification of ADM has been done by Wazwaz [27]. The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems [19–25]. Recently, the implementations of ADM for the solutions of generalized regularized long-wave (RLW) and Korteweg-de Vries (KdV) equations have been well established by the notable researchers [26–31]. 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 solution of fractional differential equation has been obtained through ADM by the researchers [32–39]. The application of ADM for the solution of nonlinear fractional differential equations has also been established by Shawagfeh, Saha Ray and Bera [35,38]. 2. Mathematical aspects 2.1. Mathematical definition The mathematical definition of fractional calculus has been the subject of several different approaches [40,41]. The most frequently encountered definition of an integral of fractional order is the Riemann–Liouville integral, in which the fractional order integral is defined as Z t dq f ðtÞ 1 f ðxÞdx q Dt f ðtÞ ¼ ¼ ; ð2:1:1Þ dtq CðqÞ 0 ðt xÞ1q while the definition of fractional order derivative is ! Z t n ðnqÞ d d f ðtÞ 1 dn f ðxÞdx q Dt f ðtÞ ¼ n ; ¼ dtðnqÞ Cðn qÞ dtn 0 ðt xÞ1nþq dt where q(q > 0 and q 2 R) is the order of the operation and n is an integer that satisfies n 1 6 q < n.
ð2:1:2Þ
296
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
3. The fractional diffusion equation model and its solution We adopt Adomian decomposition method for solving Eq. (1.1). In the light of this method we assume that 1 X un ; ð3:1Þ u¼ n¼0
to be the solution of Eq. (1.1). Now, Eq. (1.1) can be rewritten as Lt uðx; tÞ ¼ dðxÞDax uðx; tÞ þ qðx; tÞ;
ð3:2Þ Dax ðÞ
o ot
where Lt which is an easily invertible linear operator, is the Riemann–Liouville derivative of order a. Therefore, by Adomian decomposition method, we can write, !! 1 X 1 a uðx; tÞ ¼ uðx; 0Þ þ Lt dðxÞDx un ð3:3Þ þ L1 t ðqðx; tÞÞ: n¼0
Each term of series (3.1) is given by the standard Adomian decomposition method recurrence relation u0 ¼ f ; a unþ1 ¼ L1 t ðdðxÞDx un Þ; n P 0;
ð3:4Þ
where f ¼ uðx; 0Þ þ L1 t ðqðx; tÞÞ: It is worth noting that once the zeroth component u0 is defined, then the remaining components un, n P 1 can be completely determined; each term is computed by using the previous term. As a result, the components u0, u1, . . . are identified and the series solutions thus entirely determined. However, in many cases the exact solution in a closed form may be obtained. Recently, Wazwaz [27] proposed that the construction of the zeroth component u0 of the decomposition series can be defined in a slightly different way. In [27], he assumed that if the zeroth component u0 = f and the function f is possible to divide into two parts such as f1 and f2, then one can formulate the recursive algorithm for u0 and general term un+1 in a form of the modified decomposition method (MDM) recursive scheme as follows: u0 ¼ f1 ; a u1 ¼ f2 þ L1 t ðdðxÞDx un Þ;
unþ1 ¼
a L1 t ðdðxÞDx un Þ;
ð3:5Þ
n P 1:
Comparing the recursive Scheme (3.4) of the standard Adomain method with the recursive Scheme (3.5) of the modified technique leads to the conclusion that in Eq. (3.4) the zeroth component was defined by the function f, whereas in Eq. (3.5), the zeroth component u0 is defined only by a part f1 of f. The remaining part f2 of f is added to the definition of the component u1 in Eq. (3.5). Although, the modified technique needs only a slight variation from the standard Adomian decomposition method, the results are promising in that it minimizes the size of calculations needed and will accelerate the convergence. The modification could lead to a promising approach for many applications in applied science. The decomposition series solution (3.1) generally converges very rapidly in real physical problems [19,20]. The rapidity of this convergence means that few terms are required. Convergence of this method has been rigorously established by Cherruault [42], Abbaoui and Cherruault [43,44] and Himoun, Abbaoui and Cherruault [45]. The practical solution will be the n-term approximation /n /n ¼
n1 X
ui ðx; tÞ;
i¼0
with lim /n ¼ uðx; tÞ:
n!1
n P 1;
ð3:6Þ
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
Similarly, for Eq. (1.2) using Adomian decomposition method, we can obtain !! !! 1 1 X X 1 a 1 b uðx; y; tÞ ¼ uðx; y; 0Þ þ Lt dðx; yÞDx un un þ Lt eðx; yÞDy þ L1 t ðqðx; y; tÞÞ; n¼0
297
ð3:7Þ
n¼0
The standard Adomian decomposition method recurrence scheme is u0 ¼ f ; a 1 b unþ1 ¼ L1 t ðdðx; yÞDx un Þ þ Lt ðeðx; yÞDy un Þ;
n P 0;
ð3:8Þ
where f ¼ uðx; y; 0Þ þ L1 t ðqðx; y; tÞÞ. The modified decomposition method (MDM) recursive scheme is as follows: u0 ¼ f 1 ; a 1 b u1 ¼ f2 þ L1 t ðdðx; yÞDx u0 Þ þ Lt ðeðx; yÞDy u0 Þ;
unþ1 ¼
a L1 t ðdðx; yÞDx un Þ
þ
b L1 t ðeðx; yÞDy un Þ;
ð3:9Þ
n P 1:
Adomian and Rach [46] and Wazwaz [47–49] have investigated the phenomena of the self-cancelling ‘‘noise’’ terms where sum of noise terms vanishes in the limit. The noise terms are defined as the identical terms with opposite signs that arise in the components u0 and u1. Noise terms may appear if the exact solution is part of the zeroth component u0. Verification that the remaining non cancelled terms justify the equation is necessary and essential. An important observation was made that ‘‘noise’’ terms appear for inhomogeneous cases only. Further, it was formally justified that if terms in u0 are cancelled by terms in u1, even though u1 includes further terms, then the remaining non cancelled terms in u1 are cancelled by terms in u2, and so on. Finally, the exact solution of the equation is readily found for the inhomogeneous case by determining the first two or three terms of the solution u(x, t) and by keeping only the non cancelled terms of u0. 4. Implementation of modified decomposition method Example 1. Let us consider a one-dimensional fractional diffusion equation for the Eq. (1.1), as taken in [17] ouðx; tÞ o1:8 uðx; tÞ ¼ dðxÞ þ qðx; tÞ; ot ox1:8
ð4:1Þ
on a finite domain 0 < x < 1, with the diffusion coefficient dðxÞ ¼ Cð2:2Þx2:8 =6 ¼ 0:183634x2:8 ; the source/sink function qðx; tÞ ¼ ð1 þ xÞet x3 ; the initial condition uðx; 0Þ ¼ x3 ;
for 0 < x < 1;
and the boundary conditions uð0; tÞ ¼ 0;
uð1; tÞ ¼ et ;
for t > 0:
Now, Eq. (4.1) can be rewritten in operator form as Lt uðx; tÞ ¼ dðxÞD1:8 x uðx; tÞ þ qðx; tÞ;
ð4:2Þ
where Lt oto symbolizes the easily invertible linear differential operator, D1:8 x ðÞ is the Riemann–Liouville derivative of order 1.8. Rt Applying the one-fold integration inverse operator L1 t 0 ðÞdt to the Eq. (4.2) and using the specified initial condition yields
298
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
uðx; tÞ ¼ uðx; 0Þ þ
L1 t
dðxÞD1:8 x
1 X
!! un
þ L1 t ðqðx; tÞÞ
n¼0 t 3
t 4
4
¼e x þe x x þ
L1 t
dðxÞD1:8 x
1 X
!! un
:
ð4:3Þ
n¼0
The Standard Adomian Decomposition Method: u0 ¼ et x3 þ et x4 x4 ; Cð2:2Þx2:8 o1:8 u0 4ðet þ 1 tÞx5 t 4 ; u1 ¼ L1 þ 1Þx þ ¼ ðe t 6 ox1:8 2:2 t t2 2:8 1:8 t 5 80 e þ t 1 x6 2! o u1 4ðe þ t 1Þx 1 Cð2:2Þx þ u2 ¼ Lt ; ¼ 6 ox1:8 2:2 3:2 2:22 80 et þ t2 t þ 1 x6 80Cð6Þ et t3 þ t2 t þ 1 x7 2:8 1:8 2! 3! 2! Cð2:2Þx o u2 þ ; u3 ¼ L1 ¼ t 2 2 6 ox1:8 3:2 2:2 4:2 3:2 2:23 and so on. It is obvious that the ‘‘noise’’ terms appear between the components of u1, and these are all cancelled. The closed form of the solution can be found very easily by proper selection of f1 and f2 in MDM. In the case of right choice of these functions, the modified technique accelerates the convergence of the decomposition series solution by computing just u0 and u1 terms of the series. The term u0 provides the exact solution as u(x, t) = et x3 and this can be justified through substitution. The Modified Decomposition Method: We will then obtain from recursive scheme for modified decomposition method u0 ¼ et x3 ; t 4
4
u1 ¼ e x x þ
L1 t
Cð2:2Þx2:8 o1:8 u0 ; 6 ox1:8
¼ et x4 x4 ðet 1Þx4 ; ¼ 0; 2:8 1:8 o u1 1 Cð2:2Þx u2 ¼ Lt ; 6 ox1:8 ¼ 0; and so on. Therefore, the solution is uðx; tÞ ¼ et x3 :
ð4:4Þ
The solution (4.4) can be verified through substitution in Eq. (4.1). Example 2. Let us consider a two-dimensional fractional diffusion equation for the Eq. (1.2), considered in [18] ouðx; y; tÞ o1:8 uðx; y; tÞ o1:6 uðx; y; tÞ ¼ dðx; yÞ þ eðx; yÞ þ qðx; y; tÞ; ot ox1:8 oy 1:6 on a finite rectangular domain 0 < x < 1, 0 < y < 1, for 0 6 t 6 Tend with the diffusion coefficients dðx; yÞ ¼ Cð2:2Þx2:8 y=6;
ð4:5Þ
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
299
and eðx; yÞ ¼ 2xy 2:6 =Cð4:6Þ; and the forcing function qðx; y; tÞ ¼ ð1 þ 2xyÞet x3 y 3:6 ; with the initial condition uðx; y; 0Þ ¼ x3 y 3:6 ; and Dirichlet boundary conditions on the rectangle in the form u(x, 0, t) = u(0, y, t) = 0, u(x, 1, t) = et x3, and u(1, y, t) = et y3.6, for all t P 0. Now, Eq. (4.8) can be rewritten in operator form as 1:6 Lt uðx; y; tÞ ¼ dðx; yÞD1:8 x uðx; y; tÞ þ eðx; yÞDy uðx; y; tÞ þ qðx; y; tÞ;
ð4:6Þ
1:6 where Lt oto symbolizes the easily invertible linear differential operator, D1:8 x ðÞ and Dy ðÞ are the Riemann– Liouville derivatives of order 1.8 and 1.6, respectively. Rt Applying the one-fold integration inverse operator L1 t 0 ðÞdt to the Eq. (4.6) and using the specified initial condition yields !! !! 1 1 X X 1 1:8 1 1:6 uðx; y; tÞ ¼ uðx; y; 0Þ þ Lt dðx; yÞDx un un þ Lt eðx; yÞDy þ L1 t ðqðx; y; tÞÞ n¼0
dðx; yÞD1:8 ¼ x3 y 3:6 et þ 2x4 y 4:6 et 2x4 y 4:6 þ L1 t x
n¼0
1 X
!! un
n¼0
eðx; yÞD1:6 þ L1 t y
1 X
!! un
:
n¼0
ð4:7Þ The Standard Adomian Decomposition Method: u0 ¼ x3 y 3:6 et þ 2x4 y 4:6 et 2x4 y 4:6 ; 1:8 2:8 2xy 2:6 o1:6 u0 1 Cð2:2Þx y o u0 1 u1 ¼ L t þ Lt ; 6 ox1:8 Cð4:6Þ oy 1:6 8 2 4:6 5 5:6 þ x y ðet þ 1 tÞ; ¼ 2x4 y 4:6 ðet þ 1Þ þ 2:2 3 1106 5 5:6 x y ðet þ 1 tÞ; ¼ 2x4 y 4:6 ðet þ 1Þ þ 165 1:8 2:8 2xy 2:6 o1:6 u1 1 Cð2:2Þx y o u1 1 u2 ¼ L t þ Lt ; 6 ox1:8 Cð4:6Þ oy 1:6 1106 5 5:6 t 9101827 6 6:6 t t2 x y ðe 1 þ tÞ þ xy e 1þt ¼ ; 165 272250 2 and so on. In this case, the ‘‘noise’’ terms appear between the components of u1, and these are all cancelled. In the above, one obtains self-cancelling ‘‘noise’’ terms appearing between various components of u0, u1, u2, u3, . . ., and keeping the non cancelled terms, and using (3.1) lead to the exact solution as u(x, y, t) = x3y3.6et. It is worth noting that noise terms between components of the decomposition series will be cancelled, and the sum of these ‘‘noise’’ terms will vanish in the limit [46–49]. In the present analysis, the closed form of the solution can be found very easily by proper selection of f1 and f2 in MDM. In the case of right choice of these functions, the modified technique accelerates the convergence of the decomposition series solution by computing just u0 and u1 terms of the series. The term u0 provides the exact solution as u(x, y, t) = x3y3.6et and this can be justified through substitution.
300
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
The Modified Decomposition Method: The recursive scheme for modified decomposition method gives u0 ¼ x3 y 3:6 et ; u1 ¼ 2x4 y 4:6 et 2x4 y 4:6 þ L1 t
Cð2:2Þx2:8 y o1:8 u0 6 ox1:8
þ L1 t
2xy 2:6 o1:6 u0 ; Cð4:6Þ oy 1:6
¼ 2x4 y 4:6 et 2x4 y 4:6 2ðet 1Þx4 y 4:6 ; ¼ 0; Cð2:2Þx2:8 y o1:8 u1 2xy 2:6 o1:6 u1 1 u2 ¼ L1 þ L ; t t 6 ox1:8 Cð4:6Þ oy 1:6 ¼ 0; and so on. Therefore, the solution is uðx; y; tÞ ¼ x3 y 3:6 et
ð4:8Þ
The solution (4.8) can be verified through substitution in Eq. (4.5). 5. Conclusion This paper presents an analytical scheme to obtain the solutions of the one-dimensional and two-dimensional fractional diffusion equations. Two typical examples have been discussed as demonstrations. In our previous papers [36–39] we have already established as well as successfully exhibited the applicability of ADM to obtain the solutions of different types of fractional differential equations. In this work we demonstrate that this method is also well suited to solve fractional diffusion-wave equation. The decomposition method is straightforward, without restrictive assumptions and the components of the series solution can be easily computed using any mathematical symbolic package. Moreover, this method does not change the problem into a convenient one for the use of linear theory. It, therefore, provides more realistic series solutions that generally converge very rapidly in real physical problems. Moreover, no linearization or perturbation is required. It can avoid the difficulty of finding the inverse of Laplace Transform and can reduce the labour of perturbation method. Furthermore, as the decomposition method does not require discretization of the variables, i.e. time and space, it is not affected by computational round off errors and one is not faced with necessity of large computer memory and time. Consequently, the computational size will be reduced. It is also worth noting that the advantage of the decomposition methodology is that it displays a fast convergence of the solution. It may be achieved by observing the phenomenon of self-cancelling ‘‘noise’’ terms and the splitting of the initial term into appropriate parts in MDM. Therefore, the MDM will further accelerate the convergence of the series solution [27,48]. Acknowledgement We express our sincere thanks and gratitude to the learned reviewer for his kind comments and suggestions for the improvement of the paper. References [1] R. Metzler, E. Barkai, J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker–Planck equation approach, Phys. Rev. Lett. 82 (18) (1999) 3563–3567. [2] R. Metzler, J. Klafter, The random Walk’s guide to anomalous diffusion: a fractional dynamics approach 339 (2000) 1–77. [3] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004) R161–R208.
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
301
[4] R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional calculus and continuous time finance. III. The diffusion limit. Mathematical finance (Konstanz, 2000), Trends Math. Birkhuser Basel (2001) 171–180. [5] F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A 287 (2000) 468–481. [6] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Physica A 284 (2000) 376–384. [7] M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314 (2002) 749–755. [8] D.A. Benson, S. Wheatcraft, M.M. Meerschaert, Application of a fractional advection–dispersion equation, Water Resour. Res. 36 (2000) 1403–1412. [9] B. Baeumer, M.M. Meerschaert, D.A. Benson, S.W. Wheatcraft, Subordinated advection–dispersion equation for contaminant transport, Water Resour. Res. 37 (2001) 1543–1550. [10] D.A. Benson, R. Schumer, M.M. Meerschaert, S.W. Wheatcraft, Fractional dispersion, Le´vy motions, and the MADE tracer tests, Transp. Porous Media 42 (2001) 211–240. [11] R. Schumer, D.A. Benson, M.M. Meerschaert, S.W. Wheatcraft, Eulerian derivation of the fractional advection–dispersion equation, J. Contam. Hydrol. 48 (2001) 9–88. [12] R. Schumer, D.A. Benson, M.M. Meerschaert, B. Baeumer, Multiscaling fractional advection–dispersion equations and their solutions, Water Resour. Res. 39 (2003) 1022–1032. [13] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien New York, 1997, pp. 291–348. [14] F. Mainardi, G. Pagnini, The weight functions as solutions of the time-fractional diffusion equation, Appl. Math. Comput. 141 (2003) 51–62. [15] O.P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynam. 29 (2002) 145–155. [16] W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989) 134–144. [17] M.M. Meerschaert, H. Scheffler, C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys. 211 (2006) 249–261. [18] C. Tadjeran, M.M. Meerschaert, H. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006) 205–213. [19] G. Adomian, Nonlinear Stochastic Systems theory and Applications to Physics, Kluwer Academic Publishers, The Netherlands, 1989. [20] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994. [21] G. Adomian, An analytical solution of the stochastic Navier-Stokes system, Foundations of Physics 21 (7) (1991) 831–843. [22] G. Adomian, R. Rach, Linear and nonlinear Schro¨dinger equations, Foundations of Physics 21 (1991) 983–991. [23] G. Adomian, Solution of Physical Problems by Decomposition, Comput. Math. Appl. 27 (9/10) (1994) 145–154. [24] G. Adomian, Solutions of nonlinear P.D.E, Appl. Math. Lett. 11 (3) (1998) 121–123. [25] K. Abbaoui, Y. Cherruault, The decomposition method applied to the cauchy problem, Kybernetes 28 (1999) 103–108. [26] D. Kaya, A. Yokus, A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations, Math. Comput. Simul. 60 (6) (2002) 507–512. [27] A. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1) (1999) 77–86. [28] D. Kaya, S.M. El-Sayed, On a generalized fifth order KdV equations, Phys. Lett. A 310 (1) (2003) 44–51. [29] D. Kaya, S.M. El-Sayed, An application of the decomposition method for the generalized KdV and RLW equations, Chaos Solitons Fractals 17 (5) (2003) 869–877. [30] D. Kaya, An explicit and numerical solutions of some fifth-order KdV equation by decomposition method, Appl. Math. Comput. 144 (2–3) (2003) 353–363. [31] D. Kaya, A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl. Math. Comput. 149 (3) (2004) 833–841. [32] A.J. George, A. Chakrabarti, The Adomian method applied to some extraordinary differential equations, Appl. Math. Lett. 8 (3) (1995) 91–97. [33] H.L. Arora, F.I. Abdelwahid, Solutions of non-integer order differential equations via the Adomian decomposition method, Appl. Math. Lett. 6 (1) (1993) 21–23. [34] N.T. Shawagfeh, The decompostion method for fractional differential equations, J. Frac. Calc. 16 (1999) 27–33. [35] N.T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput. 131 (2002) 517–529. [36] S. Saha Ray, R.K. Bera, Solution of an extraordinary differential equation by Adomian decomposition method, J. Appl. Math. 4 (2004) 331–338. [37] S. Saha Ray, R.K. Bera, Analytical solution of a dynamic system containing fractional derivative of order 1/2 by Adomian decomposition method, Trans. ASME J. Appl. Mech. 72 (2) (2005) 290–295. [38] S. Saha Ray, R.K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. Math. Comput. 167 (1) (2005) 561–571. [39] S. Saha Ray, R.K. Bera, Analytical solution of the Bagley Torvik equation by Adomian decomposition method, Appl. Math. Comput. 168 (1) (2005) 398–410. [40] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999. [41] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York and London, 1974. [42] Y. Cherruault, Convergence of Adomian’s method, Kybernetes 18 (1989) 31–38.
302
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
[43] K. Abbaoui, Y. Cherruault, Convergence of Adomian’s method applied to differential equations, Comput.Math. Appl. 28 (5) (1994) 103–109. [44] K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition methods, Comput. Math. Appl. 29 (1995) 103–108. [45] N. Himoun, K. Abbaoui, Y. Cherruault, New results of convergence of Adomian’s method, Kybernetes 28 (4–5) (1999) 423–429. [46] G. Adomian, R. Rach, Noise terms in decomposition solution series, Comput. Math. Appl. 24 (11) (1992) 61–64. [47] A.M. Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput. 81 (2– 3) (1997) 265–274. [48] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, A.A. Balkema Publishers, Lisse, The Netherlands, 2002. [49] A.M. Wazwaz, The existence of noise terms for systems of inhomogeneous differential and integral equations 146 (1) (2003) 81– 92.
Applied Mathematics and Computation 196 (2008) 294–302 www.elsevier.com/locate/amc
Application of modified decomposition method for the analytical solution of space fractional diffusion equation S. Saha Ray a
a,*
, K.S. Chaudhuri b, R.K. Bera
a
Department of Mathematics, Heritage Institute of Technology, Anandapur, P.O.-East Kolkata Township, Kolkata 700107, India b Department of Mathematics, Jadavpur University, Kolkata 700032, India
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 Modified decomposition method (MDM). By using initial conditions, the explicit solutions of the equations have been presented in the closed form. The decomposition series analytic solution of the problem is quickly obtained by observing the existence of the self-cancelling ‘‘noise’’ phenomenon. 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 technique. The present method performs extremely well in terms of efficiency and simplicity. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Fractional derivative; Fractional diffusion equation; Adomian decomposition method; Modified decomposition method; The self-cancelling noise terms
1. Introduction Fractional diffusion equations are used to model problems in Physics [1–3], Finance [4–7], and Hydrology [8–12]. Fractional space derivatives may be used to formulate 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 < a < 2 replaces the second derivative in a diffusion 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 random walk, unification of diffusion and wave propagation phenomenon, see, e.g. the reviews in [1–16], and references therein. *
Corresponding author. E-mail addresses: [email protected] (S.S. Ray), [email protected] (K.S. Chaudhuri), [email protected] (R.K. Bera). 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.05.048
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
295
Consider a one-dimensional fractional diffusion equation considered in [17] ouðx; tÞ oa uðx; tÞ þ qðx; tÞ; ¼ dðxÞ ot oxa
ð1:1Þ
on a finite domain xL < x < xR with 1 < a 6 2. We assume that the diffusion coefficient (or diffusivity) d(x) > 0. We also assume an initial condition u(x,t = 0) = s(x) for xL < x < xR and Dirichlet boundary conditions of the form u(xL, t) = 0 and u(xR, t) = bR(t). Eq. (1.1) uses a Riemann fractional derivative of order a. Consider a two-dimensional fractional diffusion equation considered in [18] ouðx; y; tÞ oa uðx; y; tÞ ob uðx; y; tÞ ¼ dðx; yÞ þ eðx; yÞ þ qðx; y; tÞ; ot oxa oy b
ð1:2Þ
on a finite rectangular domain xL < x < xH and yL < y < yH, with fractional orders 1 < a 6 2 and 1 < b 6 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 represent sources and sinks. We will assume that this fractional diffusion equation has a unique and sufficiently smooth solution under the following initial and boundary conditions. Assume the initial condition u(x, y, t = 0) = f(x, y) for xL < x < xH, yL < y < yH, and Dirichlet boundary condition u(x, y, t) = B(x, y, t) on the boundary (perimeter) of the rectangular region xL 6 x 6 xH, yL 6 y 6 yH, with the additional restriction that B(xL, y, t) = B(x, yL, t) = 0. 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 a = b = 2. The values of 1 < a < 2, or 1 < b < 2 model a super diffusive process in that coordinate. Eq. (1.2) also uses Riemann fractional derivatives of order a and b. In this paper, we use the Adomian decomposition method (ADM) [19,20] to obtain the solutions of the fractional diffusion Eqs. (1.1) and (1.2). Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the ADM [19–26]. A reliable modification of ADM has been done by Wazwaz [27]. The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems [19–25]. Recently, the implementations of ADM for the solutions of generalized regularized long-wave (RLW) and Korteweg-de Vries (KdV) equations have been well established by the notable researchers [26–31]. 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 solution of fractional differential equation has been obtained through ADM by the researchers [32–39]. The application of ADM for the solution of nonlinear fractional differential equations has also been established by Shawagfeh, Saha Ray and Bera [35,38]. 2. Mathematical aspects 2.1. Mathematical definition The mathematical definition of fractional calculus has been the subject of several different approaches [40,41]. The most frequently encountered definition of an integral of fractional order is the Riemann–Liouville integral, in which the fractional order integral is defined as Z t dq f ðtÞ 1 f ðxÞdx q Dt f ðtÞ ¼ ¼ ; ð2:1:1Þ dtq CðqÞ 0 ðt xÞ1q while the definition of fractional order derivative is ! Z t n ðnqÞ d d f ðtÞ 1 dn f ðxÞdx q Dt f ðtÞ ¼ n ; ¼ dtðnqÞ Cðn qÞ dtn 0 ðt xÞ1nþq dt where q(q > 0 and q 2 R) is the order of the operation and n is an integer that satisfies n 1 6 q < n.
ð2:1:2Þ
296
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
3. The fractional diffusion equation model and its solution We adopt Adomian decomposition method for solving Eq. (1.1). In the light of this method we assume that 1 X un ; ð3:1Þ u¼ n¼0
to be the solution of Eq. (1.1). Now, Eq. (1.1) can be rewritten as Lt uðx; tÞ ¼ dðxÞDax uðx; tÞ þ qðx; tÞ;
ð3:2Þ Dax ðÞ
o ot
where Lt which is an easily invertible linear operator, is the Riemann–Liouville derivative of order a. Therefore, by Adomian decomposition method, we can write, !! 1 X 1 a uðx; tÞ ¼ uðx; 0Þ þ Lt dðxÞDx un ð3:3Þ þ L1 t ðqðx; tÞÞ: n¼0
Each term of series (3.1) is given by the standard Adomian decomposition method recurrence relation u0 ¼ f ; a unþ1 ¼ L1 t ðdðxÞDx un Þ; n P 0;
ð3:4Þ
where f ¼ uðx; 0Þ þ L1 t ðqðx; tÞÞ: It is worth noting that once the zeroth component u0 is defined, then the remaining components un, n P 1 can be completely determined; each term is computed by using the previous term. As a result, the components u0, u1, . . . are identified and the series solutions thus entirely determined. However, in many cases the exact solution in a closed form may be obtained. Recently, Wazwaz [27] proposed that the construction of the zeroth component u0 of the decomposition series can be defined in a slightly different way. In [27], he assumed that if the zeroth component u0 = f and the function f is possible to divide into two parts such as f1 and f2, then one can formulate the recursive algorithm for u0 and general term un+1 in a form of the modified decomposition method (MDM) recursive scheme as follows: u0 ¼ f1 ; a u1 ¼ f2 þ L1 t ðdðxÞDx un Þ;
unþ1 ¼
a L1 t ðdðxÞDx un Þ;
ð3:5Þ
n P 1:
Comparing the recursive Scheme (3.4) of the standard Adomain method with the recursive Scheme (3.5) of the modified technique leads to the conclusion that in Eq. (3.4) the zeroth component was defined by the function f, whereas in Eq. (3.5), the zeroth component u0 is defined only by a part f1 of f. The remaining part f2 of f is added to the definition of the component u1 in Eq. (3.5). Although, the modified technique needs only a slight variation from the standard Adomian decomposition method, the results are promising in that it minimizes the size of calculations needed and will accelerate the convergence. The modification could lead to a promising approach for many applications in applied science. The decomposition series solution (3.1) generally converges very rapidly in real physical problems [19,20]. The rapidity of this convergence means that few terms are required. Convergence of this method has been rigorously established by Cherruault [42], Abbaoui and Cherruault [43,44] and Himoun, Abbaoui and Cherruault [45]. The practical solution will be the n-term approximation /n /n ¼
n1 X
ui ðx; tÞ;
i¼0
with lim /n ¼ uðx; tÞ:
n!1
n P 1;
ð3:6Þ
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
Similarly, for Eq. (1.2) using Adomian decomposition method, we can obtain !! !! 1 1 X X 1 a 1 b uðx; y; tÞ ¼ uðx; y; 0Þ þ Lt dðx; yÞDx un un þ Lt eðx; yÞDy þ L1 t ðqðx; y; tÞÞ; n¼0
297
ð3:7Þ
n¼0
The standard Adomian decomposition method recurrence scheme is u0 ¼ f ; a 1 b unþ1 ¼ L1 t ðdðx; yÞDx un Þ þ Lt ðeðx; yÞDy un Þ;
n P 0;
ð3:8Þ
where f ¼ uðx; y; 0Þ þ L1 t ðqðx; y; tÞÞ. The modified decomposition method (MDM) recursive scheme is as follows: u0 ¼ f 1 ; a 1 b u1 ¼ f2 þ L1 t ðdðx; yÞDx u0 Þ þ Lt ðeðx; yÞDy u0 Þ;
unþ1 ¼
a L1 t ðdðx; yÞDx un Þ
þ
b L1 t ðeðx; yÞDy un Þ;
ð3:9Þ
n P 1:
Adomian and Rach [46] and Wazwaz [47–49] have investigated the phenomena of the self-cancelling ‘‘noise’’ terms where sum of noise terms vanishes in the limit. The noise terms are defined as the identical terms with opposite signs that arise in the components u0 and u1. Noise terms may appear if the exact solution is part of the zeroth component u0. Verification that the remaining non cancelled terms justify the equation is necessary and essential. An important observation was made that ‘‘noise’’ terms appear for inhomogeneous cases only. Further, it was formally justified that if terms in u0 are cancelled by terms in u1, even though u1 includes further terms, then the remaining non cancelled terms in u1 are cancelled by terms in u2, and so on. Finally, the exact solution of the equation is readily found for the inhomogeneous case by determining the first two or three terms of the solution u(x, t) and by keeping only the non cancelled terms of u0. 4. Implementation of modified decomposition method Example 1. Let us consider a one-dimensional fractional diffusion equation for the Eq. (1.1), as taken in [17] ouðx; tÞ o1:8 uðx; tÞ ¼ dðxÞ þ qðx; tÞ; ot ox1:8
ð4:1Þ
on a finite domain 0 < x < 1, with the diffusion coefficient dðxÞ ¼ Cð2:2Þx2:8 =6 ¼ 0:183634x2:8 ; the source/sink function qðx; tÞ ¼ ð1 þ xÞet x3 ; the initial condition uðx; 0Þ ¼ x3 ;
for 0 < x < 1;
and the boundary conditions uð0; tÞ ¼ 0;
uð1; tÞ ¼ et ;
for t > 0:
Now, Eq. (4.1) can be rewritten in operator form as Lt uðx; tÞ ¼ dðxÞD1:8 x uðx; tÞ þ qðx; tÞ;
ð4:2Þ
where Lt oto symbolizes the easily invertible linear differential operator, D1:8 x ðÞ is the Riemann–Liouville derivative of order 1.8. Rt Applying the one-fold integration inverse operator L1 t 0 ðÞdt to the Eq. (4.2) and using the specified initial condition yields
298
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
uðx; tÞ ¼ uðx; 0Þ þ
L1 t
dðxÞD1:8 x
1 X
!! un
þ L1 t ðqðx; tÞÞ
n¼0 t 3
t 4
4
¼e x þe x x þ
L1 t
dðxÞD1:8 x
1 X
!! un
:
ð4:3Þ
n¼0
The Standard Adomian Decomposition Method: u0 ¼ et x3 þ et x4 x4 ; Cð2:2Þx2:8 o1:8 u0 4ðet þ 1 tÞx5 t 4 ; u1 ¼ L1 þ 1Þx þ ¼ ðe t 6 ox1:8 2:2 t t2 2:8 1:8 t 5 80 e þ t 1 x6 2! o u1 4ðe þ t 1Þx 1 Cð2:2Þx þ u2 ¼ Lt ; ¼ 6 ox1:8 2:2 3:2 2:22 80 et þ t2 t þ 1 x6 80Cð6Þ et t3 þ t2 t þ 1 x7 2:8 1:8 2! 3! 2! Cð2:2Þx o u2 þ ; u3 ¼ L1 ¼ t 2 2 6 ox1:8 3:2 2:2 4:2 3:2 2:23 and so on. It is obvious that the ‘‘noise’’ terms appear between the components of u1, and these are all cancelled. The closed form of the solution can be found very easily by proper selection of f1 and f2 in MDM. In the case of right choice of these functions, the modified technique accelerates the convergence of the decomposition series solution by computing just u0 and u1 terms of the series. The term u0 provides the exact solution as u(x, t) = et x3 and this can be justified through substitution. The Modified Decomposition Method: We will then obtain from recursive scheme for modified decomposition method u0 ¼ et x3 ; t 4
4
u1 ¼ e x x þ
L1 t
Cð2:2Þx2:8 o1:8 u0 ; 6 ox1:8
¼ et x4 x4 ðet 1Þx4 ; ¼ 0; 2:8 1:8 o u1 1 Cð2:2Þx u2 ¼ Lt ; 6 ox1:8 ¼ 0; and so on. Therefore, the solution is uðx; tÞ ¼ et x3 :
ð4:4Þ
The solution (4.4) can be verified through substitution in Eq. (4.1). Example 2. Let us consider a two-dimensional fractional diffusion equation for the Eq. (1.2), considered in [18] ouðx; y; tÞ o1:8 uðx; y; tÞ o1:6 uðx; y; tÞ ¼ dðx; yÞ þ eðx; yÞ þ qðx; y; tÞ; ot ox1:8 oy 1:6 on a finite rectangular domain 0 < x < 1, 0 < y < 1, for 0 6 t 6 Tend with the diffusion coefficients dðx; yÞ ¼ Cð2:2Þx2:8 y=6;
ð4:5Þ
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
299
and eðx; yÞ ¼ 2xy 2:6 =Cð4:6Þ; and the forcing function qðx; y; tÞ ¼ ð1 þ 2xyÞet x3 y 3:6 ; with the initial condition uðx; y; 0Þ ¼ x3 y 3:6 ; and Dirichlet boundary conditions on the rectangle in the form u(x, 0, t) = u(0, y, t) = 0, u(x, 1, t) = et x3, and u(1, y, t) = et y3.6, for all t P 0. Now, Eq. (4.8) can be rewritten in operator form as 1:6 Lt uðx; y; tÞ ¼ dðx; yÞD1:8 x uðx; y; tÞ þ eðx; yÞDy uðx; y; tÞ þ qðx; y; tÞ;
ð4:6Þ
1:6 where Lt oto symbolizes the easily invertible linear differential operator, D1:8 x ðÞ and Dy ðÞ are the Riemann– Liouville derivatives of order 1.8 and 1.6, respectively. Rt Applying the one-fold integration inverse operator L1 t 0 ðÞdt to the Eq. (4.6) and using the specified initial condition yields !! !! 1 1 X X 1 1:8 1 1:6 uðx; y; tÞ ¼ uðx; y; 0Þ þ Lt dðx; yÞDx un un þ Lt eðx; yÞDy þ L1 t ðqðx; y; tÞÞ n¼0
dðx; yÞD1:8 ¼ x3 y 3:6 et þ 2x4 y 4:6 et 2x4 y 4:6 þ L1 t x
n¼0
1 X
!! un
n¼0
eðx; yÞD1:6 þ L1 t y
1 X
!! un
:
n¼0
ð4:7Þ The Standard Adomian Decomposition Method: u0 ¼ x3 y 3:6 et þ 2x4 y 4:6 et 2x4 y 4:6 ; 1:8 2:8 2xy 2:6 o1:6 u0 1 Cð2:2Þx y o u0 1 u1 ¼ L t þ Lt ; 6 ox1:8 Cð4:6Þ oy 1:6 8 2 4:6 5 5:6 þ x y ðet þ 1 tÞ; ¼ 2x4 y 4:6 ðet þ 1Þ þ 2:2 3 1106 5 5:6 x y ðet þ 1 tÞ; ¼ 2x4 y 4:6 ðet þ 1Þ þ 165 1:8 2:8 2xy 2:6 o1:6 u1 1 Cð2:2Þx y o u1 1 u2 ¼ L t þ Lt ; 6 ox1:8 Cð4:6Þ oy 1:6 1106 5 5:6 t 9101827 6 6:6 t t2 x y ðe 1 þ tÞ þ xy e 1þt ¼ ; 165 272250 2 and so on. In this case, the ‘‘noise’’ terms appear between the components of u1, and these are all cancelled. In the above, one obtains self-cancelling ‘‘noise’’ terms appearing between various components of u0, u1, u2, u3, . . ., and keeping the non cancelled terms, and using (3.1) lead to the exact solution as u(x, y, t) = x3y3.6et. It is worth noting that noise terms between components of the decomposition series will be cancelled, and the sum of these ‘‘noise’’ terms will vanish in the limit [46–49]. In the present analysis, the closed form of the solution can be found very easily by proper selection of f1 and f2 in MDM. In the case of right choice of these functions, the modified technique accelerates the convergence of the decomposition series solution by computing just u0 and u1 terms of the series. The term u0 provides the exact solution as u(x, y, t) = x3y3.6et and this can be justified through substitution.
300
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
The Modified Decomposition Method: The recursive scheme for modified decomposition method gives u0 ¼ x3 y 3:6 et ; u1 ¼ 2x4 y 4:6 et 2x4 y 4:6 þ L1 t
Cð2:2Þx2:8 y o1:8 u0 6 ox1:8
þ L1 t
2xy 2:6 o1:6 u0 ; Cð4:6Þ oy 1:6
¼ 2x4 y 4:6 et 2x4 y 4:6 2ðet 1Þx4 y 4:6 ; ¼ 0; Cð2:2Þx2:8 y o1:8 u1 2xy 2:6 o1:6 u1 1 u2 ¼ L1 þ L ; t t 6 ox1:8 Cð4:6Þ oy 1:6 ¼ 0; and so on. Therefore, the solution is uðx; y; tÞ ¼ x3 y 3:6 et
ð4:8Þ
The solution (4.8) can be verified through substitution in Eq. (4.5). 5. Conclusion This paper presents an analytical scheme to obtain the solutions of the one-dimensional and two-dimensional fractional diffusion equations. Two typical examples have been discussed as demonstrations. In our previous papers [36–39] we have already established as well as successfully exhibited the applicability of ADM to obtain the solutions of different types of fractional differential equations. In this work we demonstrate that this method is also well suited to solve fractional diffusion-wave equation. The decomposition method is straightforward, without restrictive assumptions and the components of the series solution can be easily computed using any mathematical symbolic package. Moreover, this method does not change the problem into a convenient one for the use of linear theory. It, therefore, provides more realistic series solutions that generally converge very rapidly in real physical problems. Moreover, no linearization or perturbation is required. It can avoid the difficulty of finding the inverse of Laplace Transform and can reduce the labour of perturbation method. Furthermore, as the decomposition method does not require discretization of the variables, i.e. time and space, it is not affected by computational round off errors and one is not faced with necessity of large computer memory and time. Consequently, the computational size will be reduced. It is also worth noting that the advantage of the decomposition methodology is that it displays a fast convergence of the solution. It may be achieved by observing the phenomenon of self-cancelling ‘‘noise’’ terms and the splitting of the initial term into appropriate parts in MDM. Therefore, the MDM will further accelerate the convergence of the series solution [27,48]. Acknowledgement We express our sincere thanks and gratitude to the learned reviewer for his kind comments and suggestions for the improvement of the paper. References [1] R. Metzler, E. Barkai, J. Klafter, Anomalous diffusion and relaxation close to thermal equilibrium: a fractional Fokker–Planck equation approach, Phys. Rev. Lett. 82 (18) (1999) 3563–3567. [2] R. Metzler, J. Klafter, The random Walk’s guide to anomalous diffusion: a fractional dynamics approach 339 (2000) 1–77. [3] R. Metzler, J. Klafter, The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A 37 (2004) R161–R208.
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
301
[4] R. Gorenflo, F. Mainardi, E. Scalas, M. Raberto, Fractional calculus and continuous time finance. III. The diffusion limit. Mathematical finance (Konstanz, 2000), Trends Math. Birkhuser Basel (2001) 171–180. [5] F. Mainardi, M. Raberto, R. Gorenflo, E. Scalas, Fractional calculus and continuous-time finance II: the waiting-time distribution, Physica A 287 (2000) 468–481. [6] E. Scalas, R. Gorenflo, F. Mainardi, Fractional calculus and continuous-time finance, Physica A 284 (2000) 376–384. [7] M. Raberto, E. Scalas, F. Mainardi, Waiting-times and returns in high-frequency financial data: an empirical study, Physica A 314 (2002) 749–755. [8] D.A. Benson, S. Wheatcraft, M.M. Meerschaert, Application of a fractional advection–dispersion equation, Water Resour. Res. 36 (2000) 1403–1412. [9] B. Baeumer, M.M. Meerschaert, D.A. Benson, S.W. Wheatcraft, Subordinated advection–dispersion equation for contaminant transport, Water Resour. Res. 37 (2001) 1543–1550. [10] D.A. Benson, R. Schumer, M.M. Meerschaert, S.W. Wheatcraft, Fractional dispersion, Le´vy motions, and the MADE tracer tests, Transp. Porous Media 42 (2001) 211–240. [11] R. Schumer, D.A. Benson, M.M. Meerschaert, S.W. Wheatcraft, Eulerian derivation of the fractional advection–dispersion equation, J. Contam. Hydrol. 48 (2001) 9–88. [12] R. Schumer, D.A. Benson, M.M. Meerschaert, B. Baeumer, Multiscaling fractional advection–dispersion equations and their solutions, Water Resour. Res. 39 (2003) 1022–1032. [13] F. Mainardi, Fractional calculus: some basic problems in continuum and statistical mechanics, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer-Verlag, Wien New York, 1997, pp. 291–348. [14] F. Mainardi, G. Pagnini, The weight functions as solutions of the time-fractional diffusion equation, Appl. Math. Comput. 141 (2003) 51–62. [15] O.P. Agrawal, Solution for a fractional diffusion-wave equation defined in a bounded domain, Nonlinear Dynam. 29 (2002) 145–155. [16] W.R. Schneider, W. Wyss, Fractional diffusion and wave equations, J. Math. Phys. 30 (1989) 134–144. [17] M.M. Meerschaert, H. Scheffler, C. Tadjeran, Finite difference methods for two-dimensional fractional dispersion equation, J. Comput. Phys. 211 (2006) 249–261. [18] C. Tadjeran, M.M. Meerschaert, H. Scheffler, A second-order accurate numerical approximation for the fractional diffusion equation, J. Comput. Phys. 213 (2006) 205–213. [19] G. Adomian, Nonlinear Stochastic Systems theory and Applications to Physics, Kluwer Academic Publishers, The Netherlands, 1989. [20] G. Adomian, Solving Frontier Problems of Physics: The Decomposition Method, Kluwer Academic Publishers, Boston, 1994. [21] G. Adomian, An analytical solution of the stochastic Navier-Stokes system, Foundations of Physics 21 (7) (1991) 831–843. [22] G. Adomian, R. Rach, Linear and nonlinear Schro¨dinger equations, Foundations of Physics 21 (1991) 983–991. [23] G. Adomian, Solution of Physical Problems by Decomposition, Comput. Math. Appl. 27 (9/10) (1994) 145–154. [24] G. Adomian, Solutions of nonlinear P.D.E, Appl. Math. Lett. 11 (3) (1998) 121–123. [25] K. Abbaoui, Y. Cherruault, The decomposition method applied to the cauchy problem, Kybernetes 28 (1999) 103–108. [26] D. Kaya, A. Yokus, A numerical comparison of partial solutions in the decomposition method for linear and nonlinear partial differential equations, Math. Comput. Simul. 60 (6) (2002) 507–512. [27] A. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1) (1999) 77–86. [28] D. Kaya, S.M. El-Sayed, On a generalized fifth order KdV equations, Phys. Lett. A 310 (1) (2003) 44–51. [29] D. Kaya, S.M. El-Sayed, An application of the decomposition method for the generalized KdV and RLW equations, Chaos Solitons Fractals 17 (5) (2003) 869–877. [30] D. Kaya, An explicit and numerical solutions of some fifth-order KdV equation by decomposition method, Appl. Math. Comput. 144 (2–3) (2003) 353–363. [31] D. Kaya, A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl. Math. Comput. 149 (3) (2004) 833–841. [32] A.J. George, A. Chakrabarti, The Adomian method applied to some extraordinary differential equations, Appl. Math. Lett. 8 (3) (1995) 91–97. [33] H.L. Arora, F.I. Abdelwahid, Solutions of non-integer order differential equations via the Adomian decomposition method, Appl. Math. Lett. 6 (1) (1993) 21–23. [34] N.T. Shawagfeh, The decompostion method for fractional differential equations, J. Frac. Calc. 16 (1999) 27–33. [35] N.T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput. 131 (2002) 517–529. [36] S. Saha Ray, R.K. Bera, Solution of an extraordinary differential equation by Adomian decomposition method, J. Appl. Math. 4 (2004) 331–338. [37] S. Saha Ray, R.K. Bera, Analytical solution of a dynamic system containing fractional derivative of order 1/2 by Adomian decomposition method, Trans. ASME J. Appl. Mech. 72 (2) (2005) 290–295. [38] S. Saha Ray, R.K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. Math. Comput. 167 (1) (2005) 561–571. [39] S. Saha Ray, R.K. Bera, Analytical solution of the Bagley Torvik equation by Adomian decomposition method, Appl. Math. Comput. 168 (1) (2005) 398–410. [40] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, USA, 1999. [41] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York and London, 1974. [42] Y. Cherruault, Convergence of Adomian’s method, Kybernetes 18 (1989) 31–38.
302
S.S. Ray et al. / Applied Mathematics and Computation 196 (2008) 294–302
[43] K. Abbaoui, Y. Cherruault, Convergence of Adomian’s method applied to differential equations, Comput.Math. Appl. 28 (5) (1994) 103–109. [44] K. Abbaoui, Y. Cherruault, New ideas for proving convergence of decomposition methods, Comput. Math. Appl. 29 (1995) 103–108. [45] N. Himoun, K. Abbaoui, Y. Cherruault, New results of convergence of Adomian’s method, Kybernetes 28 (4–5) (1999) 423–429. [46] G. Adomian, R. Rach, Noise terms in decomposition solution series, Comput. Math. Appl. 24 (11) (1992) 61–64. [47] A.M. Wazwaz, Necessary conditions for the appearance of noise terms in decomposition solution series, Appl. Math. Comput. 81 (2– 3) (1997) 265–274. [48] A.M. Wazwaz, Partial Differential Equations: Methods and Applications, A.A. Balkema Publishers, Lisse, The Netherlands, 2002. [49] A.M. Wazwaz, The existence of noise terms for systems of inhomogeneous differential and integral equations 146 (1) (2003) 81– 92.