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Analytical solution of a fractional diffusion equation by Adomian decomposition method
Applied Mathematics and Computation 174 (2006) 329–336 www.elsevier.com/locate/amc
Analytical solution of a fractional diffusion equation by Adomian decomposition method S. Saha Ray a
a,*
, R.K. Bera
b
B.P. Poddar Institute of Management and Technology, Poddar Vihar, 137, V.I.P. Road, Kolkata 700052, India b Heritage Institute of Technology, Chowbaga Road, Anandapur, Kolkata 700107, India
Abstract This paper presents an analytical solution of a fractional diffusion equation by Adomian decomposition method. By using an initial value, the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically. The present method performs extremely well in terms of efficiency and simplicity. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Adomian decomposition method; Fractional derivative; Fractional differential equation; Fractional diffusion equation
*
Corresponding author. E-mail addresses: [email protected] (S. Saha Ray), [email protected] (R.K. Bera). 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.082
330
S. Saha Ray, R.K. Bera / Appl. Math. Comput. 174 (2006) 329–336
1. Introduction The fractional differential equations appear more and more frequently in different research areas and engineering applications. The fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PIkDl controller for the control of dynamical systems, etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also described by differential equations of fractional order. The solution of the differential equation containing fractional derivative is much involved. An effective and easy-to-use method for solving such equations is needed. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. In this connection, it is worthwhile to mention that the recent papers on numerical solutions of fractional differential equations are available from the notable works of Diethelm et al. [1–5]. Recently, applications have included classes of nonlinear fractional differential equations [6] and their numerical solutions have been established by Diethelm and Ford [7]. In recent years there has been growing interest in diffusion in various fields of physics, chemistry, and engineering applications. The recent papers [8–12] on fractional diffusion are valuable in this field. The free motion of the particle was modeled by the classical diffusion equation ouðx; tÞ o2 uðx; tÞ ¼D ; ot ox2
x 2 R; t > 0;
ð1:1Þ
where u(x, t) represents the probability density function of finding a particle at the point x in the time instant t. Here D is a positive constant depending on the temperature, the friction coefficient, the universal gas constant and finally on the Avogadro number. Assuming the external outside force acting towards the origin x = 0 and being proportional to the distance of the particle from the origin, Eq. (1.1) should be augmented by a drift term: ouðx; tÞ o2 uðx; tÞ o ¼D ðF ðxÞuðx; tÞÞ; ot ox2 ox
ð1:2Þ
where F(x) is the external force. Therefore, Eq. (1.2) can be interpreted as modeling the diffusion of a particle under the action of the external outside force F(x).
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331
In this paper, we shall consider the fractional diffusion equation with drift term [8]. In general approach, the fractional diffusion in time under external force, Eq. (1.2) becomes oa uðx; tÞ o2 uðx; tÞ o ¼ D ðF ðxÞuðx; tÞÞ; ota ox2 ox
0 < a 6 1; D > 0;
ð1:3Þ
a
where ootðÞ a is the Caputo derivative of order a. In the present paper, we shall consider the fractional diffusion coefficient D = 1, a ¼ 12 and external force F(x) = x. In this paper, we use the Adomian decomposition method [13,14] to obtain a solution of a fractional diffusion equation (1.3). Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the Adomian decomposition method [13–20]. A reliable modification of Adomian decomposition method has been done by Wazwaz [21]. The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems [14–17]. Recently, the implementations of Adomian decomposition method for the solutions of generalized regularized long-wave (RLW) and Korteweg-de Vries (KdV) equations have been well established by the notable researchers [22–25]. 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 solution of fractional differential equation has been obtained through Adomian decomposition method by the researchers [26–33]. The application of Adomian decomposition method for the solution of nonlinear fractional differential equations has also been established by Shawagfeh [29] and Saha Ray and Bera [32].
2. Mathematical aspects 2.1. Mathematical definition The mathematical definition of fractional calculus has been the subject of several different approaches [34,35]. 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 ðxÞ dx Dq f ðtÞ ¼ ¼ ; ð2:1:1Þ t q dt CðqÞ 0 ðt xÞ1q
332
S. Saha Ray, R.K. Bera / Appl. Math. Comput. 174 (2006) 329–336
while the definition of fractional order derivative is ! Z t dn dðnqÞ f ðtÞ 1 dn f ðxÞ dx q Dt f ðtÞ ¼ n ; ¼ Cðn qÞ dtn 0 ðt xÞ1nþq dt dtðnqÞ
ð2:1:2Þ
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. The fractional derivative of f(t) in the Caputo sense is defined by Z t ðnÞ 1 f ðxÞ dx C a Dt f ðtÞ ¼ ; Cðn aÞ 0 ðt xÞaþ1n where a (a > 0 and a 2 R) is the order of the operation and n is an integer that satisfies n 1 < a < n. 2.2. Definition: Mittag–Leffler function A one-parameter function of the Mittag–Leffler type is defined by the series expansion [34] 1 X zk Ea ðzÞ ¼ ða > 0Þ. ð2:2:1Þ Cðak þ 1Þ k¼0
3. Fractional diffusion model and the solution We consider the equation o1=2 uðx; tÞ o2 uðx; tÞ oðxuðx; tÞÞ ; ¼ þ ot1=2 ox2 ox
ð3:1Þ 1=2
with initial condition u(x, 0) = f(x). Here, oot1=2ðÞ is the Caputo derivative of order 1/2. We adopt Adomian decomposition method for solving Eq. (3.1). In the light of this method we assume that 1 X un u¼ n¼0
to be the solution of Eq. (3.1), Now, Eq. (3.1) can be written as 1=2 Lt uðx; tÞ ¼ Dt1=2 ðLxx uðx; tÞÞ þ D1=2 t ðxLx uðx; tÞÞ þ Dt uðx; tÞ;
ð3:2Þ
where Lt oto which is an easily invertible linear operator, D1=2 t ðÞ is the Rie2 mann–Liouville derivative of order 1/2, Lxx ¼ oxo 2 and Lx ¼ oxo .
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333
Therefore, by Adomian decomposition method, we can write, 1=2 1=2 Dt ðLxx uðx; tÞÞ þ L1 Dt ðxLx uðx; tÞÞ uðx; tÞ ¼ uðx; 0Þ þ L1 t t 1=2 Dt uðx; tÞ ; þ L1 t
ð3:3Þ
where u0 ¼ uðx; 0Þ; 1=2 1=2 1=2 u1 ¼ L1 Dt ðLxx u0 Þ þ L1 Dt ðxLx u0 Þ þ L1 Dt u0 ; t t t 1 Dt1=2 ðLxx u1 Þ þ L1 D1=2 D1=2 u2 ¼ L1 t t t ðxLx u1 Þ þ Lt t u1 ; 1=2 1=2 1=2 Dt ðLxx u2 Þ þ L1 Dt ðxLx u2 Þ þ L1 Dt u2 ; u3 ¼ L1 t t t and so on.
4. Illustrative example Let us consider f(x) = x for Eq. (3.1). We will then obtain u0 ¼ uðx; 0Þ ¼ x; 1=2 1=2 1=2 2xt1=2 Dt ðLxx u0 Þ þ L1 Dt ðxLx u0 Þ þ L1 Dt u0 ¼ ; u1 ¼ L1 t t t Cð3=2Þ 1=2 1=2 1=2 22 xt 1 1 ; D D Dt u1 ¼ u2 ¼ L1 ðL u Þ þ L ðxL u Þ þ L xx 1 x 1 t t t t t Cð2Þ 1=2 1=2 1=2 23 xt3=2 1 1 u3 ¼ L1 ; ðL u Þ þ L ðxL u Þ þ L D D Dt u2 ¼ xx 2 x 2 t t t t t Cð5=2Þ 1=2 1=2 1=2 24 xt2 1 1 D D Dt u3 ¼ ; u4 ¼ L1 ðL u Þ þ L ðxL u Þ þ L xx 3 x 3 t t t t t Cð3Þ 1=2 1=2 1=2 25 xt5=2 1 1 D D Dt u4 ¼ ; u5 ¼ L1 ðL u Þ þ L ðxL u Þ þ L xx 4 x 4 t t t t t Cð7=2Þ and so on. Therefore, the solution is uðx; tÞ ¼
1 X 2r xtr=2 ¼ xE1 ð2t1=2 Þ; 2 C 2r þ 1 r¼0
where Ek(z) is the Mittag–Leffler function in one parameter. The solution (4.1) can be verified through substitution in Eq. (3.1).
ð4:1Þ
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S. Saha Ray, R.K. Bera / Appl. Math. Comput. 174 (2006) 329–336 u(x,t) 100 80 60 40 20 0.2
0.4
0.6
0.8
1
t
Fig. 1. The solution u(x, t) vs. time t when x = 1.
60 1
u(x,t) 40 20 0 0
0.75 0.5 t 0.25
0.25 0.5 x
0.75 10
Fig. 2. Three-dimensional figure for u(x, t) with respect to x and time t.
5. Numerical results and discussions In the present numerical experiment, Eq. (4.1) has been used to draw the graph as shown in Fig. 1, assuming x = 1. Fig. 2 shows three-dimensional figure for u(x, t) with respect to x and time t. In the present analysis, Eq. (4.1) also has been used to draw the graph as shown in Fig. 2. Figs. 1 and 2 have been drawn using the Mathematica software [36].
6. Conclusion This present analysis exhibits the applicability of the decomposition method to solve a fractional diffusion equation. In our previous papers [31,33] we have already as well as successfully exhibit the applicability of Adomian decomposition method to obtain a solution for dynamic system containing factional derivative of order 1/2 and 3/2, respectively. In this work, we demonstrate that
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335
this method is also well suited to solve fractional diffusion 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. When solutions are computed numerically, the rapid convergence is obvious. Moreover, no linearization or perturbation is required. It can avoid the difficulty of finding the inverse of Laplace Transform and can reduce the labor of perturbation method. This paper presents an analytical scheme to obtain the solution of a fractional diffusion equation. Acknowledgements We take this opportunity to express our sincere thanks and gratitude to Prof. F. Mainardi for providing us with his reprints [10,12]. The authors are grateful to Prof. F. Mainardi for his kind co-operation. References [1] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Elec. Trans. Numer. Anal. 5 (1997) 1–6. [2] K. Diethelm, A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, in: S. Heinzel, T. Plesser (Eds.), GWDG-Berichte, Forschung und wissenschaftliches Rechnen: Beitra¨ge zum Heinz-Billing-Preis 1998 (No. 52), Gesellschaft fu¨r wissenschaftliche Datenverarbeitung, Go¨ttingen, 1999, pp. 57–71. [3] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [4] K. Diethelm, N.J. Ford, A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2002) 3–22. [5] K. Diethelm, N.J. Ford, Numerical solution of the Bagley–Torvik equation, Bit Numer. Math. 42 (3) (2002) 490–507. [6] K. Diethelm, A.D. Ford, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, in: F. Keil, W. Mackens, H. Voß, J. Werther (Eds.), Scientific Computing in Chemical Engineering II. Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp. 217–224. [7] K. Diethelm, N.J. Ford, The numerical solution of linear and nonlinear fractional differential equations involving fractional derivatives of several orders, Numerical Analysis Report, vol. 379, Manchester Centre for Computational Mathematics, Manchester, England, 2001. [8] R. Gorenflo, E.A. Abdel-Rehim, Approximation of time-fractional diffusion with central drift by difference schemes, Berlin Free University, 2003. Available from:. [9] R. Gorenflo, F. Mainardi, Fractional calculus and stable probability distributions, Arch. Mech. 50 (1998) 377–388. [10] F. Mainardi, Fractional relaxation–oscillation and fractional diffusion-wave phenomena, Chaos Soliton Fract. 7 (9) (1996) 1461–1477.
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[11] R. Gorenflo, Y. Luchko, F. Mainardi, Wright functions as scale-invariant solutions of the diffusion-wave equation, J. Comput. Appl. Math. 118 (2000) 175–191. [12] F. Mainardi, G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equation, Appl. Math. Comput. 141 (2003) 51–62. [13] G. Adomian, Nonlinear Stochastic Systems Theory and Applications to Physics, Kluwer Academic Publishers, Netherlands, 1989. [14] G. Adomian, Solving Frontier Problems of Physics: the Decomposition Method, Kluwer Academic Publishers, Boston, 1994. [15] G. Adomian, An analytical solution of the stochastic Navier–Stokes system, Found. Phys. 21 (7) (1991) 831–843. [16] G. Adomian, R. Rach, Linear and nonlinear Schro¨dinger equations, Found. Phys. 21 (1991) 983–991. [17] G. Adomian, Solution of physical problems by decomposition, Comput. Math. Appl. 27 (9/10) (1994) 145–154. [18] G. Adomian, Solutions of nonlinear P.D.E., Appl. Math. Lett. 11 (3) (1998) 121–123. [19] K. Abbaoui, Y. Cherruault, The decomposition method applied to the Cauchy problem, Kybernetes 28 (1999) 103–108. [20] 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. [21] A. Wazwaz, A reliable modification of Adomian decomposition method, Appl. Math. Comput. 102 (1) (1999) 77–86. [22] D. Kaya, S.M. El-Sayed, On a generalized fifth order KdV equations, Phys. Lett. A 310 (1) (2003) 44–51. [23] D. Kaya, S.M. El-Sayed, An application of the decomposition method for the generalized KdV and RLW equations, Chaos Soliton Fract. 17 (5) (2003) 869–877. [24] 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. [25] D. Kaya, A numerical simulation of solitary-wave solutions of the generalized regularized long-wave equation, Appl. Math. Comput. 149 (3) (2004) 833–841. [26] A.J. George, A. Chakrabarti, The Adomian method applied to some extraordinary differential equations, Appl. Math. Lett. 8 (3) (1995) 91–97. [27] 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. [28] N.T. Shawagfeh, The decomposition method for fractional differential equations, J. Frac. Calc. 16 (1999) 27–33. [29] N.T. Shawagfeh, Analytical approximate solutions for nonlinear fractional differential equations, Appl. Math. Comput. 131 (2002) 517–529. [30] S. Saha Ray, R.K. Bera, Solution of an extraordinary differential equation by Adomian decomposition method, J. Appl. Math. 4 (2004) 331–338. [31] 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. [32] S. Saha Ray, R.K. Bera, An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. Math. Comput., in press, doi:10.1016/ j.amc.2004.07.020. [33] S. Saha Ray, R.K. Bera, Analytical solution of the Bagley Torvik equation by Adomian decomposition method, Appl. Math. Comput., in press, doi:10.1016/j.amc.2004.09.006. [34] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999. [35] K.B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York and London, 1974. [36] S. Wolfram, Mathematica for Windows, Version 5.0, Wolfram Research, 2003..
Analytical solution of a fractional diffusion equation by Adomian decomposition method S. Saha Ray a
a,*
, R.K. Bera
b
B.P. Poddar Institute of Management and Technology, Poddar Vihar, 137, V.I.P. Road, Kolkata 700052, India b Heritage Institute of Technology, Chowbaga Road, Anandapur, Kolkata 700107, India
Abstract This paper presents an analytical solution of a fractional diffusion equation by Adomian decomposition method. By using an initial value, the explicit solution of the equation has been presented in the closed form and then its numerical solution has been represented graphically. The present method performs extremely well in terms of efficiency and simplicity. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Adomian decomposition method; Fractional derivative; Fractional differential equation; Fractional diffusion equation
*
Corresponding author. E-mail addresses: [email protected] (S. Saha Ray), [email protected] (R.K. Bera). 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.04.082
330
S. Saha Ray, R.K. Bera / Appl. Math. Comput. 174 (2006) 329–336
1. Introduction The fractional differential equations appear more and more frequently in different research areas and engineering applications. The fractional derivative has been occurring in many physical problems such as frequency dependent damping behavior of materials, motion of a large thin plate in a Newtonian fluid, creep and relaxation functions for viscoelastic materials, the PIkDl controller for the control of dynamical systems, etc. Phenomena in electromagnetics, acoustics, viscoelasticity, electrochemistry and material science are also described by differential equations of fractional order. The solution of the differential equation containing fractional derivative is much involved. An effective and easy-to-use method for solving such equations is needed. Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. For that reason we need a reliable and efficient technique for the solution of fractional differential equations. In this connection, it is worthwhile to mention that the recent papers on numerical solutions of fractional differential equations are available from the notable works of Diethelm et al. [1–5]. Recently, applications have included classes of nonlinear fractional differential equations [6] and their numerical solutions have been established by Diethelm and Ford [7]. In recent years there has been growing interest in diffusion in various fields of physics, chemistry, and engineering applications. The recent papers [8–12] on fractional diffusion are valuable in this field. The free motion of the particle was modeled by the classical diffusion equation ouðx; tÞ o2 uðx; tÞ ¼D ; ot ox2
x 2 R; t > 0;
ð1:1Þ
where u(x, t) represents the probability density function of finding a particle at the point x in the time instant t. Here D is a positive constant depending on the temperature, the friction coefficient, the universal gas constant and finally on the Avogadro number. Assuming the external outside force acting towards the origin x = 0 and being proportional to the distance of the particle from the origin, Eq. (1.1) should be augmented by a drift term: ouðx; tÞ o2 uðx; tÞ o ¼D ðF ðxÞuðx; tÞÞ; ot ox2 ox
ð1:2Þ
where F(x) is the external force. Therefore, Eq. (1.2) can be interpreted as modeling the diffusion of a particle under the action of the external outside force F(x).
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331
In this paper, we shall consider the fractional diffusion equation with drift term [8]. In general approach, the fractional diffusion in time under external force, Eq. (1.2) becomes oa uðx; tÞ o2 uðx; tÞ o ¼ D ðF ðxÞuðx; tÞÞ; ota ox2 ox
0 < a 6 1; D > 0;
ð1:3Þ
a
where ootðÞ a is the Caputo derivative of order a. In the present paper, we shall consider the fractional diffusion coefficient D = 1, a ¼ 12 and external force F(x) = x. In this paper, we use the Adomian decomposition method [13,14] to obtain a solution of a fractional diffusion equation (1.3). Large classes of linear and nonlinear differential equations, both ordinary as well as partial, can be solved by the Adomian decomposition method [13–20]. A reliable modification of Adomian decomposition method has been done by Wazwaz [21]. The decomposition method provides an effective procedure for analytical solution of a wide and general class of dynamical systems representing real physical problems [14–17]. Recently, the implementations of Adomian decomposition method for the solutions of generalized regularized long-wave (RLW) and Korteweg-de Vries (KdV) equations have been well established by the notable researchers [22–25]. 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 solution of fractional differential equation has been obtained through Adomian decomposition method by the researchers [26–33]. The application of Adomian decomposition method for the solution of nonlinear fractional differential equations has also been established by Shawagfeh [29] and Saha Ray and Bera [32].
2. Mathematical aspects 2.1. Mathematical definition The mathematical definition of fractional calculus has been the subject of several different approaches [34,35]. 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 ðxÞ dx Dq f ðtÞ ¼ ¼ ; ð2:1:1Þ t q dt CðqÞ 0 ðt xÞ1q
332
S. Saha Ray, R.K. Bera / Appl. Math. Comput. 174 (2006) 329–336
while the definition of fractional order derivative is ! Z t dn dðnqÞ f ðtÞ 1 dn f ðxÞ dx q Dt f ðtÞ ¼ n ; ¼ Cðn qÞ dtn 0 ðt xÞ1nþq dt dtðnqÞ
ð2:1:2Þ
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. The fractional derivative of f(t) in the Caputo sense is defined by Z t ðnÞ 1 f ðxÞ dx C a Dt f ðtÞ ¼ ; Cðn aÞ 0 ðt xÞaþ1n where a (a > 0 and a 2 R) is the order of the operation and n is an integer that satisfies n 1 < a < n. 2.2. Definition: Mittag–Leffler function A one-parameter function of the Mittag–Leffler type is defined by the series expansion [34] 1 X zk Ea ðzÞ ¼ ða > 0Þ. ð2:2:1Þ Cðak þ 1Þ k¼0
3. Fractional diffusion model and the solution We consider the equation o1=2 uðx; tÞ o2 uðx; tÞ oðxuðx; tÞÞ ; ¼ þ ot1=2 ox2 ox
ð3:1Þ 1=2
with initial condition u(x, 0) = f(x). Here, oot1=2ðÞ is the Caputo derivative of order 1/2. We adopt Adomian decomposition method for solving Eq. (3.1). In the light of this method we assume that 1 X un u¼ n¼0
to be the solution of Eq. (3.1), Now, Eq. (3.1) can be written as 1=2 Lt uðx; tÞ ¼ Dt1=2 ðLxx uðx; tÞÞ þ D1=2 t ðxLx uðx; tÞÞ þ Dt uðx; tÞ;
ð3:2Þ
where Lt oto which is an easily invertible linear operator, D1=2 t ðÞ is the Rie2 mann–Liouville derivative of order 1/2, Lxx ¼ oxo 2 and Lx ¼ oxo .
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333
Therefore, by Adomian decomposition method, we can write, 1=2 1=2 Dt ðLxx uðx; tÞÞ þ L1 Dt ðxLx uðx; tÞÞ uðx; tÞ ¼ uðx; 0Þ þ L1 t t 1=2 Dt uðx; tÞ ; þ L1 t
ð3:3Þ
where u0 ¼ uðx; 0Þ; 1=2 1=2 1=2 u1 ¼ L1 Dt ðLxx u0 Þ þ L1 Dt ðxLx u0 Þ þ L1 Dt u0 ; t t t 1 Dt1=2 ðLxx u1 Þ þ L1 D1=2 D1=2 u2 ¼ L1 t t t ðxLx u1 Þ þ Lt t u1 ; 1=2 1=2 1=2 Dt ðLxx u2 Þ þ L1 Dt ðxLx u2 Þ þ L1 Dt u2 ; u3 ¼ L1 t t t and so on.
4. Illustrative example Let us consider f(x) = x for Eq. (3.1). We will then obtain u0 ¼ uðx; 0Þ ¼ x; 1=2 1=2 1=2 2xt1=2 Dt ðLxx u0 Þ þ L1 Dt ðxLx u0 Þ þ L1 Dt u0 ¼ ; u1 ¼ L1 t t t Cð3=2Þ 1=2 1=2 1=2 22 xt 1 1 ; D D Dt u1 ¼ u2 ¼ L1 ðL u Þ þ L ðxL u Þ þ L xx 1 x 1 t t t t t Cð2Þ 1=2 1=2 1=2 23 xt3=2 1 1 u3 ¼ L1 ; ðL u Þ þ L ðxL u Þ þ L D D Dt u2 ¼ xx 2 x 2 t t t t t Cð5=2Þ 1=2 1=2 1=2 24 xt2 1 1 D D Dt u3 ¼ ; u4 ¼ L1 ðL u Þ þ L ðxL u Þ þ L xx 3 x 3 t t t t t Cð3Þ 1=2 1=2 1=2 25 xt5=2 1 1 D D Dt u4 ¼ ; u5 ¼ L1 ðL u Þ þ L ðxL u Þ þ L xx 4 x 4 t t t t t Cð7=2Þ and so on. Therefore, the solution is uðx; tÞ ¼
1 X 2r xtr=2 ¼ xE1 ð2t1=2 Þ; 2 C 2r þ 1 r¼0
where Ek(z) is the Mittag–Leffler function in one parameter. The solution (4.1) can be verified through substitution in Eq. (3.1).
ð4:1Þ
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S. Saha Ray, R.K. Bera / Appl. Math. Comput. 174 (2006) 329–336 u(x,t) 100 80 60 40 20 0.2
0.4
0.6
0.8
1
t
Fig. 1. The solution u(x, t) vs. time t when x = 1.
60 1
u(x,t) 40 20 0 0
0.75 0.5 t 0.25
0.25 0.5 x
0.75 10
Fig. 2. Three-dimensional figure for u(x, t) with respect to x and time t.
5. Numerical results and discussions In the present numerical experiment, Eq. (4.1) has been used to draw the graph as shown in Fig. 1, assuming x = 1. Fig. 2 shows three-dimensional figure for u(x, t) with respect to x and time t. In the present analysis, Eq. (4.1) also has been used to draw the graph as shown in Fig. 2. Figs. 1 and 2 have been drawn using the Mathematica software [36].
6. Conclusion This present analysis exhibits the applicability of the decomposition method to solve a fractional diffusion equation. In our previous papers [31,33] we have already as well as successfully exhibit the applicability of Adomian decomposition method to obtain a solution for dynamic system containing factional derivative of order 1/2 and 3/2, respectively. In this work, we demonstrate that
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this method is also well suited to solve fractional diffusion 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. When solutions are computed numerically, the rapid convergence is obvious. Moreover, no linearization or perturbation is required. It can avoid the difficulty of finding the inverse of Laplace Transform and can reduce the labor of perturbation method. This paper presents an analytical scheme to obtain the solution of a fractional diffusion equation. Acknowledgements We take this opportunity to express our sincere thanks and gratitude to Prof. F. Mainardi for providing us with his reprints [10,12]. The authors are grateful to Prof. F. Mainardi for his kind co-operation. References [1] K. Diethelm, An algorithm for the numerical solution of differential equations of fractional order, Elec. Trans. Numer. Anal. 5 (1997) 1–6. [2] K. Diethelm, A.D. Freed, The FracPECE subroutine for the numerical solution of differential equations of fractional order, in: S. Heinzel, T. Plesser (Eds.), GWDG-Berichte, Forschung und wissenschaftliches Rechnen: Beitra¨ge zum Heinz-Billing-Preis 1998 (No. 52), Gesellschaft fu¨r wissenschaftliche Datenverarbeitung, Go¨ttingen, 1999, pp. 57–71. [3] K. Diethelm, N.J. Ford, Analysis of fractional differential equations, J. Math. Anal. Appl. 265 (2002) 229–248. [4] K. Diethelm, N.J. Ford, A.D. Freed, A predictor–corrector approach for the numerical solution of fractional differential equations, Nonlinear Dyn. 29 (2002) 3–22. [5] K. Diethelm, N.J. Ford, Numerical solution of the Bagley–Torvik equation, Bit Numer. Math. 42 (3) (2002) 490–507. [6] K. Diethelm, A.D. Ford, On the solution of nonlinear fractional-order differential equations used in the modeling of viscoplasticity, in: F. Keil, W. Mackens, H. Voß, J. Werther (Eds.), Scientific Computing in Chemical Engineering II. Computational Fluid Dynamics, Reaction Engineering, and Molecular Properties, Springer-Verlag, Heidelberg, 1999, pp. 217–224. [7] K. Diethelm, N.J. Ford, The numerical solution of linear and nonlinear fractional differential equations involving fractional derivatives of several orders, Numerical Analysis Report, vol. 379, Manchester Centre for Computational Mathematics, Manchester, England, 2001. [8] R. Gorenflo, E.A. Abdel-Rehim, Approximation of time-fractional diffusion with central drift by difference schemes, Berlin Free University, 2003. Available from:
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