§1 Introduction

Appl. Math. J. Chinese Univ. Ser. B 2007, 22(1): 7-12

SOLUTION OF SYSTEM OF FRACTIONAL DIFFERENTIAL EQUATIONS BY ADOMIAN DECOMPOSITION METHOD Duan Junsheng1

An Jianye1

Xu Mingyu2

Abstract. The aim of this paper is to apply the relatively new Adomian decomposition method to solving the system of linear fractional, in the sense of Riemann-Liouville and Caputo respectively, differential equations. The solutions are expressed in terms of Mittag-Leffler functions of matric argument. The Adomian decomposition method is straightforward, applicable for broader problems and avoids the difficulties in applying integral transforms. As the order is 1, the result here is simplified to that of first order differential equation.

§1 Introduction The fractional calculus is used in many fields of science and engineering[1-3] . The solution of differential equation containing fractional derivatives is much involved and its classical analytic methods are mainly integral transforms, such as Laplace transform, Fourier transform, Mellin transform, etc.[1,2,4] . In recent years Adomian decomposition method is applied to solving fractional differential equations. This method efficiently works for initial value or boundary value problems, for linear or nonlinear, ordinary or partial differential equations, and even for stochastic systems[5] as well. By using this method Saha Ray et al[6-8] solved linear differential equations containing fractional derivative of order 1/2 or 3/2, and nonlinear differential equation containing fractional derivative of order 1/2. Recently, Atanackovic and Stankovic[9] introduced the system of fractional differential equations into the analysis of lateral motion of an elastic column fixed at one end and loaded at the other. Daftardar-Gejji and Babakhani[10] studied the system of linear fractional differential equations with constant coefficients using methods of linear algebra and proved existence and uniqueness theorems for the initial value problem. In this paper we apply the relatively new Adomian decomposition method to the solution of the system of linear fractional, in the sense of Riemann-Liouville and Caputo respectively, Received: 2006-05-22. MR Subject Classification: 26A33, 45K05. Keywords: fractional calculus, Adomian decomposition method, Mittag-Leffler function. Supported by the NNSF of China(10272067;10461005) and the Scientific Research Foundation of Tianjin Education Committee(20050404).

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differential equations.

§2 Preliminaries First we give the definitions of the fractional integral and derivatives. For more details the reader can refer to monographs, e.g.,[1,2]. Definition 2.1. Let f (t) be piecewise continuous on (0, +∞) and integrable (convergent if improper integral) on any finite subinterval of (0, +∞). Then fractional integral of f (t) of order β is defined as the convolution Z t tβ−1 (t − τ )β−1 f (τ )dτ, t > 0, β > 0. (2.1) J β f (t) := + ∗ f (t) = Γ(β) Γ(β) 0 We use the symbol C in [2] to denote the class of functions described in Definition 2.1. The functions of the form tλ η(t) or tλ (ln t)η(t), where λ > −1 and η(t) is an entire function, are of class C [2] . In this paper we will use the following equality J ν tµ =

Γ(µ + 1) µ+ν t , ν > 0, µ > −1, t > 0. Γ(µ + ν + 1)

(2.2)

Definition 2.2. Let f (t) be a function of class C and 0 < α < 1. Then Riemann-Liouville fractional derivative of f (t) of order α is defined as (if it exists) Dα f (t) :=


d J 1−α f (t) , t > 0. dt

(2.3)

Definition 2.3. Let f ′ (t) be a function of class C and 0 < α < 1. Then Caputo fractional derivative of f (t) of order α is defined as Dα f (t) := J 1−α f ′ (t).

(2.4)

The solutions of many fractional differential equations are involved with a class of important special functions – Mittag-Leffler functions. The Mittag-Leffler function with two parameters is defined by the series expansion[1,11], Eλ,ρ (z) =

∞ X

k=0

zk , λ > 0, ρ > 0, z ∈ C. Γ(λk + ρ)

(2.5)

The Mittag-Leffler function of matric argument Eλ,ρ (A) =

∞ X

k=0

Ak , λ > 0, ρ > 0 Γ(λk + ρ)

(2.6)

makes sense for arbitrary nth-order matrix A over complex field, where A0 = I, the nth-order identity matrix. In this paper by using Adomian decomposition method we solve the following fractional differential equation: Dα x(t) = Ax(t) + f (t), 0 < α < 1, 0 < t < T,

(2.7)

Duan Junsheng,et al.

SOLUTION OF SYSTEM OF FDE BY ADOMIAN ...

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where A is an nth-order real matrix, f (t) is a given n-dimensional column vector function, x (t) = [x1 (t), x2 (t), . . . , xn (t)]T is the unknown vector function, Dα denotes the operator of RiemannLiouville fractional derivative or Caputo fractional derivative, D α x (t) is defined componentwise. The entries of f (t) are not only of class C, but sufficiently smooth on (0, T ) to guarantee the existence of unique solution of Eq.(2.7).

§3 Decomposition method in the case of Riemann-Liouville fractional derivative Consider the fractional differential equation (2.7) in the case of Riemann-Liouville fractional derivative Dα x (t) = Ax (t) + f (t), 0 < α < 1, 0 < t < T (3.1) with the initial condition 

 J 1−α x (t) t=0+ = c,

(3.2)

where c is a specific initial vector. The initial value of fractional integral is involved in the case of Riemann-Liouville fractional derivative[1] . Further we suppose there exist ǫ > 0, M > 0 and β > −1 such that |fi (t)| ≤ M tβ holds on the interval 0 < t < ǫ for each entry of f (t). Operating the two sides of Eq.(3.1) with J α and using the relation[1] J α Dα f (t) = f (t) −

 tα−1  1−α J f (t) t=0+ Γ(α)

(3.3)

and the initial condition (3.2), we obtain x (t) =

tα−1 c + J α f (t) + J α Ax (t). Γ(α)

(3.4)

In the light of Adomian decomposition method we decompose x (t) into x (t) =

∞ X

xm (t)

(3.5)

m=0

with x0 (t) =

tα−1 c + J α f (t). Γ(α)

(3.6)

Then we can write by Eq.(3.4) x1 (t) = J α Ax0 (t), x2 (t) = J α Ax1 (t), . . . , xm (t) = J α Axm−1 (t), m ≥ 1.

(3.7)

Thus we have xm (t) = (J α A)m x0 (t), and using Eq.(3.6) yields xm (t) = Am J mα Let xh (t) =

∞ X

m=0

Am J mα

tα−1 c + Am J mα+α f (t). Γ(α)

∞ X tα−1 c, xp (t) = Am J mα+α f (t). Γ(α) m=0

(3.8)

(3.9)

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With the help of (2.2) and (2.1) we have xh (t) =

∞ ∞ X X Am tmα+α−1 Am tmα+α−1 c, xp (t) = ∗ f (t). Γ(mα + α) Γ(mα + α) m=0 m=0

(3.10)

In terms of Mittag-Leffler functions of matric argument (2.6), Eq. (3.10) can be written as xh (t) = tα−1 Eα,α (Atα )c, xp (t) = tα−1 Eα,α (Atα ) ∗ f (t).

(3.11)

Thus the solution of initial value problem (3.1),(3.2) is given x (t) = xh (t) + xp (t) = tα−1 Eα,α (Atα )c + tα−1 Eα,α (Atα ) ∗ f (t).

(3.12)

We note that xh (t) is the solution of the homogeneous equation corresponding to Eq.(3.1), and satisfies the initial condition (3.2), while xp (t) is the solution of Eq.(3.1), satisfying the homogeneous initial condition. If the dimension is one, Eq.(3.12) coincides with the Podlubny’s result([1],p140) obtained by using Laplace transform.

§4 Decomposition method in the case of Caputo fractional derivative Consider the initial value problem for fractional differential equation in the case of Caputo fractional derivative Dα x (t) = Ax (t) + f (t), 0 < α < 1, 0 < t < T, (4.1) x (0+ ) = c.

(4.2)

In this case the initial condition in traditional form is involved. Here we further suppose there exist ǫ > 0, M > 0 and β > −α such that |fi (t)| ≤ M tβ holds on the interval 0 < t < ǫ for each entry of f (t). According to the definitions (2.1) and (2.4) we have Z t Z (t − u)α−1 u (u − τ )−α ′ du J α Dα x (t) = x (τ )dτ. Γ(α) Γ(1 − α) 0 0 Exchanging the sequence of integrals and making use of the equality Z t Γ(a)Γ(b) (t − u)a−1 (u − τ )b−1 du = (t − τ )a+b−1 , a > 0, b > 0, Γ(a + b) τ

(4.3)

yield J α Dα x (t) = x (t) − x (0+ ).

(4.4)

We note that problem (4.1),(4.2) can be converted to problem (3.1),(3.2) and to be solved through the relation[11] t−α Dα f (t) = Dα f (t) + f (0+ ), (4.5) Γ(1 − α) but here we want to show the application of decomposition method.

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Operating Eq.(4.1) with J α and using Eq.(4.4) and initial condition (4.2), we have x (t) = c + J α Ax (t) + J α f (t). We assume x (t) =

P∞

m=0 xm (t)

(4.6)

to be the solution of Eq.(4.1), where

x0 (t) = c + J α f (t), xm (t) = J α Axm−1 (t), m ≥ 1.

Then we can write xm (t) = Am J mα x0 (t) =

Am tmα c + Am J mα+α f (t). Γ(mα + 1)

(4.7)

The summation of xm (t) gives the solution in terms of Mittag-Leffler functions of matric argument x (t) = xh (t) + xp (t) = Eα,1 (Atα )c + tα−1 Eα,α (Atα ) ∗ f (t), (4.8) where xh (t) = Eα,1 (Atα )c, xp (t) = tα−1 Eα,α (Atα ) ∗ f (t) are the solution of the homogeneous equation with initial condition (4.2) and the solution of Eq.(4.1) with homogeneous initial condition, respectively. We note that here xh (t) agrees with Daftardar-Gejji and Babakhani’s result[10] .

§5 Conclusion It is shown that the applicability of Adomian decomposition method to solve the system of fractional differential equations of fractional order α(0 < α < 1). The Adomian decomposition method is straightforward, applicable for broader problems, and gives quite satisfactory results in terms of Mittag-Leffler function of matric argument. It can avoid the difficulty of finding the inverse of Laplace transform. If α = 1, Eqs.(3.12) and (4.8) are all simplified to the result of first order differential equation x (t) = exp(At)c + exp(At) ∗ f (t).

References 1 Podlubny I. Fractional Differential Equations, San Diego: AcademicPress, 1999. 2 Miller K S, Ross B. An Introduction to the Fractional Calculus and Fractional Differential Equations, New York: Wiley, 1993. 3 Shimizu N, Zhang W. Fractional calculus approach to dynamic problems of viscoelastic materials, JSME Series C–Mechanical Systems, Machine Elements and Manufacturing, 1999, 42: 825-837. 4 Duan J S. Time- and space-fractional partial differential equations, J Math Phys, 2005, 46: 13504-13511. 5 Adomian G. Nonlinear Stochastic Operator Equations, New York: Academic Press, 1986. 6 Saha Ray S, Poddar B P, Bera R K. Analytical solution of a dynamic system containing fractional derivative of order 1/2 by Adomian decomposition method, ASME J Appl Mech, 2005, 72: 290295.

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7 Saha Ray S, Bera R K. Analytical solution of the Bagley Torvik equation by Adomian decomposition method, Appl Math Comput, 2005, 168: 398-410. 8 Saha Ray S, Bera R K. An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl Math Comput, 2005, 167: 561-571. 9 Atanackovic T M, Stankovic B. On a system of differential equations with fractional derivatives arising in rod theory, J Phys A, 2004, 37: 1241-1250. 10 Daftardar-Gejji V, Babakhani A. Analysis of a system of fractional differential equations, J Math Anal Appl, 2004, 293: 511-522. 11 Mainardi F, Gorenflo R. On Mittag-Leffler-type functions in fractional evolution processes, J Comput Appl Math, 2000, 118: 283-299.

1 College of Science, Tianjin University of Commerce, Tianjin 300134, China. 2 Institute of Mathematics and Systematical Science, Shandong University, Jinan 250100, China.