Solution to the Linear Fractional Differential Equation Using Adomian Decomposition Method

Abstract

We obtain the analytical general solution of the linear fractional differential equations with constant coefficients by Adomian decomposition method under nonhomogeneous initial value condition, which is in the sense of the Caputo fractional derivative.

1. Introduction

Fractional differential equations are hot topics both in mathematics and physics. Recently, the fractional differential equations have been the subject of intensive research. There are several methods to obtain the solution, such as the Laplace transform method, power series method, and Green function method. Many remarkable results for the fractional differential equations can be found in the literature [1–11]. In particular, the Adomian decomposition method has attracted the attention of many mathematicians [12–15].

For a better understanding of the fractional derivatives and for a physical understanding of the fractional equations, the readers can refer to the recent publications in [16, 17]. Ebaid [18] suggested a modification of the Adomian method, and a few iterations lead to exact solution. Das [19] compared the variational iteration method with the Adomian method for fractional equations and found that the variational iteration method is much more effective. For other methods of the fractional differential equations, especially the homotopy perturbation method, variational iteration method and differential transform method were presented in [20, 21].

Consider the following 𝑛-term fractional differential equation with constant coefficients:𝑎_𝑛^𝐶𝐷^𝛽𝑛𝑦(𝑡)+𝑎^𝑛−1^𝐶𝐷^𝛽𝑛−1𝑦(𝑡)+⋯+𝑎¹^𝐶𝐷^𝛽1𝑦(𝑡)+𝑎⁰^𝐶𝐷^𝛽0𝑦(𝑡)=𝑓(𝑡),(1.1) where 𝑛+1>𝛽≥𝑛>𝛽_𝑛−1>⋯>𝛽₁>𝛽₀ and 𝑎_𝑖(𝑖=0,1,…,𝑛) is a real constant. In [12], the authors obtain the particular solution of (1.1) of the homogeneous initial value problem of the form 𝑦^𝑖(0)=0,𝑖=0,1,…,𝑛.(1.2)

However, it seems also more meaningful and more complicated for solving general solution of (1.1) under nonhomogeneous initial value condition. Therefore, in this paper, we will remove the restriction of the homogeneous initial value, consider the nonhomogeneous initial value problems of the form 𝑦^𝑗𝑖(0)=𝑐_𝑖𝑗𝑖,𝑖=0,1,…,𝑛,𝑗_𝑖=1,2,…,𝑙_𝑖,𝑙_𝑖−1⩽𝛽_𝑖<𝑙_𝑖,(1.3) and obtain the analytical general solution of (1.1), which generalizes the result in [12].

We organize the paper as follows. In Section 2, we give some basic definitions and properties. In Section 3, we obtain the analytical general solution of the linear fractional differential equations by Adomian decomposition method. Some explicit examples are given in Section 4.

2. Basic Definitions and Notations

Definition 2.1 (see [1]). The Riemann-Liouville integral of order 𝑝 is defined by _𝑎𝐷^𝑡−𝑝𝑓(𝑡)=1Γ(𝑝)^𝑡𝑎(𝑡−𝜏)^𝑝−1𝑓(𝜏)𝑑𝜏,𝑝>0.(2.1) From Definition 2.1, we clearly see that _𝑎𝐷^𝑡−𝛼(𝑡−𝑎)^𝜈=Γ(1+𝜈)Γ(1+𝜈+𝛼)(𝑡−𝑎)^𝜈+𝛼,(2.2)_𝑎𝐷^𝛼𝑡(𝑡−𝑎)^𝜈=Γ(1+𝜈)Γ(1+𝜈−𝛼)(𝑡−𝑎)^𝜈−𝛼,(2.3) where 𝛼>0 and 𝜈 is a real number.

Definition 2.2 (see [1]). For 𝑓(𝑡)∈𝐶^𝑚(𝑚∈𝑁), the Caputo fractional derivative of 𝑓(𝑡) is defined by ^𝐶𝑎𝐷^𝑝𝑡𝑓(𝑡)=⎧⎪⎨⎪⎩𝐷^𝑚−𝑝𝑑𝑓^(𝑚)(𝑡),𝑚−1<𝑝<𝑚,_(𝑚)𝑑𝑡^𝑚𝑓(𝑡),𝑝=𝑚,_𝑎𝐷^𝑝𝑡𝑓(𝑡),𝑝⩽0.(2.4) Therefore, ^𝑐𝑎𝐷^𝑡−𝛼(𝑡−𝑎)^𝜈=_𝑎𝐷^𝑡−𝛼(𝑡−𝑎)^𝜈=Γ(1+𝜈)Γ(1+𝜈+𝛼)(𝑡−𝑎)^𝜈+𝛼.(2.5)

Lemma 2.3 (see [1]). If 𝑓(𝑡) is continuous, then ^𝐶𝑎𝐷^𝑡−𝑛𝑓^(𝑛)(𝑡)=_𝑎𝐷^𝑡−𝑛𝑓^(𝑛)(𝑡)=𝑓(𝑡)−^𝑛−1_𝑗=0𝑓^(𝑗)(𝑎)(𝑡−𝑎)^𝑗Γ(𝑗+1),𝑛∈𝑍.(2.6)

Lemma 2.4 (see [1]). If 𝑓(𝑡) is continuous, then ^𝐶𝑎𝐷^𝑡−𝑝^𝐶𝑎𝐷^𝑡−𝑞𝑓(𝑡)=_𝑎𝐷^𝑡−𝑝_𝑎𝐷^𝑡−𝑞𝑓(𝑡)=_𝑎𝐷^{𝑡−𝑝−𝑞}𝑓(𝑡)=^𝐶𝑎𝐷^{𝑡−𝑝−𝑞}𝑓(𝑡)=^𝐶𝑎𝐷^𝑡−𝑞^𝐶𝑎𝐷^𝑡−𝑝𝑓(𝑡).(2.7)

Lemma 2.5. If 𝑓(𝑡) is continuous, then ^𝐶𝑎𝐷^𝑡−𝑝^𝐶𝑎𝐷^𝑞𝑡𝑓(𝑡)=_𝑎𝐷^{𝑡𝑞−𝑝}𝑓(𝑡)−^𝑚−1_𝑗=0𝑓^(𝑗)(𝑎)(𝑡−𝑎)^{𝑗−𝑞+𝑝}Γ(𝑗−𝑞+𝑝+1),(2.8) where 𝑚−1<𝑞<𝑚 and 𝑞⩽𝑝.

Proof. From Definition 2.2 and Lemmas 2.3–2.4, we get ^𝐶𝑎𝐷^𝑡−𝑝^𝐶𝑎𝐷^𝑞𝑡𝑓(𝑡)==_𝑎𝐷^𝑡−𝑝_𝑎𝐷^{𝑡𝑞−𝑚}𝑓^(𝑚)(𝑡)=_𝑎𝐷^{𝑡−𝑝+𝑞}_𝑎𝐷^𝑡−𝑞_𝑎𝐷^{𝑡𝑞−𝑚}𝑓^(𝑚)(𝑡)_𝑎𝐷^{𝑡−𝑝+𝑞}_𝑎𝐷^𝑡−𝑚(𝑡)𝑓^(𝑚)(𝑡)=_𝑎𝐷^{𝑡−𝑝+𝑞}𝑓(𝑡)−^𝑚−1_𝑗=0𝑓^(𝑗)(𝑎)(𝑡−𝑎)^𝑗=Γ(𝑗+1)_𝑎𝐷^{𝑡𝑞−𝑝}𝑓(𝑡)−^𝑚−1_𝑗=0𝑓^(𝑗)(𝑎)(𝑡−𝑎)^{𝑗−𝑞+𝑝}Γ(𝑗−𝑞+𝑝+1).(2.9) It is easy to see that ^𝐶𝑎𝐷^𝑡−𝑝^𝐶𝑎𝐷^𝑝𝑡𝑓(𝑡)=𝑓(𝑡)−^𝑚−1_𝑗=0𝑓^(𝑗)(𝑎)(𝑡−𝑎)^𝑗Γ(𝑗+1).(2.10)

Proposition 2.6 (see [12]). One has ^∞_𝑘1=0⋯^∞_𝑘_𝑛=0𝑎_{𝑘1,𝑘2⋯𝑘𝑛−1𝑘𝑛}=^∞_𝑚₌₀_{𝑘1,…,𝑘𝑛−1,𝑘𝑛𝑘⩾01+⋯+𝑘𝑛−1+𝑘𝑛=𝑚}𝑎_{𝑘1,𝑘2⋯𝑘𝑛−1𝑘𝑛}.(2.11)

Proposition 2.7 (see [12]). More over, one has ^∞_𝑚=0_{𝑘1,…,𝑘𝑛−1,𝑘𝑛𝑘⩾01+⋯+𝑘𝑛−1+𝑘𝑛=𝑚}𝑎_{𝑘1,𝑘2⋯𝑘𝑛−1𝑘𝑛}=^∞_𝑠=0_{𝑘1,…,𝑘𝑛−1𝑘⩾01+⋯+𝑘𝑛−1∞=𝑠}_𝑘𝑛=0𝑎_{𝑘1,𝑘2⋯𝑘𝑛−1𝑘𝑛}.(2.12)

3. The Analytical Solution of the Linear Constant Coefficient Fractional Differential Equation

For simplicity, if 𝑎=0, then we denote ^𝐶𝑎𝐷^𝑝𝑡 or ^𝐶𝑎𝐷^𝑡−𝑝 by ^𝐶𝐷^𝑝 or ^𝐶𝐷^−𝑝.

In this section, we use Adomian decomposition method to discuss the general form of the linear fractional differential equations with constant coefficients, and apply and some basic transformation and integration to obtain the solution of the equations.

Let us consider the following 𝑛-term linear fractional differential equations with constant coefficients: 𝑎_𝑛^𝐶𝐷^𝛽𝑛𝑦𝑦(𝑡)+𝑎^𝑛−1^𝐶𝐷^𝛽𝑛−1𝑦(𝑡)+⋯+𝑎¹^𝐶𝐷^𝛽1𝑦(𝑡)+𝑎⁰^𝐶𝐷^𝛽0𝑦(𝑡)=𝑓(𝑡),_(𝑗𝑖)(0)=𝑐_𝑖𝑗𝑖,𝑖=0,1,…,𝑛,𝑗_𝑖=1,2,…,𝑙_𝑖,𝑙_𝑖−1⩽𝛽_𝑖<𝑙_𝑖,(3.1) where 𝑛+1>𝛽_𝑛⩾𝑛>𝛽_𝑛−1⋯>𝛽₁>𝛽₀,𝑎_𝑖and𝑐_𝑖𝑗𝑖 are real constants, ^𝐶𝐷^𝑝=^𝐶0𝐷^𝑝𝑡 denotes Caputo fractional derivative of order 𝛼.

Applying ^𝐶𝐷^−𝛽𝑛 to both sides of (1.1) and utilizing Lemma 2.5, we get 𝑦(𝑡)+𝑎^𝑛−1𝑎_𝑛𝐶𝐷^{𝛽𝑛−1−𝛽𝑛}𝑦(𝑡)+⋯+𝑎⁰𝑎_𝑛𝐶=1𝐷^{𝛽0−𝛽𝑛}𝑦(𝑡)𝑎_𝑛𝐶𝐷^−𝛽𝑛𝑓(𝑡)+1𝑎_{𝑛𝑛𝑖=0}𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑦^(𝑗𝑖)(0)𝑡^{𝛽𝑛−𝛽𝑖+𝑗𝑖}Γ1+𝛽_𝑛−𝛽_𝑖+𝑗_𝑖.(3.2)

By the Adomian decomposition method, we obtain the recursive relationship as follows: 𝑦₀(𝑡)=1𝑎_𝑛𝐶𝐷^−𝛽𝑛𝑓(𝑡)+1𝑎_{𝑛𝑛𝑖=0}𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑦^(𝑗𝑖)(0)𝑡^{𝛽𝑛−𝛽𝑖+𝑗𝑖}𝑦₁Γ1+𝛽_𝑛−𝛽_𝑖+𝑗_𝑖,(𝑡)=−𝑎^𝑛−1𝑎_𝑛𝐶𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐶𝐷^{𝛽0−𝛽𝑛}𝑦⁰𝑦(𝑡),₂(𝑡)=(−1)²𝑎𝑎^𝑛−1_𝑛𝐶𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐶𝐷^{𝛽0−𝛽𝑛}²𝑦⋮𝑦₀(𝑡),_𝑠(𝑡)=(−1)^𝑠𝑎𝑎^𝑛−1_𝑛𝐶𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐶𝐷^{𝛽0−𝛽𝑛}^𝑠⋮𝑦₀(𝑡),(3.3)

By Adomian decomposition method, adding all terms of the recursion, we obtain the solution of (1.1) as 𝑦(𝑡)=^∞_𝑠=0=𝑦_𝑠(𝑡)_∞_𝑠=0(−1)^𝑠𝑎𝑎^𝑛−1_𝑛𝐶𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐶𝐷^{𝛽0−𝛽𝑛}^𝑠=1𝑦₀(𝑡)𝑎_{𝑛∞𝑠=0}(−1)^𝑠𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐷^{𝛽0−𝛽𝑛}^𝑠+1𝐷^−𝛽𝑛𝑓(𝑡)𝑎_{𝑛∞𝑠=0}(−1)^𝑠𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐷^{𝛽0−𝛽𝑛}^𝑠𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑦^(𝑗𝑖)(0)𝑡^{𝛽𝑛−𝛽𝑖+𝑗𝑖}Γ1+𝛽_𝑛−𝛽_𝑖+𝑗_𝑖.(3.4) Let 𝐼₁=1𝑎_{𝑛∞𝑠=0}(−1)^𝑠𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐷^{𝛽0−𝛽𝑛}^𝑠𝐼𝐷^−𝛽𝑛𝑓(𝑡)₂=1𝑎_{𝑛∞𝑠=0}(−1)^𝑠𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐷^{𝛽0−𝛽𝑛}^𝑠𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑦^(𝑗𝑖)(0)𝑡^{𝛽𝑛−𝛽𝑖+𝑗𝑖}Γ1+𝛽_𝑛−𝛽_𝑖+𝑗_𝑖.(3.5) Then,𝑦(𝑡)=𝐼₁+𝐼₂.(3.6)

Next, we estimate 𝐼₁ and 𝐼₂, respectively.

For 𝐼₁, by [12] we obtain 1𝑎_{𝑛∞𝑠=0}(−1)^𝑠𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐷^{𝛽0−𝛽𝑛}^𝑠=𝐷^−𝛽𝑛𝑓(𝑡),_𝑡01𝑎_{𝑛∞𝑚=0}(−1)^𝑚𝑚!_{𝑘0,𝑘1,…,𝑘𝑛−2𝑘⩾00+𝑘1+⋯+𝑘𝑛−2=𝑚}𝑚;𝑘₀×_{𝑛−2𝑝=0},𝑘₁,…,𝑘_𝑛−2𝑎𝑎^𝑝_𝑛^𝑘𝑝(𝑡−𝜏)^{(𝛽𝑛−𝛽𝑛−1)𝑚+𝛽𝑛+∑𝑛−2𝑗=0(𝛽𝑛−1−𝛽𝑗)𝑘𝑗−1}×𝐸^𝛽(𝑚)_{𝑛−𝛽𝑛−1,𝛽𝑛+∑𝑛−2𝑗=0𝛽𝑛−1−𝛽𝑗𝑘𝑗}−𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−𝛽𝑛−1}𝑓(𝜏)𝑑𝜏.(3.7)

For 𝐼₂, by the initial conditions (2.10) we get 1𝑎_{𝑛∞𝑠=0}(−1)^𝑠𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−1−𝛽𝑛}+⋯+𝑎⁰𝑎_𝑛𝐷^{𝛽0−𝛽𝑛}^𝑠𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑦^(𝑗𝑖)(0)𝑡^{𝛽𝑛−𝛽𝑖+𝑗𝑖}=1Γ1+𝛽_𝑛−𝛽_𝑖+𝑗_𝑖𝑎_{𝑛∞𝑠=0}(−1)^𝑠_{𝑘0,𝑘1,…,𝑘𝑛−1}_{𝑘⩾00+𝑘1+⋯+𝑘𝑛−1=𝑠}𝑠!𝑘₀!𝑘₁!⋯𝑘_𝑛−1!𝑎𝑎^𝑛−1_𝑛^𝑘𝑛−1𝑎𝑎^𝑛−2_𝑛^𝑘𝑛−2⋯𝑎⁰𝑎_𝑛^𝑘0×𝐷^{𝑘𝑛−1(𝛽𝑛−1−𝛽𝑛)+𝑘𝑛−2(𝛽𝑛−2−𝛽𝑛)+⋯+𝑘0(𝛽0−𝛽𝑛)𝑛}_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖𝑡^{𝛽𝑛−𝛽𝑖+𝑗𝑖}Γ1+𝛽_𝑛−𝛽_𝑖+𝑗_𝑖.(3.8)

Using formulas (2.2) and (2.3), the above expression can be written as1𝑎_{𝑛∞𝑠=0}(−1)^𝑠_{𝑘0,𝑘1,…,𝑘𝑛−1}_{𝑘⩾00+𝑘1+⋯+𝑘𝑛−1=𝑠}𝑠!𝑘₀!𝑘₁!⋯𝑘_𝑛−1!𝑎𝑎^𝑛−1_𝑛^𝑘𝑛−1𝑎𝑎^𝑛−2_𝑛^𝑘𝑛−2⋯𝑎⁰𝑎_𝑛^𝑘0×^𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖Γ(𝒜)𝑡^{𝛽𝑛−𝛽𝑖+𝑗𝑖+𝑘𝑛−1(𝛽𝑛−𝛽𝑛−1)+𝑘𝑛−2(𝛽𝑛−𝛽𝑛−2)+⋯+𝑘0(𝛽𝑛−𝛽0)}=1Γ(𝒜)⋅Γ𝒜+𝑘_𝑛−1𝛽_𝑛−𝛽_𝑛−1+𝑘_𝑛−2𝛽_𝑛−𝛽_𝑛−2+⋯+𝑘₀𝛽_𝑛−𝛽₀𝑎_{𝑛∞𝑠=0}(−1)^𝑠_{𝑘0,𝑘1,…,𝑘𝑛−2}_{𝑘⩾00+𝑘1+⋯+𝑘𝑛−1=𝑠}𝑠!𝑘₀!𝑘₁!⋯𝑘_𝑛−1!𝑎𝑎^𝑛−1_𝑛^𝑘𝑛−1𝑎𝑎^𝑛−2_𝑛^𝑘𝑛−2⋯𝑎⁰𝑎_𝑛^𝑘0×^𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖𝑡^{𝛽𝑛−𝛽𝑖+𝑗𝑖+𝑘𝑛−1(𝛽𝑛−𝛽𝑛−1)+𝑘𝑛−2(𝛽𝑛−𝛽𝑛−2)+⋯+𝑘0(𝛽𝑛−𝛽0)}=Γ𝒜+𝑘_𝑛−1𝛽_𝑛−𝛽_𝑛−1+𝑘_𝑛−2𝛽_𝑛−𝛽_𝑛−2+⋯+𝑘₀𝛽_𝑛−𝛽₀_𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖𝐷^{(𝛽𝑖−𝑗𝑖−1)}⎡⎢⎢⎢⎢⎣1𝑎_{𝑛∞𝑠=0}(−1)^𝑠_{𝑘0,𝑘1,…,𝑘𝑛−1}_{𝑘⩾00+𝑘1+⋯+𝑘𝑛−1=𝑠}𝑠!×𝑎_𝑛−1𝑘₀!𝑘₁!⋯𝑘_𝑛−1!𝑎_𝑛^𝑘𝑛−1𝑎𝑎^𝑛−2_𝑛^𝑘𝑛−2⋯𝑎⁰𝑎_𝑛^𝑘0×𝑡^{𝛽𝑛−1+𝑘𝑛−1(𝛽𝑛−𝛽𝑛−1)+𝑘𝑛−2(𝛽𝑛−𝛽𝑛−2)+⋯+𝑘0(𝛽𝑛−𝛽0)}Γ𝛽_𝑛+𝑘_𝑛−1𝛽_𝑛−𝛽_𝑛−1+𝑘_𝑛−2𝛽_𝑛−𝛽_𝑛−2+⋯+𝑘₀𝛽_𝑛−𝛽₀⎤⎥⎥⎥⎥⎦,(3.9) where 𝒜 denotes 1+𝛽_𝑛−𝛽_𝑖+𝑗_𝑖.

Using Propositions 2.6 and 2.7, the above solution is equivalent to the following form: ^𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖𝐷^{(𝛽𝑖−𝑗𝑖−1)}⎡⎢⎢⎢⎢⎣1𝑎_{𝑛∞𝑘𝑛−1∞=0}_𝑚=0_{𝑘0,𝑘1,…,𝑘𝑛−2𝑘⩾00+𝑘1+⋯+𝑘𝑛−2=𝑚}(−1)^{𝑘𝑛−1+𝑚}𝑘_𝑛−1+𝑚!×𝑎_𝑛−1𝑘₀!𝑘₁!⋯𝑘_𝑛−1!𝑎_𝑛^𝑘𝑛−1𝑎𝑎^𝑛−2_𝑛^𝑘𝑛−2⋯𝑎⁰𝑎_𝑛^𝑘0×𝑡^{𝛽𝑛−1+𝑘𝑛−1(𝛽𝑛−𝛽𝑛−1)+𝑘𝑛−2(𝛽𝑛−𝛽𝑛−2)+⋯+𝑘0(𝛽𝑛−𝛽0)}=Γ𝛽_𝑛+𝑘_𝑛−1𝛽_𝑛−𝛽_𝑛−1+𝑘_𝑛−2𝛽_𝑛−𝛽_𝑛−2+⋯+𝑘₀𝛽_𝑛−𝛽₀_𝑛⎤⎥⎥⎥⎥⎦_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖𝐷^{(𝛽𝑖−𝑗𝑖−1)}⎡⎢⎢⎢⎢⎣1𝑎_{𝑛∞𝑚=0}(−1)^𝑚𝑚!_{𝑘0,𝑘1,…,𝑘𝑛−2𝑘⩾00+𝑘1+⋯+𝑘𝑛−2=𝑚}𝑚!×_{𝑛−2𝑟=0}𝑘₀!𝑘₁!⋯𝑘_𝑛−2!𝑎𝑎^𝑟_𝑛^𝑘𝑟𝑡^{(𝛽𝑛−𝛽𝑛−1∑)𝑚+𝑛−2𝜉=0(𝛽𝑛−1−𝛽𝜉)𝑘𝜉+𝛽𝑛−1}×^∞_𝑘𝑛₋₁₌₀(−1)^𝑘𝑛−1𝑎𝑎^𝑛−1_𝑛^𝑘𝑛−1𝑘_𝑛−1+𝑚!×𝑘𝑡_𝑛−1!_{𝑘𝑛−𝑙(𝛽𝑛−𝛽𝑛−1)}=_𝑛Γ𝑘_𝑛−1𝛽_𝑛−𝛽_𝑛−1+𝛽_𝑛−𝛽_𝑛−1𝑚+∑^{𝑛−2𝛿=0}𝛽_𝑛−1−𝛽_𝛿𝑘_𝛿+𝛽_𝑛⎤⎥⎥⎥⎥⎦_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖𝐷^{(𝛽𝑖−𝑗𝑖−1)}⎡⎢⎢⎢⎢⎣1𝑎_{𝑛∞𝑚=0}(−1)^𝑚𝑚!_{𝑘0,𝑘1,…,𝑘𝑛−2𝑘⩾00+𝑘1+⋯+𝑘𝑛−2=𝑚}𝑚;𝑘₀×_{𝑛−2𝑟=0},𝑘₁,…,𝑘_𝑛−2𝑎𝑎^𝑟_𝑛^𝑘𝑟𝑡^{(𝛽𝑛−𝛽𝑛−1∑)𝑚+𝑛−2𝜉=0(𝛽𝑛−1−𝛽𝜉)𝑘𝜉+𝛽𝑛−1}×𝐸^𝛽(𝑚)_{𝑛−𝛽𝑛−1,∑𝑛−2𝛿=0𝛽𝑛−1−𝛽𝑙𝑘𝛿+𝛽𝑛}−𝑎𝑎^𝑛−1_𝑛𝑡^{𝛽𝑛−𝛽𝑛−1}⎤⎥⎥⎥⎥⎦.(3.10)

Therefore, 𝑦(𝑡)=𝐼₁+𝐼₂=^𝑡01𝑎_{𝑛∞𝑚=0}(−1)^𝑚𝑚!_{𝑘0,𝑘1,…,𝑘𝑛−2𝑘⩾00+𝑘1+⋯+𝑘𝑛−2=𝑚}𝑚;𝑘₀×_{𝑛−2𝑝=0},𝑘₁,…,𝑘_𝑛−2𝑎𝑎^𝑝_𝑛^𝑘𝑝(𝑡−𝜏)^{(𝛽𝑛−𝛽𝑛−1)𝑚+𝛽𝑛+∑𝑛−2𝑗=0(𝛽𝑛−1−𝛽𝑗)𝑘𝑗−1}×𝐸^𝛽(𝑚)_{𝑛−𝛽𝑛−1,𝛽𝑛+∑𝑛−2𝑗=0(𝛽𝑛−1−𝛽𝑗)𝑘𝑗}−𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−𝛽𝑛−1}+𝑓(𝜏)𝑑𝜏_𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖𝐷^{(𝛽𝑖−𝑗𝑖−1)}⎡⎢⎢⎢⎢⎣1𝑎_{𝑛∞𝑚=0}(−1)^𝑚𝑚!_{𝑘0,𝑘1,…,𝑘𝑛−2𝑘⩾00+𝑘1+⋯+𝑘𝑛−2=𝑚}𝑚;𝑘₀×_{𝑛−2𝑟=0},𝑘₁,…,𝑘_𝑛−2𝑎𝑎^𝑟_𝑛^𝑘𝑟𝑡^{(𝛽𝑛−𝛽𝑛−1∑)𝑚+𝑛−2𝜉=0(𝛽𝑛−1−𝛽𝜉)𝑘𝜉+𝛽𝑛−1}×𝐸^𝛽(𝑚)_{𝑛−𝛽𝑛−1,𝑛−2∑𝛿=0𝛽𝑛−1−𝛽𝑙𝑘𝛿+𝛽𝑛}−𝑎𝑎^𝑛−1_𝑛𝑡^{𝛽𝑛−𝛽𝑛−1}⎤⎥⎥⎥⎥⎦,(3.11) where 𝑚;𝑘₀,𝑘₁,…,𝑘_𝑛−2=𝑘₀!𝑘₁!𝑚⋯!𝑘_𝑛−2!,(3.12) and 𝐸^{(𝑖)𝜆,𝜇}(𝑦) is the Mittag-Leffler function 𝐸^{(𝑖)𝜆,𝜇}(𝑦)=𝑑^𝑖𝑑𝑦^𝑖𝐸_𝜆,𝜇(𝑦)=^∞_𝑗=0(𝑖+𝑗)!𝑦^𝑗𝑗!Γ(𝜆𝑗+𝜆𝑖+𝜇).(3.13) Substituting the Green function 𝐺_𝑛(𝑡)=1𝑎_{𝑛∞𝑚=0}(−1)^𝑚𝑚!_{𝑘0,𝑘1,…,𝑘𝑛−2𝑘⩾00+𝑘1+⋯+𝑘𝑛−2=𝑚}𝑚;𝑘₀×_𝑛−2,𝑘₁,…,𝑘_𝑛−2_𝑝=0𝑎𝑎^𝑝_𝑛^𝑘𝑝𝑡^{(𝛽𝑛−𝛽𝑛−1)𝑚+𝛽𝑛+∑𝑛−2𝑗=0(𝛽𝑛−1−𝛽𝑗)𝑘𝑗−1}×𝐸^𝛽(𝑚)_{𝑛−𝛽𝑛−1,𝛽𝑛+∑𝑛−2𝑗=0(𝛽𝑛−1−𝛽𝑗)𝑘𝑗}−𝑎𝑎^𝑛−1_𝑛𝐷^{𝛽𝑛−𝛽𝑛−1}(3.14) into the above expression, we know that𝑦(𝑡)=^𝑡0𝐺_𝑛(𝑡−𝜏)𝑓(𝜏)𝑑𝜏+^𝑛_𝑖=0𝑎_𝑖𝑙^𝑖−1_𝑗_𝑖=0𝑐_𝑖𝑗𝑖𝐺^{(𝛽𝑖−𝑗𝑖𝑛−1)}(𝑡)(3.15) is the analytical general solution of (1.1).

4. Illustrative Examples

In order to verify our conclusions, we give some examples.

(1). Consider an initial value problem for the relaxation-oscillation equation (see [1]) 𝑦𝐷^𝐶𝛼𝑦(𝑡)+𝐴𝑦(𝑡)=𝑓(𝑡),𝑡>0,_(𝑗)(0)=𝑏_𝑗,𝑗=1,2,…,𝑚−1,𝑚−1<𝛼<𝑚,(4.1) where 𝑏_𝑗 are real constants.

Utilizing Lemma 2.5 and applying ^𝐶𝐷^−𝛼 to both sides of (4.1), we obtain 𝑦(𝑡)+𝐴𝐷^−𝛼𝑦(𝑡)=𝐷^−𝛼𝑓(𝑡)+^𝑚−1_𝑗=0𝑦^(𝑗)(0)𝑡^𝑗Γ(𝑗+1).(4.2)

According to the above procedure of solving the linear fractional differential equations with constant coefficients and using the Adomian decomposition method, let 𝑦₀(𝑡)=𝐷^−𝛼𝑓(𝑡)+^𝑚−1_𝑗=0𝑦^(𝑗)(0)𝑡^𝑗𝑦Γ(𝑗+1)₁,𝑦(𝑡)=−𝐴𝐷^−𝛼𝑦₀(𝑡),₂𝑦(𝑡)=⋮−𝐴𝐷^−𝛼𝑦₁(𝑡)=(−𝐴)²𝐷^−2𝛼𝑦₀(𝑡),_𝑠(𝑡)=⋮−𝐴𝐷^−𝛼𝑦_𝑠−1(𝑡)=(−𝐴)^𝑠𝐷^−𝑠𝛼𝑦₀(𝑡),(4.3)

Adding all of the above terms, we obtain the solution of the equation by Adomian decomposition method as follows: 𝑦(𝑡)=^∞_𝑠=0=𝑦_𝑠(𝑡)_∞_𝑠=0=(−𝐴)^𝑠𝐷^−𝑠𝛼𝑦₀(𝑡)_∞_𝑠=0(−𝐴)^𝑠𝐷^{(−𝑠−1)𝛼}𝑓(𝑡)+^∞_𝑠=0(−𝐴)^𝑠𝐷^{−𝑠𝛼𝑚−1}_𝑗=0𝑦^(𝑗)(0)𝑡^𝑗=Γ(𝑗+1)_∞_𝑠=0(−𝐴)^𝑠1Γ((𝑠+1)𝛼)^𝑡0(𝑡−𝜏)^{(𝑠+1)𝛼−1}𝑓(𝜏)𝑑𝜏+^𝑚−1_𝑗=0𝑏_𝑗∞_𝑠=0(−𝐴)^𝑠𝑡^{𝑗+𝑠𝛼}=Γ(𝑗+𝑠𝛼+1)_𝑡0∞_𝑠=0(𝑡−𝜏)^𝛼−1⋅[−𝐴(𝑡−𝜏)^𝛼]^𝑠Γ(𝑠𝛼+𝛼)𝑓(𝜏)𝑑𝜏+^𝑚−1_𝑗=0𝑏_𝑗𝐷^{(𝛼−𝑗−1)}^∞_𝑠=0𝑡^𝛼−1(−𝐴𝑡^𝛼)^𝑠=Γ(𝑠𝛼+𝛼)_𝑡0(𝑡−𝜏)^𝛼−1𝐸_𝛼,𝛼(−𝐴(𝑡−𝜏)^𝛼)𝑓(𝜏)𝑑𝜏+^𝑚−1_𝑗=0=_𝑡0𝑏_𝑗𝐷^{(𝛼−𝑗−1)}𝑡^𝛼−1𝐸_𝛼,𝛼(−𝐴𝑡^𝛼)𝐺₂(𝑡−𝜏)𝑓(𝜏)𝑑𝜏+^𝑚−1_𝑗=0𝑏_𝑗𝐷^{(𝛼−𝑗−1)}𝐺₂(𝑡),(4.4) where 𝐺₂(𝑡)=𝑡^𝛼−1𝐸_𝛼,𝛼(−𝐴𝑡^𝛼).

It is easy to see that 𝑦(𝑡)=^𝑡0𝐺₂(𝑡−𝜏)𝑓(𝜏)𝑑𝜏+𝑏₀𝐷^(𝛼−1)𝐺₂(𝑡),(0<𝛼<1).(4.5)

(2). Consider an initial value problem for the nonhomogeneous Bagley-Torvik equation (see [5]) 𝐴𝑦^(𝑡)+𝐵^𝐶𝑦𝐷^3/2𝑦(𝑡)+𝐶𝑦(𝑡)=𝑓(𝑡),𝑡>0,_(𝑖)(0)=𝑎_𝑖,𝑖=0,1,(4.6) where 𝑎_𝑖are real constants.

Utilizing Lemma 2.5 and applying ^𝐶𝐷⁻² to both sides of (4.6), we obtain 𝑦(𝑡)+1𝐴𝐵𝐷^−1/2=1𝑦(𝑡)+𝐶𝐷⁻²𝑦(𝑡)𝐴𝐷⁻²𝑓(𝑡)+¹_𝑖=0𝑦^(𝑖)(0)𝑡^𝑖Γ(𝑖+1)+𝐵𝐴¹_𝑖=0𝑦^(𝑖)(0)𝑡^1/2+𝑖=1Γ(3/2+𝑖)𝐴𝐷⁻²𝑓(𝑡)+¹_𝑖=0𝑦^(𝑖)(0)𝑡^𝑖Γ(𝑖+1)+𝐵𝐴⋅Γ(3𝑡/2+^1/2+𝑖𝑖)(4.7)

According to the above procedure of solving the linear fractional differential equation with constant coefficients and using the Adomian decomposition method, let 𝑦₀(𝑡)=1𝐴𝐷⁻²𝑓(𝑡)+¹_𝑖=0𝑦^(𝑖)(0)𝑡^𝑖𝑦Γ(𝑖+1)+𝐵𝐴⋅Γ(3𝑡/2+^1/2+𝑖𝑖)₁,(𝑡)=−1𝐴𝐵𝐷^−1/2𝑦₀(𝑡)+𝐶𝐷⁻²𝑦₀(𝑡)=−𝐶𝐴𝐶𝐵𝐼+𝐷^−3/2𝐷^−1/2𝑦𝑦₀(𝑡),₂(𝑡)=−𝐶𝐴𝐶𝐵𝐼+𝐷^−3/2𝐷^−1/2𝑦₁(𝑡)=−𝐶𝐴²𝐶𝐵𝐼+𝐷^−3/2²𝑦⋮𝐷^−2/2𝑦₀(𝑡),_𝑛(𝑡)=−𝐶𝐴𝐶𝐵𝐼+𝐷^−3/2𝐷^−1/2𝑦_𝑛−1(𝑡)=−𝐶𝐴^𝑛𝐶𝐵𝐼+𝐷^−3/2^𝑛⋮𝐷^−𝑛/2𝑦₀(𝑡),(4.8)

Adding all of the above terms, we obtain the solution of the equation by Adomian decomposition method as follows: 𝑦(𝑡)=^∞_𝑛=0=𝑦_𝑛(𝑡)_∞_𝑛=0−𝐶𝐴^𝑛𝐶𝐵𝐼+𝐷^−3/2^𝑛=𝐷^−𝑛/2𝑦₀(𝑡)_∞_𝑛=0−𝐶𝐴^𝑛𝐶𝐵𝐼+𝐷^−3/2^𝑛𝐷^−𝑛/21𝐴𝐷⁻²+𝑓(𝑡)_∞_𝑛=0−𝐶𝐴^𝑛𝐶𝐵𝐼+𝐷^−3/2^𝑛𝐷^1−𝑛/2_𝑖=0𝑦^(𝑖)(0)𝑡^𝑖=1Γ(𝑖+1)+𝐵𝐴⋅Γ(3𝑡/2+^1/2+𝑖𝑖)𝐴^∞_𝑛=0−𝐶𝐴^𝑛𝑛_𝑘=0𝐶^{𝑛𝑛−𝑘}𝐵𝐶𝐼^𝑛−𝑘+𝐷^−3𝑘/2𝐷^−𝑛/2𝐷⁻²𝑓(𝑡)₁_𝑖=0𝑎_𝑖∞_𝑛=0−𝐶𝐴^𝑛𝑛_𝑘=0𝐶^{𝑛𝑛−𝑘}𝐵𝐶𝐼^𝑛−𝑘𝐷^−3𝑘/2𝐷^−𝑛/2=1Γ(𝑖𝑡+^𝑖1)+𝐵𝐴⋅Γ(3𝑡/2+^1/2+𝑖𝑖)𝐴^∞_𝑛=0−𝐶𝐴^𝑛𝑛_𝑘=0𝑛!(𝑛−𝑘)!𝑘!𝐶𝐵𝐼^𝑛−𝑘+𝐷^{−(3𝑘/2+𝑛/2+2)}𝑓(𝑡)₁_𝑖=0𝑎_𝑖∞_𝑛=0−𝐶𝐴^𝑛𝑛_𝑘=0𝑛!(𝑛−𝑘)!𝑘!𝐶𝐵𝐼^𝑛−𝑘𝐷^{−(3𝑘/2+𝑛/2)}=1Γ(𝑖𝑡+^𝑖1)+𝐵𝐴⋅Γ(3𝑡/2+^1/2+𝑖𝑖)𝐴^𝑡0∞_𝑛=0−𝐶𝐴^𝑛𝑛_𝑘=0𝑛!(𝑛−𝑘)!𝑘!𝐶𝐵𝐼^𝑛−𝑘1Γ(3𝑘/2+𝑛/2+2)(𝑡−𝜏)^{3𝑘/2+𝑛/2+2−1}+𝑓(𝜏)𝑑𝜏₁_𝑖=0𝑎_𝑖∞_𝑛=0−𝐶𝐴^𝑛𝑛_𝑘=0𝑛!(𝑛−𝑘)!𝑘!𝐶𝐵𝐼+𝐵^𝑛−𝑘Γ(𝑖+𝑡3𝑘/2+𝑛^{𝑖+3𝑘/2+𝑛/2}/2+1)=1𝐴⋅Γ(𝑖+𝑡3𝑘/2+𝑛/2+^{1/2+𝑖+(3𝑘/2)+(𝑛/2)}3/2)𝐴^𝑡0∞_∞𝑘=0_𝑚=0−𝐶𝐴^𝑘+𝑚(𝑘+𝑚)!𝑚!𝑘!𝐶𝐵𝐼^𝑚1×(𝑡−𝜏)_{3𝑘/2+(𝑘+𝑚)/2+1}Γ(3/2𝑘+(𝑘+𝑚/2)+2)+𝑓(𝜏)𝑑𝜏₁_𝑖=0𝑎_𝑖𝐷^(1−𝑖)^∞_∞𝑘=0_𝑚=0−𝐶𝐴^𝑘+𝑚(𝑘+𝑚)!𝑚!𝑘!𝐶𝐵𝐼^𝑚𝑡^{3𝑘/2+(𝑘+𝑚)/2+1}+Γ(3/2𝑘+(𝑘+𝑚)/2+2)₁_𝑖=0𝑎_𝑖𝐷^(1/2−𝑖)^∞_∞𝑘=0_𝑚=0−𝐶𝐴^𝑘+𝑚(𝑘+𝑚)!𝑚!𝑘!𝐶𝐵𝐼^𝑚𝑡^{3𝑘/2+(𝑘+𝑚)/2+1}=1Γ(3/2𝑘+(𝑘+𝑚)/2+2)𝐴^𝑡0∞_𝑘=0(−𝐶/𝐴)^𝑘𝑘!(𝑡−𝜏)^∞2𝑘+1_𝑚=0(𝑘+𝑚)!𝑚!⋅(−(𝐵/𝐴)(𝑡−𝜏)^1/2)^𝑚+𝐴₁Γ(𝑘/2+𝑚/2+3𝑘/2+2)𝑓(𝜏)𝑑𝜏_𝑖=0𝑎_𝑖𝐷^(1−𝑖)𝐴1^∞_𝑘=0(−𝐶/𝐴)^𝑘𝑘!(𝑡−𝜏)^∞2𝑘+1_𝑚=0(𝑘+𝑚)!𝑚!⋅−(𝐵/𝐴)(𝑡−𝜏)^1/2^𝑚+𝐴₁Γ(𝑘/2+𝑚/2+3𝑘/2+2)_𝑖=0𝑎_𝑖𝐷^(1/2−𝑖)𝐴1^∞_𝑘=0(−𝐶/𝐴)^𝑘𝑘!(𝑡−𝜏)^2𝑘+1×^∞_𝑚₌₀(𝑘+𝑚)!𝑚!⋅−(𝐵/𝐴)(𝑡−𝜏)^1/2^𝑚=1Γ((𝑘/2)+(𝑚/2)+(3𝑘/2)+2)𝐴^𝑡0∞_𝑘=0(−𝐶/𝐴)^𝑘𝑘!(𝑡−𝜏)^2𝑘+1𝐸^{(𝑘)1/2,3𝑘/2+2}−𝐵+𝐴₁𝐴(𝑡−𝜏)^1/2𝑓(𝜏)𝑑𝜏_𝑖=0𝑎_𝑖𝐷^(1−𝑖)𝐴1^∞_𝑘=0(−𝐶/𝐴)^𝑘𝑘!𝑡^2𝑘+1𝐸^{(𝑘)1/2,3𝑘/2+2}−𝐵+𝐴₁𝐴𝑡^1/2_𝑖=0𝑎_𝑖𝐷^(1/2−𝑖)𝐴1^∞_𝑘=0(−𝐶/𝐴)^𝑘𝑘!𝑡^2𝑘+1𝐸^{(𝑘)1/2,3𝑘/2+2}−𝐵=_𝑡0𝐴𝑡^1/2𝐺₃(𝑡−𝜏)𝑓(𝜏)𝑑𝜏+𝐴¹_𝑖=0𝑎_𝑖𝐷^(1−𝑖)𝐺₃(𝑡)+𝐷^(1/2−𝑖)𝐺₃(𝑡),(4.9) where 𝐺₃(𝑡)=(1/𝐴)∑^∞𝑘=0((−𝐶/𝐴)^𝑘/𝑘!)𝑡^2𝑘+1𝐸^{(𝑘)1/2,3𝑘/2+2}(−(𝐵/𝐴)𝑡^1/2).

Acknowledgment

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This work was supported by the National Natural Science Foundation of China (No. 60850005), the Natural Science Foundation of Zhejiang Province (Nos. D7080080 and Y607128) and the Innovation Team Foundation of the Department of Education of Zhejiang Province (No. T200924).