Analytical Solutions of Fractional Differential Equations Using the Convenient Adomian Series

and Applied Analysis 3

In fact, these methods are developed from original versions for ordinary differential equation of the integer order equation.Compared with the ordinary calculus, the fractional calculus has the long interaction traits or the socalled memory effects; this characteristic can better depict various nonlinear dynamics in both theories and engineering mathematical modeling.However, it also results in finding the solution to the fractional models.The challenge is analyzed in the analytical methods in [17].
The ADM has been extensively applied to fractional differential equations due to its convenience.The Adomian series should be calculated in each iteration which greatly affects the efficiency and accuracy of the analytical approximation.In order to solve this problem, Duan very recently suggested a new way to calculate the Adomian series in [22][23][24][25] and successfully extended it to the fractional differential equation [26].
In this paper, we adopt Duan's way to calculate the Adomian series and apply it to FDEs for 1 <  < 2. We define the residual function and give the error analysis and investigate the validness of the iteration formulae.

Algorithm of the Fractional Differential Equations
Definition 1 (see [2]).The Caputo derivative is defined as where Γ is the Gamma function.
Now we present our analytical schemes using the convenient Adomian series, Laplace transform, and Pade approximation.We adopt the steps in [27,28].Considering the following general fractional differential equations (FDEs), where [] is a linear operator with respect to  such as . ., we show the following iteration schemes.
(a) Take Laplace transform L to both sides: We can have iteration formula (4) through inverse of Laplace transform L−1 : where λ() and () can be determined by calculation of Laplace transform to [], ().This step makes (3) in time domain equivalently defined in the Laplace domain.This idea is illustrated in the solution of differential equations [27][28][29].Here the λ() similarly plays a role as the Lagrange multipliers in the VIM [29].(b) Through the Picard successive approximation, we can obtain the following iteration formula: (c) Let   = ∑  =0 V  and apply the Adomian series to expand the term [] as ∑ ∞ =0   .Then the iteration formula reads according to [24], where   is calculated by (d) The th-order approximation is explicitly given as The   depends on the V 0 , V 1 , . . ., V  which is denoted as This characteristic allows us to obtain the approximate solutions from V 0 and  0 = (V 0 ).

Numerical Example and Error Analysis
Example 1.Consider the following nonlinear FDE: with the initial conditions Following to the steps (a) to (d) in Section 2, we can have where V 0 can be identified using the Laplace as V 0 = 1.
The iteration formula is written as For  = 1.9, the Adomian series and the approximate solutions can be calculated as We find the new way to calculate the solution and the series is very convenient compared with the classical one.Set  = 20 and plot the defined residual function in Figure 1: From Figure 2, we can conclude that iteration formula (14) and the approximate solutions are correct.
Example 2. The second example is given as We obtain the following formulae based on the Adomian series: Abstract and Applied Analysis  For  = 1.5, we can have the approximate solutions successively as We plot the  100 in Figure 3.

Conclusions
This study applied the convenient Adomian series to fractional differential equations whose order is between one and two.Two nonlinear examples are used to illustrate the basics of the steps.We found the calculation of the solutions is more convenient and more rapid compared with the classical version.The approximate order here can be chosen as  = 100 while this choice is impossible for the classical definition.This merit is particularly nice for the fractional differential equations and the results show this purpose.