The Adomian Decomposition Method for a Type of Fractional Differential Equations

Fractional differential equations are widely used in many fields. In this paper, we discussed the fractional differential equation and the applications of Adomian decomposition method. Where the fractional operator is in Caputo sense. Through the numerical test, we can find that the Adomian decomposition method is a powerful tool for solving linear and nonlinear fractional differential equations. The numerical results also show the efficiency of this method.

Fractional calculus is found to be more suitable modeling the process with long range interaction and physical problems described by fractional equations, but sometimes it's difficult to get the solution of fractional differential equations. For that reason, we need a reliable and efficient technique for solving fractional differential equations. In [13], Tamsir and Srivastava give an analytical study for time fractional Klein-Gordon equation. Chen et al. use the discrete method to study the time fractional Klein-Gordon equation [14]. Fewer researchers consider giving an approximate solution. In this paper, we give an analytical solution of the time fractional differential equation of the following form ( )  This paper is organized as follows. In Section 2, we discuss some basic properties about fractional derivative and fractional integral which will be used in the following part. In Section 3, we introduce the Adomian decomposition method, and the detailed Scheme about the time fractional differential Equation (1.1) will be discussed. In Section 3, a numerical test will be showed, the approximate solution will be compared with the exact solution, and the error analysis will be given.

Fractional Integral and Fractional Derivative
First, we will give some definitions about fractional calculus including fractional integral and fractional derivative. For fractional derivative there are already exist several different definitions and in general these different definitions are not equivalent to each other. Here we only give the most common definition. (2.4) (2.5) Here, we only give some basic properties about Caputo fractional derivative and fractional integral which we will use in the following part. For some other properties about Caputo fractional derivative and other definitions about fractional calculus we can refer to [4].

Adomian Decomposition Method
The Adomian decomposition method [15] [16] is powerful tool for solving linear or nonlinear equations. For every nonlinear differential equation can be decomposed into the following form where L is the highest order differential operator, Ru is the remainder of the linear part, Nu represents the nonlinear part and g is a given function. In general, the operator L is invertible. If we take 1 L − on both sides of Equation (3.1), an equivalent expression can be given, where ϕ satisfy For clarity, first few several items of the Adomian polynomials will be listed   [18]. From some numerical tests of the following part, we can find that the sum of the first three or four terms has high accuracy. The more terms we calculate, the higher the accuracy.

Numerical Examples
In order to verify the accuracy of the method which described in the last section, two numerical examples will be considered. Example 1. First, we take 0,t D α − on both sides of the example 1, the following relation is given With the scheme we discussed in the last part, we have . Figure 1 shows the exact solution and the approximate solution with the first four terms. Table 1 shows the error of the exact solution and the approximate solution. In this example, we only use the forst four terms to approximate the exact solution. From the error column we can find that the absolute error is very small, the Adomian decomposition method has a high convergence order. The more terms we use, the higher accuracy we get.
Similarly, with the procedure we used in the first example, we have the following result about i u ( ) In this example we use the sum of the first three terms as the approximate solution of the problem we discussed. When we consider 2 α = , the exact Figure 2 shows the exact solution and the approximate solution. Table 2 shows the exact solution and approximate solution of the nonlinear fractional differential equation. In the last column we can find that the absolute error is small, here, we only use the first three terms to approximate the solution. If we use more terms, the approximation works better.

Conclusion
In this work, the Adomian decomposition method is applied to solving a time fractional differential equation. Both the linear and nonlinear type of fractional    high precision. In general, some differential equations are hard to deal with because of the nonlinear terms. The Adomian decomposition method is a powerful tool to cope with this problem. Moreover, no linearization or perturbation is required in this method.