Nonlinear fractional differential equations (NFDEs) are an important tool in modeling many real-world problems that arise in fluid mechanics, elasticity, signal processing, chemical reactions, electromagnetism, biology, biomedical, biomathematics and so on. See for example [1, 2, 3, 8, 9, 14].

The difficulty of solving some NFDEs exactly, has necessitated the developement of efficient numerical methods to solve them. In recent years, solutions of NFDEs have been discussed by many researchers using various numerical techniques such as: the Laplace decomposition method [11], the homotopy perturbation transform method [12], the optimal homotopy analysis method [5], the fractional variational iteration method [13], the new iterative method [6], and the residual power series method [7].

The aim of this article is to use a new approach known as the modified fractional Taylor series method (MFTSM) to obtain an analytical series solution for the nonlinear Caputo fractional Lienard equation in the form

with

Here D2η denotes the fractional derivative operator, in the Caputo sense, of order 2η with 1/2<η≤1 and a,b,c,Ψ0 and Ψ1 are real numbers.

The MFTSM is an iterative algorithm. It is effective and makes it easy to obtain a power series solution for linear and nonlinear fractional differential equations without resorting to linearization, perturbation, or discretization. Unlike other series methods, the MFTSM does not require matching the coefficients of similar conditions, and no repeated connection is needed. The present method computes the coefficients of the power series by a bond of algebraic equations. In addition, the MFTSM does not need any transformation during the change from low order to higher order, thus it is possible to work with the present method directly on a given example by choosing an suitable initial estimate approximation.

The rest of the article is structured as follows. In Section 2, we provide some definitions and preliminary concepts of fractional calculus theory. Section 3 is devoted to the basic idea of the MFTSM. In Section 4, we apply the above-mentioned method to two numerical examples of a nonlinear Caputo fractional Lienard equation and discuss the applicability and reliability of the method through tables and graphs. Section 5 is devoted to the conclusion.

In this section we recall the basic definitions and concepts of fractional calculus theory that are used in the present article.

Definition 2.1.([9]) Let v:ℝ+→ℝ be a continous function. The fractional integral in the Riemann-Liouville sense of order η≥0, is defined as

Here, Γ(.) denotes the gamma function.

Definition 2.2.([9]) Let v(n):ℝ+→ℝ be a continous function. The fractional derivative in the Caputo sense of order n−1<η≤n, n∈ℕ∗, is defined as

Some properties of Dη are as follows

• 1) Dη(λ)=0, where λ∈ℝ.

• 2) Dηζγ=Γ(γ+1)Γ(γ−η+1) ζγ−η ,γ>n−1,0,              γ≤n−1.

• 3) Dηvn(ζ)=nvn−1(ζ)Dηv(ζ).

Theorem 3.1. Suppose we have the nonlinear Caputo fractional Lienard equation (1.1) with (1.2). Using MFTSM, the solution of (1.1)-(1.2) can be expressed as

Here, (3.1) is an infinite series which converges rapidly to the exact solution, Ψi are real coefficients and R is the radius of convergence.

Proof. To prove this result, we assume that the solution of equation (1.1) takes the following form

Therefore, the nth−order approximate solution of equation (1.1), can be written as

Applying the operator D2η on equation (3.3), we get the following formula

Then, by replacing equations (3.3) and (3.4) in equation (1.1), we obtain the following iterative relation

To determine the coefficient Ψn,n=2,3,4,..., we follow the same methodology used to obtain the coefficients of the Taylor series. To achieve this, we must solve the following equation

where

We now determine the terms of the sequence Ψn2N.

For n=2 we have

Solving F(0,η,2)=0, yields

To determine Ψ3, we consider

Then, we solve DηF(ζ,η,3)↓ζ=0=0, to get

In general, to determine Ψr, we consider

Then, we solve Dr−2ηF(ζ,η,r)↓ζ=0=0, to get

Therefore, the solution of equations (1.1)-(1.2) is

Theorem 3.2. The series solution given in equation (3.1) converges to the exact solution if there exists a constant 0<τ<1 such that

Proof. For every 0<ζ<R, we have

Because 0<τ<1 and v0 is bounded, so we get

In this section, we demonstrate the accurateness and effectiveness of the proposed method by presenting two different examples of nonlinear Caputo fractional Lienard equations.

Example 4.1. Let us take the following nonlinear Caputo fractional Lienard equation

with

Using the same procedure of the MFTSM given in Section 3, we have

and

Therefore, the solution of equations (4.1)-(4.2), is given by

If we take η=1 in equation (4.3), the solution becomes

which is the exact solution for equations (4.1)-(4.2), when η=1 (See.[4]).

Example 4.2. Let us take the following nonlinear Caputo fractional Lienard equation

with

Using the same procedure of the MFTSM given in Section 3, we have

and

Therefore, the solution of equations (4.4)-(4.5), is given by

If we take η=1 in equation (4.6), the solution becomes

which is the exact solution for equations (4.4)-(4.5), when η=1 (See.[4]).

Figure 1 and 2 show the graphs of the exact solutions and the 3^rd order approximate solutions using the MFTSM at η=0.7,0.8,0.9,1 for equations (4.1) and (4.4), respectively. The figures show that for various fractional-order values, the proposed method is reliable, accurate and efficient. Tables 1 and 2 shows the comparison between the exact solutions, approximate soluions using FHATM at η =2 (See.[10]) and approximate solutions using MFTSM at η =1. The tables show that there is a very good agreement between the solutions obtained and those available in the literature.

Table 1 . Comparison between the exact solution, FHATM solution and MFTSM solution for Example 4.1.

		η=2	η=1	Absolute error
ζ	vexact	vFHATM	vMFTSM	vexact−vMFTSM
0.00	0.70711	0.70711	0.70711	0
0.02	0.71414	0.71414	0.71414	5.0793×10−9
0.04	0.72110	0.72110	0.72110	8.2374×10−8
0.06	0.72799	0.72799	0.72799	4.2249×10−7
0.08	0.73479	0.73479	0.73479	1.3522×10−6
0.1	0.74151	0.74151	0.74151	3.3415×10−6

Table 2 . Comparison between the exact solution, FHATM solution and MFTSM solution for Example 4.2.

		η=2	η=1	Absolute error
ζ	vexact	vFHATM	vMFTSM	vexact−vMFTSM
0.00	0.64359	0.64359	0.64359	0.0
0.02	0.64344	0.64344	0.64344	3.2888×10−8
0.04	0.64299	0.64299	0.64299	5.2585×10−7
0.06	0.64224	0.64224	0.64224	2.6590×10−6
0.08	0.64119	0.64118	0.64118	8.3902×10−6
0.1	0.63984	0.63982	0.63982	2.0441×10−5

Figure 1. Graph of the exact solution and MFTSM-solution for Example 4.1

Figure 2. Graph of the exact solution and MFTSM-solution for Example 4.2

In this article, we used a new approach known as the modified fractional Taylor series method (MFTSM) to obtain an analytical series solution of the nonlinear fractional Lienard equation with the Caputo fractional derivative. Numerical results have been presented to demonstrate the accuracy and efficiency of the MFTSM. From the obtained results, it is clear that the MFTSM provided highly accurate series solutions, which converge very rapidly to the exact solution. In addition, it has been observed that there exists a very good agreement between the solutions obtained and those available in the literature. Finally, we can conclude that the proposed method is extremely methodical, more effective and very accurate, and which can be applied to solve various classes of nonlinear fractional differential equations.

The author would like to express his gratitude to the anonymous referees. I am very grateful for your helpful comments and careful reading, which have led to the improvement of the article.