Revised Variational Iteration Method for Solving Systems of Nonlinear Fractional-Order Differential Equations

Abstract

A modification of the variational iteration method (VIM) for solving systems of nonlinear fractional-order differential equations is proposed. The fractional derivatives are described in the Caputo sense. The solutions of fractional differential equations (FDE) obtained using the traditional variational iteration method give good approximations in the neighborhood of the initial position. The main advantage of the present method is that it can accelerate the convergence of the iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. Illustrative examples are presented to show the validity of this modification.

1. Introduction

Recently, fractional-order calculus has been studied as an alternative calculus in mathematics. Numerous problems in physics, chemistry, biology, and engineering can be modeled with fractional derivatives [1–12]. On the other hand, in control society, fractional-order dynamic systems and controls have gained an increasing attention [13–17], and also motion of an elastic column fixed at one end loaded at the other can be formulated in terms of a system of fractional differential equations [18]. Since most fractional differential equations do not have exact analytic solutions, approximate and numerical techniques, therefore, are used extensively.

The variational iteration method is relatively new approaches to provide approximate solutions to linear and nonlinear problems. The variational iteration method, which is proposed by He [19], was successfully applied to find the solutions of several classes of variational problems. Some research works in this field are [20–25]. Recently, the application of the method is extended for fractional differential equations [26, 27].

Daftardar-Gejji and Jafari [28] have explored the Adomian decomposition method to obtain solution of a system of linear and nonlinear fractional differential equations. Further in [29], they have suggested a modification (termed as “revised ADM”) of this method and applied revised method for solving systems of linear/nonlinear ordinary and fractional differential equations [30].

The objective of this paper is the use of revised variational iteration method (RVIM) for solving systems of nonlinear fractional-order differential equations. We demonstrate that the approximate solution thus obtained converges faster relative to the approximate solutions by standard variational iteration method. Several illustrative examples have been presented.

2. Definitions and Preliminaries

In this section, we give some definitions and properties of the fractional calculus [9] which are used further in this paper.

Definition 1. A real function 𝑓(𝑡),𝑡>0 is said to be in the space 𝐶_𝛼,𝛼∈ℜ if there exists a real number p (>𝛼), such that 𝑓(𝑡)=𝑡^𝑝𝑓₁(𝑡), where 𝑓₁∈𝐶[0,∞].

Definition 2. A function 𝑓(𝑡),𝑡>0 is said to be in the space 𝐶^𝑚_𝛼,𝑚∈𝑁⋃{0}if𝑓^(𝑚)∈𝐶_𝛼.

Definition 3. The left-sided Riemann-Liouville fractional integral of order 𝜇≥0, of a function 𝑓∈𝐶_𝛼,𝛼≥−1 is defined as 𝐼^𝜇_𝑡𝑓(𝑡)=1Γ𝜇∫^𝑡₀𝑓(𝜏)(𝑡−𝜏)^1−𝜇𝑓(𝑡)𝑑𝜏,𝜇>0,𝑡>0,,𝜇=0. (1)

Definition 4. The left-sided Caputo fractional derivative of 𝑓, 𝑓∈𝐶^𝑚₋₁, 𝑚∈𝑁∪{0}, is defined as 𝐷^𝜇_𝑡𝑓(𝑡)=𝑑^𝜇𝑓(𝑡)𝑑𝑡^𝜇=𝐼^𝑚−𝜇𝑑^𝑚𝑓(𝑡)𝑑𝑡^𝑚𝑑,𝑚−1<𝜇<𝑚,𝑚∈𝑁,_𝑚𝑓(𝑡)𝑑𝑡^𝑚,𝜇=𝑚. (2)

3. The VIM for FDE

The principles of the variational iteration method and its applicability for various kinds of differential equations are given in [20, 31]. In [26], He showed that the variational iteration method is also valid for fractional differential equations. In this section, following the discussion presented in [26], we extend the application of the variational iteration method to solve the fractional differential equation as follows: 𝐷^𝛼_𝑡𝑦(𝑡)+𝑁𝑦(𝑡)=𝑓(𝑡),0<𝛼≤1, (3) where 𝑁 is an operator with respect to 𝑦(𝑡) and 𝑓(𝑡) is a known function. According to the variational iteration method, we can construct the correction functional for (3) as follows: 𝑦_𝑛+1(𝑡)=𝑦_𝑛(𝑡)+𝐼^𝛼_𝑡𝜆𝐷^𝛼_𝑡𝑦_𝑛(𝑡)+𝑁𝑦_𝑛(𝑡)−𝑓(𝑡)=𝑦_𝑛(𝑡)+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝜆(𝑠)×𝐷^𝛼_𝑠𝑦_𝑛(𝑠)−𝑁𝑦_𝑛(𝑠)−𝑓(𝑠)𝑑𝑠, (4) where 𝜆 is the general Lagrange multiplier, which can be identified optimally via variational theory [32].

To identify approximately Lagrange multiplier, some approximations must be made. The correction functional equation (4) can be approximately expressed as follows: 𝑦_𝑛+1(𝑡)=𝑦_𝑛(𝑡)+^𝑡₀𝜆(𝑠)𝑦^′_𝑛(𝑠)+𝑁𝑦_𝑛(𝑠)−𝑓(𝑠)𝑑𝑠. (5) Here we apply restricted variations 𝑦_𝑛 to the term 𝑁(𝑦), and in this case, we can easily determine the multiplier. Making the aforementioned functional stationary, noticing that 𝛿𝑦_𝑛=0, 𝛿𝑦_𝑛+1(𝑡)=𝛿𝑦_𝑛(𝑡)+𝛿^𝑡₀𝜆(𝑠)𝑦^′_𝑛(𝑠)−𝑓(𝑠)𝑑𝑠 (6) yields the Lagrange multiplier 𝜆=−1, and substituting into the functional equation (4), we obtain the following iteration formula: 𝑦_𝑛+1(𝑡)=𝑦_𝑛(𝑡)−𝐼^𝛼_𝑡𝐷^𝛼_𝑡𝑦_𝑛(𝑡)+𝑁𝑦_𝑛(𝑡)−𝑓(𝑡). (7) The initial approximation 𝑦₀(𝑡) can be freely chosen if it satisfies the initial conditions of the problem. Finally, we approximate the solution 𝑦(𝑡)=lim_𝑛→1𝑦_𝑛(𝑡) by the 𝑁th term 𝑦_𝑁(𝑡).

4. The System of FDE and Revised VIM

Let us consider the following system of fractional differential equations: 𝐷^𝛼𝑖_𝑡𝑦_𝑖(𝑡)+𝑁_𝑖𝑦₁(𝑡),…,𝑦_𝑚(𝑡)=𝑓_𝑖(𝑡),0<𝛼_𝑖≤1,𝑖=1,2,…,𝑚, (8) where 𝑁_𝑖 are operators with respect to 𝑦_𝑖(𝑡) and 𝑓_𝑖(𝑡) are known functions. In this case, the correction functionals are obtained as follows: 𝑦_𝑖(𝑛+1)(𝑡)=𝑦_𝑖𝑛(𝑡)−𝐼^𝛼𝑖_𝑡𝐷^𝛼𝑖_𝑡𝑦_𝑖𝑛(𝑡)+𝑁_𝑖𝑦_1𝑛(𝑡),…,𝑦_𝑚𝑛(𝑡)−𝑓_𝑖(𝑡)𝑖=1,2,…,𝑚. (9) Here we construct the following iteration formula instead of the iteration formula obtained with the standard variational iteration method equation (9): 𝑦_𝑖(𝑛+1)(𝑡)=𝑦_𝑖𝑛(𝑡)−𝐼^𝛼𝑖_𝑡𝐷^𝛼𝑖_𝑡𝑦_𝑖𝑛(𝑡)+𝑁_𝑖𝑦_1(𝑛+1)(𝑡),…,𝑦_{(𝑖−1)(𝑛+1)}(𝑡),𝑦_(𝑖)𝑛(𝑡),…,𝑦_𝑚𝑛(𝑡)−𝑓_𝑖(𝑡). (10) In fact, the updated values 𝑦_1(𝑛+1)(𝑡),…,𝑦_{(𝑖−1)(𝑛+1)}(𝑡) are used for finding 𝑦_𝑖(𝑛+1)(𝑡). We called it the revised variational iteration method (RVIM). This technique can accelerate the convergence of iterative approximate solutions relative to the approximate solutions obtained using the traditional variational iteration method. The effect of this correction is clear in 𝑦_𝑖(𝑛+1)(𝑡) because the updated values are used to compute it. Tatari and Dehghan have employed this technique for systems of ordinary differential equations [33].

5. Test Examples

In this section, we illustrate the applicability of revised variational iteration method to systems of nonlinear fractional-order differential equations.

Example 1. Consider the following system of nonlinear fractional-order Van der pol: 𝐷^𝛼_𝑡𝑥(𝑡)(𝑡),(𝑡)=𝑦𝑑𝑦(𝑡)(𝑡)𝑑𝑡=−𝑥−.5𝑥²𝑦(𝑡)(𝑡),()+.5𝑦0<𝛼≤1 (11) with the initial condition 𝑥(0)=0,𝑦(0)=1. (12) The standard VIM for (11) leads to the following iteration formula: 𝑥_(𝑛+1)(𝑡)=𝑥_𝑛(𝑡)−𝐼^𝛼_𝑡𝐷^𝛼_𝑡𝑥_𝑛(𝑡)−𝑦_𝑛(𝑡,𝑦_(𝑛+1))(𝑡)=𝑦_𝑛(𝑡)−^𝑡₀𝑑𝑦_𝑛(𝑠)𝑑𝑠+𝑥_𝑛(𝑠)+.5𝑥_𝑛(𝑠)²𝑦_𝑛(𝑠)−.5𝑦_𝑛(𝑠)𝑑𝑠. (13) The use of the revised VIM for (11) results in the following formula: 𝑥_(𝑛+1)(𝑡)=𝑥_𝑛(𝑡)−𝐼^𝛼_𝑡𝐷^𝛼_𝑡𝑥_𝑛(𝑡)−𝑦_𝑛(𝑡,𝑦_(𝑛+1))(𝑡)=𝑦_𝑛(𝑡)−^𝑡₀𝑑𝑦_𝑛(𝑠)𝑑𝑠+𝑥_(𝑛+1)(𝑠)+.5𝑥_(𝑛+1)(𝑠)²𝑦_𝑛(𝑠)−.5𝑦_𝑛(𝑠)𝑑𝑠. (14) Starting with the initial approximations 𝑥₀(𝑡)=0 and 𝑦₀(𝑡)=1, we can easily obtain the results using (13) and (14).

In Figures 1 and 2, the results from VIM and RVIM are shown. Figures 1(a) and 1(b) show comparison between the approximate solutions (𝑥(𝑡)≅𝑥₄(𝑡)), ((𝑦(𝑡)≅𝑦₄(𝑡)) of (11) obtained using VIM and RVIM for the special case 𝛼=.98 and the numerical solutions for the special case 𝛼=1, respectively. Figures 2(a) and 2(b), show approximate solutions (𝑥(𝑡)≅𝑥₅(𝑡)), ((𝑦(𝑡)≅𝑦₄(𝑡)) of (11) using VIM and RVIM for the special case 𝛼=1 and the numerical solutions, respectively.

(a)

(b)

Figure 1 

Simulation of the approximate results for 𝛼=.98 and exact solution for 𝛼=1 of the Van der pol system.

(a)

(b)

Figure 2 

Simulation of the approximate results for 𝛼=1 and exact solution of the Van der pol system.

Example 2. Consider the system of nonlinear fractional-order Brusselator: 𝐷^𝛼1𝑦(𝑡)=.1𝑥(𝑡)−𝑥(𝑡)²𝑦(𝑡),𝐷^𝛼2𝑥(𝑡)=.5−1.1𝑥(𝑡)+𝑥(𝑡)²𝑦(𝑡)0<𝛼₁,𝛼₂≤1 (15) with the initial conditions 𝑥(0)=.4,𝑦(0)=1.5. (16) In this example, the use of the variational iteration method leads to 𝑦_𝑛+1(𝑡)=𝑦_𝑛(𝑡)−𝐼^𝛼1_𝑡𝐷^𝛼1_𝑡𝑦_𝑛(𝑡)−.1𝑥_𝑛(𝑡)+𝑥²_𝑛(𝑡)𝑦_𝑛(𝑡,𝑥_𝑛+1)(𝑡)=𝑥_𝑛(𝑡)−𝐼^𝛼2_𝑡𝐷^𝛼2_𝑡𝑥_𝑛(𝑡)−.5+1.1𝑥_𝑛(𝑡)−𝑥²_𝑛(𝑡)𝑦_𝑛(𝑡). (17) Using the RVIM, we obtain 𝑦_𝑛+1(𝑡)=𝑦_𝑛(𝑡)−𝐼^𝛼1_𝑡𝐷^𝛼1_𝑡𝑦_𝑛(𝑡)−.1𝑥_𝑛(𝑡)+𝑥²_𝑛(𝑡)𝑦_𝑛(𝑡,𝑥_𝑛+1)(𝑡)=𝑥_𝑛(𝑡)−𝐼^𝛼2_𝑡𝐷^𝛼2_𝑡𝑥_𝑛(𝑡)−.5+1.1𝑥_𝑛(𝑡)−𝑥²_𝑛(𝑡)𝑦_(𝑛+1)(𝑡). (18) We consider the initial approximations 𝑥₀(𝑡)=.4+(.5×(𝑡^𝛼2/Γ[𝛼₂+1])) and 𝑦₀(𝑡)=1.5. Results are shown in Figure 3. Figures 3(a) and 3(b) show comparison between the approximate solutions (𝑥(𝑡)≅𝑥₂(𝑡)), (𝑦(𝑡)≅𝑦₃(𝑡)) of (15) obtained using VIM and RVIM for the special case 𝛼₁=𝛼₂=.98 and the numerical solutions for the special case 𝛼₁=𝛼₂=1, respectively.

(a)

(b)

Figure 3 

Simulation of the approximate results for 𝛼1=𝛼2=.98 and exact solution for 𝛼1=𝛼2=1 of the Brusselator system.

Example 3. Consider the system of nonlinear fractional-order Genesio-Tesi 𝐷^𝛼1𝑥(𝑡)(𝑡),𝐷=𝑦_𝛼2𝑦(𝑡)(𝑡),𝐷=𝑧_𝛼3𝑧(𝑡)=−𝛽₁𝑥(𝑡)−𝛽₂𝑦(𝑡)−𝛽₃𝑧(𝑡)+𝛽₄𝑥(𝑡)²,0<𝛼₁,𝛼₂,𝛼₃≤1, (19) where 𝛽₁, 𝛽₂, 𝛽₃, and 𝛽₄ are system parameters. With the initial conditions, 𝑥(0)=−2,𝑦(0)=.5,𝑧(0)=2. (20) In view of the variational iteration method, we set 𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)−𝐼^𝛼1_𝑡𝐷^𝛼1_𝑡𝑥_𝑛(𝑡)−𝑦_𝑛(𝑡,𝑦_𝑛+1)(𝑡)=𝑦_𝑛(𝑡)−𝐼^𝛼2_𝑡𝐷^𝛼2_𝑡𝑦_𝑛(𝑡)−𝑧_𝑛(𝑡,𝑧_𝑛+1)(𝑡)=𝑧_𝑛(𝑡)−𝐼^𝛼3_𝑡𝐷^𝛼3_𝑡𝑧_𝑛(𝑡)+𝛽₁𝑥_𝑛(𝑡)+𝛽₂𝑦_𝑛(𝑡)+𝛽₃𝑧_𝑛(𝑡)−𝛽₄𝑥_𝑛(𝑡)². (21) The revised variational iteration method would lead to 𝑥_𝑛+1(𝑡)=𝑥_𝑛(𝑡)−𝐼^𝛼1_𝑡𝐷^𝛼1_𝑡𝑥_𝑛(𝑡)−𝑦_𝑛(𝑡,𝑦_𝑛+1)(𝑡)=𝑦_𝑛(𝑡)−𝐼^𝛼2_𝑡𝐷^𝛼2_𝑡𝑦_𝑛(𝑡)−𝑧_𝑛(𝑡,𝑧_𝑛+1)(𝑡)=𝑧_𝑛(𝑡)−𝐼^𝛼3_𝑡𝐷^𝛼3_𝑡𝑧_𝑛(𝑡)+𝛽₁𝑥_(𝑛+1)(𝑡)+𝛽₂𝑦_(𝑛+1)(𝑡)+𝛽₃𝑧_𝑛(𝑡)−𝛽₄𝑥_(𝑛+1)(𝑡)². (22) Beginning with the initial approximations 𝑥₀(𝑡)=−2 and 𝑦₀(𝑡)=.5,𝑧₀(𝑡)=2, we can easily obtain the results. Results are shown in Figures 4 and 5. Figures 4(a), 4(b), and 4(c) show comparison between the approximate solutions (𝑥(𝑡)≅𝑥₉(𝑡)), (𝑦(𝑡)≅𝑦₉(𝑡)) and (𝑧(𝑡)≅𝑧₉(𝑡)) of (19) obtained using VIM and RVIM for the special case 𝛼₁=𝛼₂=𝛼₃=.98 and the numerical solutions for the special case 𝛼₁=𝛼₂=𝛼₃=1, respectively.

(a)

(b)

(c)

Figure 4 

Simulation of the approximate results for 𝛼1=𝛼2=𝛼3=.98 and exact results for 𝛼1=𝛼2=𝛼3=1 of the Genesio-Tesi system for parameters 𝛽1=6.5, 𝛽2=2.92, 𝛽3=1.2, and 𝛽4=1.0.

(a)

(b)

(c)

Figure 5 

Simulation of the approximate results for 𝛼1=𝛼2=𝛼3=1 and exact results of the Genesio-Tesi system for parameters 𝛽1=6.5, 𝛽2=2.92, 𝛽3=1.2, and 𝛽4=1.0.

Figures 5(a), 5(b), and 5(c), show approximate solutions (𝑥(𝑡)≅𝑥₈(𝑡)), (𝑦(𝑡)≅𝑦₇(𝑡)), and (𝑧(𝑡)≅𝑧₇(𝑡)) of (19) using VIM and RVIM for the special case 𝛼₁=𝛼₂=𝛼₃=1 and the numerical solutions, respectively.

6. Conclusion

The variational iteration method is an efficient method for solving various kinds of problems. In this paper, we have suggested a modification of this method which is called “revised variational iteration method.” We employ the revised VIM for solving a systems of nonlinear fractional-order differential equations. The revised method yields a series solution which converges faster than the series obtained by standard VIM. Illustrative examples presented clear support for this claim.

Mathematica has been used for computation and graphs presented in this paper.