On the Solutions Fractional Riccati Differential Equation with Modified Riemann-Liouville Derivative

Fractional variational iteration method FVIM is performed to give an approximate analytical solution of nonlinear fractional Riccati differential equation. Fractional derivatives are described in the Riemann-Liouville derivative. A new application of fractional variational iteration method FVIM was extended to derive analytical solutions in the form of a series for these equations. The behavior of the solutions and the effects of different values of fractional order α are indicated graphically. The results obtained by the FVIM reveal that the method is very reliable, convenient, and effective method for nonlinear differential equations with modified RiemannLiouville derivative


Introduction
In recent years, fractional calculus used in many areas such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry of corrosion, chemical physics, optics, engineering, acoustics, viscoelasticity, material science and signal processing can be successfully modelled by linear or nonlinear fractional order differential equations 1-10 .As it is well known, Riccati differential equations concerned with applications in pattern formation in dynamic games, linear systems with Markovian jumps, river flows, econometric models, stochastic control, theory, diffusion problems, and invariant embedding 11-17 .Many studies have been conducted on solutions of the Riccati differential equations.Some of them, the approximate solution of ordinary Riccati differential equation obtained from homotopy perturbation method HPM 18-20 , homotopy analysis method HAM 21 , and variational iteration method proposed by He 22 .The He's homotopy perturbation method proposed by  the variational iteration method 26 and International Journal of Differential Equations Adomian decomposition method ADM 27 to solve quadratic Riccati differential equation of fractional order.
The variational iteration method VIM , which proposed by He 28,29 , was successfully applied to autonomous ordinary and partial differential equations and other fields.He 30 was the first to apply the variational iteration method to fractional differential equations.In recent years, a new modified Riemann-Liouville left derivative is suggested by  Recently, the fractional Riccati differential equation is solved with help of new homotopy perturbation method HPM 23 .In this paper, we extend the application of the VIM in order to derive analytical approximate solutions to nonlinear fractional Riccati differential equation: subject to the initial conditions where α is fractional derivative order, n is an integer, A x , B x , and C x are known real functions, and d k is a constant.The goal of this paper is to extend the application of the variational iteration method to solve fractional nonlinear Riccati differential equations with modified Riemann-Liouville derivative.
The paper is organized as follows: In Section 2, we give definitions related to the fractional calculus theory briefly.In Section 3, we define the solution procedure of the fractional variational iteration method to show inefficiency of this method, we present the application of the FVIM for the fractional nonlinear Riccati differential equations with modified Riemann-Liouville derivative and numerical results in Section 4. The conclusions are then given in the final Section 5.

Basic Definitions
Here, some basic definitions and properties of the fractional calculus theory which can be found in 31-35 .
Definition 2.1.Assume f : R → R, x → f x denote a continuous but not necessarily differentiable function, and let the partition h > 0 in the interval 0, 1 .Jumarie's derivative is defined through the fractional difference 34 : where FWf x f x h .Fractional derivative is defined as the following limit form 1, 7 : This definition is close to the standard definition of derivatives calculus for beginners , and as a direct result, the αth derivative of a constant, 0 < α < 1, is zero.
Definition 2.2.The left-sided Riemann-Liouville fractional integral operator of order α ≥ 0, of a function f ∈ C μ , μ ≥ −1 is defined as The properties of the operator J α can be found in 1, 7, 36 .
In addition, we want to give as in the following some properties of the fractional modified Riemann-Liouville derivative.
Fractional Leibniz product law: Fractional Leibniz Formulation: Fractional the integration of part: Definition 2.4.Fractional derivative of compounded functions 33, 34 is defined as Definition 2.5.The integral with respect to dx α 33, 34 is defined as the solution of the fractional differential equation: 2.9 Lemma 2.6.Let f x denote a continuous function [33,34] then the solution of 2.5 is defined as

International Journal of Differential Equations
For example, f x x β in 2.10 one obtains

2.11
Definition 2.7.Assume that the continuous function f : R → R, x → f x has a fractional derivative of order kα, for any positive integer k and any α, 0 < α ≤ 1; then the following equality holds, which is On making the substitution h → x and x → 0, we obtain the fractional Mc-Laurin series:

Fractional Variational Iteration Method
To describe the solution procedure of the fractional variational iteration method 31-35 , we consider the following fractional Riccati differential equation: According to the VIM, we can build a correct functional for 3.1 as follows:

3.2
Using 2.3 , we obtain a new correction functional:

3.3
It is obvious that the sequential approximations y k , k ≥ 0 can be established by determining λ, a general Lagrange's multiplier, which can be identified optimally with the variational theory.The function y n is a restricted variation which means δ y n 0. Therefore, we first designate the Lagrange multiplier λ that will be identified optimally via integration by parts.The successive approximations y n 1 x , n ≥ 0 of the solution y x will be readily obtained upon using the obtained Lagrange multiplier and by using any selective function y 0 .The initial values are usually used for choosing the zeroth approximation y 0 .With λ determined, then several approximations y k , k ≥ 0 follow immediately 37 .Consequently, the exact solution may be procured by using

Applications
In this section, we present the solution of two examples of the Riccati differential equations as the applicability of FVIM.
Example 4.1.Let usconsider the fractional Riccati differential equation, we get with initial conditions: Construction the following functional: we have

4.4
Similarly, we can get the coefficients of δy n to zero: The generalized Lagrange multiplier can be identified by the above equations: International Journal of Differential Equations substituting 4.6 into 4.3 produces the iteration formulation as follows: Taking the initial value y 0 x 0, we can derive  Then, the approximate solutions in a series form are

4.9
As α 1 is The exact solution of 4.1 is y x e 2x − 1 / e 2x 1 , when α 1. Figure 1 indicates the solution obtained using FVIM versus the exact solution when α 1. Figure 2 is plotted for approximate solution of time-fractional Riccati differential equation for α 0.7, 0.8, 0.9, and 1. Equation 4.1 is solved by using the homotopy perturbation method HPM 24 .FVIM solutions indicate that the present algorithm performs extreme efficiency, simplicity, and reliability.The results obtained from FVIM are fully compatible with those of the HPM.
Table 1 shows the approximate solutions for 4.1 obtained for different values of α using the variational iteration method and HPM 24 .From the numerical results in Table 1, it is clear that the approximate solutions are in high agreement with the exact solutions, when α 1, and the solution continuously depends on the time-fractional derivative.Construction the following functional: we have dτ α δy n τ dτ α .

4.14
Similarly, we can get the coefficients of δy n to zero: The generalized Lagrange multiplier can be identified by the above equations: substituting 4.16 into 4.13 produces the iteration formulation as follows: 4.17 Taking the initial value y 0 x 0, we can derive

4.18
Then, the approximate solutions in a series form are

4.19
As α 1 is The exact solution of 4.11 is Figure 3 is plotted for approximate solution of time-fractional Riccati differential equation found in Example 4.2.In Figure 4, we have shown the graphic of approximate solution of 4.11 for α 0.7, 0.8, 0.9, and 1. Figures 2 and 4 show that a decrease in the fractional order α causes an increase in the function.
Table 2 indicates the approximate solutions for 4.11 obtained for different values of α using the variational iteration method and HPM 24 .From the numerical results in Table 2, it is clear that the approximate solutions are in high agreement with the exact solutions, when α 1, and the solution continuously depends on the time-fractional derivative.Construction the following functional: we have Similarly, we can get the coefficients of δy n to zero:

4.24
The generalized Lagrange multiplier can be identified by the above equations: 4.26 substituting 4.26 into 4.23 produces the iteration formulation as follows: Taking the initial value y 0 x 1, we can derive  Then, the approximate solutions in a series form are

4.30
The exact solution of 4.21 is where J v t is the Bessel function of first kind, when α 1. Figure 5 is plotted for approximate solution of time-fractional Riccati differential equation found in Example 4.3.In Figure 6, we have shown the graphic of approximate solution of 4.21 for α 0.5, 0.65, 0.75, and 1. Figures 2, 4, and 6 show that a decrease in the fractional order α causes an increase in the function.
Table 3 indicates the approximate solutions for 4.21 obtained for different values of α using the HPM 23 .From the numerical results in Table 3, it is clear that the approximate solutions are in substantial agreement with the exact solutions, when α 1, and the solution continuously depends on the time-fractional derivative.

Figure 1 :
Figure 1: The graph indicates the solution y x for 4.1 , when α 1.

Figure 2 :
Figure 2: Plots of approx.solution y 3 x for different values of α.

Figure 3 :
Figure 3: The graph indicates the solution y x for 4.11 , when α 1.

2 Figure 4 :
Figure 4: Plots of approx.solution y 3 x for different values of α.

Figure 5 :
Figure 5: The graph indicates the solution y x for 4.21 , when α 1.

InternationalFigure 6 :
Figure 6: Plots of approx.solution y 3 x for different values of α.

Table 3 :
Approximate solutions for 4.21 .In this paper, variational iteration method having integral w.r.t.dτ α has been successfully implemented to finding approximate analytical solution of fractional Riccati differential equations.Variational iteration method known as very powerful and an effective method for solving fractional Riccati differential equation.It is also a promising method to solve other nonlinear equations.In this paper, we have discussed modified variational iteration method having integral w.r.t.dτ α used for the first time by Jumarie.The obtained results indicate that this method is powerful and meaningful for solving the nonlinear fractional differential equations.Three examples indicate that the results of variational iteration method having integral w.r.t.dτ α are in excellent agreement with those obtained by HPM, ADM, and HAM, which is available in the literature.