ON THE FRACTIONAL RICCATI DIFFERENTIAL EQUATION

In this paper, We tried to find an analytical solution of nonlinear Riccati conformable fractional differential equation. Fractional derivatives are described in the conformable derivative. The behavior of the solutions and the effects of different values of fractional order α are presented graphically and table. The results obtained by the CFD(conformable fractional derivative) are compared with homotopy perturbation method(HPM), fractional variational iteration method(FVIM).


Introduction
In recent years, it has worked that many phenomena in biology, chemistry, acoustics, control theory, psychology and other areas of science can be productively modeled by the use of fractional-order derivatives.The subject of fractional derivative is as old as calculus.In 1695, L'Hopital asked if the expression d f has any meaning.Since then, many researchers have been trying to generalize the concept of the usual derivative to fractional derivatives.
Liouville derivative: Liouville left-sided derivative: ( Liouville fight-sided derivative: Riemann-Liouville left-sided derivative: Riemann-Liouville right-sided derivative: Caputo right-sided derivative: Grünwald-Letnikov left-sided derivative: Grünwald-Letnikov right-sided derivative: Recently, Khalil at al. give a new definition of fractional derivative and fractional integral [33].This new definition benefit from a limit form as in usual derivatives.This new theory is improved by Abdeljawad [34].This paper is planned as follows: In Section 2, we briefly give definitions related to the conformable fractional calculus theory.In Section 3, we define the fractional nonlinear Riccati differential equations with conformable derivative.We present the application of the fractional nonlinear Riccati differential equations and numerical results in Section 4. The conclusions are then given in the final Section 5.
One can easily show that T α satisfies all the features in the following theorem [33].

The Conformable Fractional Riccati Equation
In this paper, we have achieved the analytical solutions to conformable fractional Riccati differential equation[1] subject to the initial conditions where α is fractional derivative order, n is an integer, P (x) , Q (x) and R (x) are known real functions, and d k is a constant.

Applications
In this section, we present the solution of three examples of the Riccati differential equations as the applicability of conformable fractional differential equations.
Example 4.1.Let us consider the fractional Riccati differential equation: We get with initial conditions The same holds true for the Riccati equation.In fact, if one particular solution y 1 can be found, the general solution is obtained as substituting A set of solutions to the Riccati equation is then given by where u is the general solution to the aforementioned linear equation.
The same holds true for the Riccati equation.In fact, if one particular solution y 1 can be found, the general solution is obtained as substituting A set of solutions to the Riccati equation is then given by Table 2: Approximate solutions of (19).Values from the stated references are given in the table for the corresponding α values.
The same holds true for the Riccati equation.In fact, if one particular solution y 1 can be found, the general solution is obtained as substituting A set of solutions to the Riccati equation is then given by where u is the general solution to the aforementioned linear equation.For T α (1) = 0, and y 1 (x) = 1 is particular solution.
Table 3 indicates the approximate solutions for Eq. ( 23) obtained for different values of a using the CDF [33] and new homotopy perturbation method (NHPM) [38].From the numerical results in Table 3, it is clear that the approximate solutions are in substantially agreement with the exact solutions, when α = 1, and the solution continuously depends on the time-fractional derivative.

Conclusions
In this paper, analytical and numerical solutions of Riccati conformable fractional differential equation successfully obtained.It is also a promising method to solve other nonlinear equations.In this paper, we have discussed fractional Riccati equation having conformable fractional derivative(CFD) used for the  3: Approximate results for (23).5-term values from the stated references are given in the table for the corresponding α values.

Figure 3 :
Figure 3: Plots of approx.solution y(x) for different values of αTable 3: Approximate results for (23).5-term values from the stated references are given in the table for the corresponding α values.

Table 1 :
(16)oximate solutions for(16).Values from the stated references are given in the table for the corresponding α values.
[33] by Khalil et al[33].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 CFD are agreement with those obtained by HPM, ADM, HAM,FVIM which is available in the literature. first