Reproducing Kernel Method for Fractional Riccati Differential Equations

and Applied Analysis 3 1 2 3 4 0.2 0.4 0.6 0.8 1.0

Li [11] presented a numerical method for fractional differential equations based on Chebyshev wavelets.Hosseinnia et al. [12] applied an enhanced homotopy perturbation method for fractional Riccati differential equations.Yüzbas ¸ı [13] introduced a numerical method for fractional Riccati differential equations using the Bernstein polynomial.Khader [14] developed the fractional Chebyshev finite difference method for fractional Riccati differential equations.Yang and Baleanu [15], Yang et al. [16], and Baleanu et al. [17] proposed local fractional variation iteration for fractional heat conduction and wave equations.
The aim of this paper is to present a new method for fractional Riccati differential equations, based on the RKM and the quasilinearization technique.
The rest of the paper is organized as follows.In the next section, the quasilinearization technique is applied to fractional Riccati differential equation.The RKM for reduced linear fractional differential equations is introduced in Section 3. The numerical examples are presented in Section 4. Section 5 ends this paper with a brief conclusion.Therefore, the following iteration formula for (1) can be derived:

𝑓 (𝑥, 𝑢
where and  0 () is the initial approximation.Clearly, to solve (1), it suffices for us to solve the series of linear problem (5).
=1 is dense on [0, ] and the solution of (12) is unique, then the solution of (12) is Now, an approximate solution   () of ( 6) can be obtained by the N-term intercept of the exact solution V() and Similarly, the approximate solutions   () can be obtained: where   = ∑  =1     (  ).

Numerical Examples
Example 1.Consider the following fractional Riccati differential equation [10][11][12][13]: The exact solution for  = 1 can be easily determined to be ) . ( Applying the proposed method, taking  = 1,  = 3,  = 30,  = 50, the numerical results compared with other methods are listed in Tables 1 and 2. Taking  = 4,  = 3,  = 30,  = 50, the numerical results on [0, 4] are listed in Table 3. From Table 3, it is easily found that the present approximations are effective for a larger interval, rather than a local vicinity of the initial position.
Example 2. Consider the following fractional Riccati differential equation [10][11][12][13][14]: The exact solution for  = 1 can be easily determined to be According to the present method, taking  = 1,  = 3,  = 50,  = 50, the numerical results compared with other methods are given in Tables 4, 5, and 6.Taking  = 4,  = 5,  = 50,  = 80, the numerical results on [0, 4] are shown in Figures 1 and 2. From these figures we can conclude that the obtained numerical solutions are in excellent agreement with the exact solution for a larger interval.

Conclusion
In this paper, combining the RKM, the numerical integral, and quasilinearization techniques, a new numerical method is proposed for fractional Riccati differential equations.The main advantage of this method is that it can provide accurate numerical approximations on a larger interval.Numerical results compared with the existing methods show that the present method is a powerful method for solving fractional Riccati differential equations.

Table 1 :
Comparison of the numerical solutions with the other methods for  = 0.75.

Table 2 :
Comparison of the numerical solutions with the other methods for  = 0.90.

Table 4 :
Comparison of the numerical solutions with the other methods for  = 0.75.

Table 5 :
Comparison of the numerical solutions with the other methods for  = 0.90.

Table 6 :
Comparison of the numerical solutions with the other methods for  = 1.0.