New Implementation of Residual Power Series for Solving Fuzzy Fractional Riccati Equation

This paper reveals a computational method using a Residual Power Series Method (RPSM) for the solution of fuzzy fractional riccati equation under caputo fractional differentiability. An analytical solution of fuzzy fractional riccati equation is obtained as a convergent fractional power series. The procedure produces solutions of high accuracy, and some illustrative examples are solved with a different value of orders to show the efficiency of the RPSM.


Introduction
Riccati equation was established after the Italian Nobleman Count Jacopo Francesco Riccati (1676-1754) [1].In the past decades, this type of equation had wide applications in optimal control, random processes, and diffusion problems [2][3][4][5][6][7][8][9].Currently, in light of the growing fractional concept for the order of differential equation, fractional riccati equation has appeared as a more comprehensive form with a different value of derivative order in many studies [10][11][12][13][14][15].Many studies refer to the overlap between the differential equations and the foggy logic, whereby these studies have developed solutions to several fuzzy differential equations by using traditional numerical methods as Laplace transforms [16], transformation method [17], Taylor method [18], Homotopy [19] and Adomian decomposition [20].Recently, the broader formula of differential equations has included the fractional logic of derivative and the fuzzy logic in its terms.Hence, several studies have developed traditional methods to find solutions to fuzzy fractional differential equations [21][22][23].Residual Power Series Method (RPSM) was developed by [24] to solve the fuzzy differential equations and [25] used the same method to solve the Fractional Riccati Equation (FRE) and was extended for the implementation of RPS.Hence, this paper will solve the Fuzzy Fractional Reccati Equation (FFRE).Therefore, we consider the next form of FFRE as follows: Where ,  and  are constants, d is a fuzzy triangular number and    is the Caputo fractional derivative for order .It can be observed that Eq. ( 1) is a general formulation of FFRE, whereby the initial value d is a fuzzy number.
The rest of the paper is organized as follows: A general introduction about RPS and FFRE is introduced.Section 2 provides the main definitions about fractional calculus, fuzzy numbers.Section 3 presents RPSM for solving FFRE.Section 4 introduces two numerical examples to demonstrate the effectiveness of RPSM.The conclusion of the study is given in section 5.

Main definitions
This section contains briefly the main definitions of caputo fractional derivatives and fuzzy numbers.
Next, the details of the derivation of residual power series solution to the Fractional Riccati Equation are presented.

Residual power series method for solving fuzzy fractional Riccati equations
For solving FFREs, RPSM is used to solve FREs [25].Firstly, we consider the new form of general FFREs as follows: Where 0< ≤ 1,0 ≤  <R, and  = (  1 ,   2 ,   3 ) is a fuzzy triangular.By applying the fuzzy set theory, the next form for Eq. ( 5) is obtained: The RPSM proposes the solution for Eq (6) as Fuzzy Fractional Power Series (FFPS) about the initial point  = 0 of the form.
To solve the Eq. ( 16), we used the proposed steps in previous section with 10 terms.Then the approximate results obtained are compared with the exact solutions at β =1 that are presented in Table 3 and 4, where the exact solutions for   1 () and   2 () whereby β =1 given by:   1 () = 0.686141(−1.+1×2.7182811.436141 ) 0.313859+2.7182821.436141   ,   2 () = 0.686141(−0.649129+2.718281821.436141 ) 0.2037351+2.7182821.436141   .Tables 3 and 4 show a comparison of the approximate solution of   1 ,10 () and   2 ,10 () of different values of the fractional Caputo derivative order 0 <  ≤ 1.0 with exact solutions at β =1.0.It is clear through Tables 3  and 4 that the approximate solution is in high agreement with the exact solution at β =1

Conclusions
In this paper, we have studied the solutions of FFREs with Caputo derivatives by RPSM.The proposed steps are considerably convenient since it requires less effort and does not need a complex software for the application proposed procedure of solution.The accuracy of results obtained in this paper from the illustrated two examples indicates the effectiveness of the method.It also refers to the possibility of future research to find solutions to various forms of fuzzy fractional equations by using RPSM.