The Bernstein Operational Matrices for Solving the Fractional Quadratic Riccati Differential Equations with the Riemann-Liouville Derivative

and Applied Analysis 3 Substituting (13) in (1) and (2), we have an initial-value problem as follows:


Introduction
The Riccati differential equation is named after the Italian Nobleman Count Jacopo Francesco Riccati (1676-1754). The book of Reid [1] contains the fundamental theories of the Riccati equation, with applications to random processes, optimal control, and diffusion problems. Moreover, it is well known that the one-dimensional static Schrödinger equation is closely related to a Riccati differential equation [2]. Solitary wave solution of a nonlinear partial differential equation can be represented as a polynomial in two elementary functions satisfying a projective Riccati equation [3].
The general response expression (1) contains a parameter , the order of the fractional derivative that can be varied to obtain various responses. In the case that is integer, then (1) is reduced to the classical Riccati differential equation.
The aim of this work is using the Bernstein polynomials for solving the problem (1) and (2). We notice that the problem presented [18] was in the Caputo sense but in our work, the problem is with the Riemann-Liouville derivative; therefore we considered a more general space of functions. Also, in [18], the authors used the polynomials in the form of , ( ) = ( ) (1 − ) − ( = 0, 1, . . . , ) that is different from the standard Bernstein polynomials. So, the operational matrices in this work are different from those in [18].
The organization of this paper is as follows. In Section 2, the Bernstein polynomials are introduced. Some basic definitions and properties of the fractional calculus and also the BPs operational matrix for the Riemann-Liouville fractional integration are presented in Section 3. In Section 4, by BPs operational matrices, we solve the fractional quadratic Riccati differential equation. In Section 5, we discuss the convergence of the proposed method. In Section 6, several examples are considered to evaluate the power and effectiveness of the presented method. Some conclusions are summarized in the last section.

The Bernstein Polynomials and Their Properties
On the interval [0, 1] we define the Bernstein polynomials (BPs) of mth degree as follows [19]: Set ] [20].
As a result, any polynomial of degree can be expanded in terms of linear combination of , ( ) ( = 0, 1, . . . , ) as given below: where The approximation of functions within the Bernstein polynomials and convergence analysis can be found in [20,21].

BPs Operational Matrix for the Riemann-Liouville Fractional Integration
In this section, firstly, we give some basic definitions and properties of the fractional calculus which are used further in this paper.
. The operator , defined by is called the Caputo fractional derivative operator of order .

BPs for Solving the Fractional Quadratic Riccati Differential Equation
Firstly, we use the initial conditions to reduce a given initialvalue problem to a problem with zero initial conditions. So, we define wherê( ) is some known function that satisfied the initial conditions (2) and ( ) is a new unknown function.

Convergence Analysis
In this section, we investigate the convergence analysis for the method presented in Section 4. The problem (14) changes to the following problems since ( ) = (16) ( ) .

Illustrative Numerical Examples
In this section, we apply our method with = 10 (BPs of degree = 10) to solve the following examples. We define ( ) and ( ) for the approximate solution and the exact solution, respectively. Example 1. Consider the nonlinear Riccati differential equation [14]: subject to the initial condition as (0) = 0. The exact solution of the equation for = 1 is given as Numerical results compared to [14] are given in Table 1 and also Figure 1 shows the absolute error for our method for = 1 and Figure 2 shows behavior 10 ( ) for different values of .
Example 2. Consider the following quadratic Riccati differential equation of fractional order [14] 0 ( ) = 2 ( ) − ( ) 2 + 1, 0 < ≤ 1, subject to the initial condition as (0) = 0. The exact solution of the equation for = 1 is given as Numerical results compared to [14] are given in Table 2 and also, Figure 3 shows the absolute error for our method for = 1 and Figure 4 shows behavior 10 ( ) for different values of .
Abstract and Applied Analysis 5 This problem has been studied by using ADM [25], FDTM [26], and BPFs [27]. Our results with = 1.5, = 2.5 are compared to [25][26][27] in Tables 3 and 4. Therefore, we see that our method is very effective and obtained solutions that are in good agreement with the results in [25][26][27]. Also, Figure 5 shows behavior ( ) for different values of .

Conclusion
In this paper, we proposed a numerical method for solving the fractional quadratic Riccati differential equations by the operational matrices of the Bernstein polynomials. We applied operational matrix for fractional integration in the Riemann-Liouville sense. Then by using this matrix and operational matrix of product, we reduced the fractional quadratic Riccati differential equation to a system of algebraic equations that can be solved easily. Finally, examples have been simulated to demonstrate the high performance of the proposed method. We saw that the results were in good