Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials

In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system. We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations. Two illustrative examples are given in order to demonstrate the accuracy and simplicity of the proposed techniques.

Chaos theory is considered an important tool for viewing and understanding our universe and different techniques are utilized in order to reduce problems produced by the unusual behaviours of chaotic systems including chaos control (cf. [14,15]). In the literature, several authors have considered the chaotic system known as the fractional-order Brusselator system (FOBS) recently (cf. [7,16]). For example, Gafiychuk and Datsko investigate the stability of fractional-order Brusselator system in [17]. In [18], Wang and Li proved that the solution of fractional-order Brusselator system has a limit cycle using numerical method. Jafari et al. used the variational iteration method to investigate the approximate solutions of this system [19].
In this paper, we are interested in obtaining the numerical solution of the nonlinear fractional-order Brusselator system given by with initial conditions by means of operational matrices of fractional-order integration and multiplication of Bernstein polynomials, provided that > 0, > 0, , ∈ (0, 1], and 1 , 2 are constants.
In Section 2, we discuss the Bernstein polynomials and their properties. Also, we give the approximation of functions via Bernstein polynomials. In Section 3, we discuss operational matrices for fractional integration and multiplication via Bernstein polynomials. In Section 4, we give a numerical scheme for the Brusselator system based on Bernstein polynomials. In Section 5, illustrative examples are given which demonstrate the accuracy of our scheme based on the operational matrices for fractional-order integration of Bernstein polynomials. In the final section, a summary of the paper is presented.

Bernstein Polynomials and Their Properties
By using the binomial expansion of (1 − ) − , Bernstein polynomials can be shown in terms of linear combination of the basis functions: We can write the Bernstein polynomials in the form , Now if we introduce ( + 1) × ( + 1) matrix in the form where | | = Π =0 ( ). Thus is an invertible matrix.

Approximation of Function.
The set of Bernstein polynomials { 0, , 1, , . . . , , } in Hilbert space 2 [0, 1] is a complete basis (cf. [20]). Therefore, any function can be represented by Bernstein polynomials by means of The where and ⟨ , ⟩ is called dual matrix of which is showed by where where is the symmetric ( Proof. See [9].

Operational Matrix for Fractional Integration Based on
Bernstein Polynomials. The operational matrices of fractional integration of the vector Φ( ) can be approximated (cf. [21]) as follows: where is the ( + 1) × ( + 1) Riemann-Liouville fractional operational matrix of integration for Bernstein polynomials. By the use of (7), we have where the operator * denotes the convolution product. By substituting ( ) = ( ) and from (5) we get where is ( + 1) × ( + 1) matrix and and are given by ] .
(24) Now we approximate + by + 1 terms of the Bernstein basis: We have The Scientific World Journal Then is ( + 1) × ( + 1) matrix that has vector −1 for th columns. Therefore, we can write Finally, we obtain is called fractional integration within the operational matrix.

Numerical Solution of Nonlinear Fractional-Order Brusselator Systems Using Bernstein Polynomials
In this paper, we employ the Bernstein polynomials for solving the nonlinear fractional-order Brusselator systems given in (1). Firstly, we expand the fractional derivative in (1) by the Bernstein basis as follows. Taking are unknowns, and using initial conditions (2), (6), and (30), we approximate ( ) by where ( + ) = and is the fractional operational matrix of integration of order and The Scientific World Journal Similarly, we approximate ( ) from (1) by Bernstein polynomials as where ( + ) = and is the fractional operational matrix of integration of order and Solving this system for the vectors , , we can approximate ( ) and ( ) from (40) and (42) respectively.

Illustrative Examples
Below we use the presented approach to solve two examples.

Conclusion
Due to the applications of fractional differential equations in the daily life of so many scientific disciplines as discussed in Section 1, we see many interesting results for its numerical solutions in the available literature as cited in the references via different mathematical tools. We have also been attracted towards the numerical solutions of fractional differential equations and have presented a numerical solution of the fractional-order Brusselator system given in (1) and (2) using the operational matrices of fractional integration and multiplication based on Bernstein polynomials. The proposed method is used due to the simplicity and accurateness in most of the cited work in which the fractional-order differential equations were expressed in the system of algebraic equations which were easily handled for their numerical solutions. For testing the accurateness of the scheme, we give two illustrative examples which show that the results are in agreement with the exact solutions. The numerical simulations were carried out using Mathematica.