Abstract
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.
1. Introduction
Fractional calculus has applications in many scientific disciplines based on mathematical modeling including signal and image processing, physics, aerodynamics, chemistry, economics, electrodynamics, polymer rheology, economics, biophysics, control theory, and blood flow phenomena (cf. [1–7]). Researchers are investigating and developing fractional calculus in different ways including the numerical solutions of fractional-order differential equations using different numerical tools. There is interesting and valuable work in the literature for the numerical solutions of fractional-order differential equations using Bernstein polynomials (BPs). This work has interested many researchers recently (see, e.g., [8–13]).
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 , , , and , are constants. Moreover, , represent Caputo's derivative of order , , respectively [1, 6], namely, Note that where denotes the fractional Riemann-Liouville integral [1, 6], namely, Detailed explanations regarding the properties of the fractional operators may be found in [1, 6].
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.
2. Bernstein Polynomials and Their Properties
2.1. Definition of Bernstein Polynomials
We consider the Bernstein polynomials of the th degree on the interval on (cf. [11]) given by The Bernstein polynomials satisfy the recursive definition given by By using the binomial expansion of , Bernstein polynomials can be shown in terms of linear combination of the basis functions: We can write the Bernstein polynomials in the form , for , where Now if we introduce matrix in the form then we have , where and matrix is an upper triangular matrix given by where . Thus is an invertible matrix.
2.2. Approximation of Function
The set of Bernstein polynomials in Hilbert space is a complete basis (cf. [20]). Therefore, any function can be represented by Bernstein polynomials by means of where and . Then can be obtained by where and is called dual matrix of which is showed by where Thus where is the symmetric matrix where
Lemma 1. Suppose that the function is -times continuously differentiable, and . If is the best approximation out of then where .
Proof. See [9].
3. Operational Matrix of Bernstein Polynomials
3.1. Operational Matrix for Fractional Integration Based on Bernstein Polynomials
The operational matrices of fractional integration of the vector can be approximated (cf. [21]) as follows: where is the 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 matrix and and are given by Now we approximate by terms of the Bernstein basis: We have where and Then is matrix that has vector for th columns. Therefore, we can write Finally, we obtain where is called fractional integration within the operational matrix.
3.2. Operational Matrix of Multiplication
It is always necessary to assess the product of and , which is called the product matrix for the Bernstein polynomial basis. The operational matrices for the product are given by where is matrix. So we have Now, we approximate all functions in terms of for . From (14), we have where . Then we obtain the components of the vector of where Thus we obtain where is an matrix that has vectors given for each column. If we choose an matrix , then from (32) and (35) we can write and therefore we obtain the operational matrix of product, .
Corollary 2. If , consequently one can get the approximate function for , using Bernstein polynomials by where and is operational matrix of product using Bernstein polynomials.
Proof. This arises obviously from [8].
4. 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 where 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 Similarly, we approximate from (1) by Bernstein polynomials as where and is the fractional operational matrix of integration of order and Substituting (38), (40), and (42) into (1), we get Now using matrix of multiplication (36) in (44) we have which yields the system Using the independent property of Bernstein polynomials we obtain Solving this system for the vectors , , we can approximate and from (40) and (42) respectively.
5. Illustrative Examples
Below we use the presented approach to solve two examples.
Example 3. We consider fractional-order Brusselator system given in [19] by with initial conditions and .
Figure 1 presents comparison between exact solution and approximate solution obtained by the help of Bernstein polynomials for , at , when . Figure 2 presents comparison between the exact solution and our approximate solution by Bernstein polynomials for , at and different values of and .
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Example 4. We demonstrate accuracy of the presented numerical scheme by considering the fractional-order Brusselator system given in [19] by with initial conditions and .
Figure 3 demonstrates the exact solution together with the approximate solutions , for and different values of . Definitely, by increasing the value of of Bernstein basis, the approximate values of , converge to the exact solutions. From the approximate solutions , together with the exact solution for and different values of , plotted in Figure 4 we see that as approaches 1, the numerical solution converges to exact solution.
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6. 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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.