Research Article
Numerical Solutions of the Nonlinear Fractional-Order
Brusselator System by Bernstein Polynomials
Hasib Khan,
1,2
Hossein Jafari,
3
Rahmat Ali Khan,
1
Haleh Tajadodi,
4
and Sarah Jane Johnston
3
1
Department of Mathematics, University of Malakand, Dir Lower, Khyber Pakhtunkhwa 18000, Pakistan
2
Shaheed Benazir Bhutto University, Sheringal, Dir Upper, Khyber Pakhtunkhwa 18000, Pakistan
3
Department of Mathematical Sciences, University of South Africa, P.O. Box 392, UNISA 0003, South Africa
4
Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran
Correspondence should be addressed to Hossein Jafari; jafari@umz.ac.ir
Received 28 July 2014; Revised 2 September 2014; Accepted 2 September 2014; Published 17 November 2014
Academic Editor: Dumitru Baleanu
Copyright © 2014 Hasib Khan et al. Tis is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 scientifc dis-
ciplines based on mathematical modeling including signal
and image processing, physics, aerodynamics, chemistry,
economics, electrodynamics, polymer rheology, economics,
biophysics, control theory, and blood fow phenomena (cf. [1–
7]). Researchers are investigating and developing fractional
calculus in diferent ways including the numerical solu-
tions of fractional-order diferential equations using diferent
numerical tools. Tere is interesting and valuable work in
the literature for the numerical solutions of fractional-order
diferential equations using Bernstein polynomials (BPs).
Tis 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 diferent 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, Gafychuk
and Datsko investigate the stability of fractional-order Brus-
selator 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
()=− (+1) ()+
2
()(),
()=()−
2
()(),
(1)
with initial conditions
(0)=
1
, (0)=
2
(2)
by means of operational matrices of fractional-order integra-
tion and multiplication of Bernstein polynomials, provided
that >0, >0, ,∈(0,1], and
1
,
2
are constants.
Hindawi Publishing Corporation
e Scientific World Journal
Volume 2014, Article ID 257484, 7 pages
http://dx.doi.org/10.1155/2014/257484