Approximate Analytical Solutions of the Fractional-Order Brusselator System Using the Polynomial Least Squares Method

In recent years, in many practical applications in various fields such as physics,mechanics, chemistry, and biology (see, e.g., [1–6]), the problems being studied are modeled using fractional nonlinear equations. Formost of such fractional nonlinear equations, the exact solutions cannot be found and, as a consequence, a numerical solution or, if possible, an analytical approximate solution of these equations is sought. Due to the complexity of this type of problems, a general approximation algorithm does not exist and, thus, various approximation methods, each with its strong and weak points, were proposed, including, among others:

For most of such fractional nonlinear equations, the exact solutions cannot be found and, as a consequence, a numerical solution or, if possible, an analytical approximate solution of these equations is sought.
The objective of our paper is to present the Polynomial Least Squares Method (PLSM), which allows us to compute approximate analytical solutions for the Brusselator system.
The fractional order Brusselator system was recently studied by several authors [21][22][23] and can be expressed as follows.
We consider the following Brusselator system: together with the initial conditions: where  > 0,  > 0, 0 <  1 ≤ 1, 0 <  2 ≤ 1,  1 ,  2 are real constants, and    denotes Caputo's fractional derivative [15]: In the next section we will introduce PLSM for the Brusselator system and in the third section we will compare the approximate solutions obtained by using PLSM with the approximate solutions from [20].The computations show that the approximations computed by using our method present an error smaller than the error of the corresponding solutions from [20].

Application: the Fractional-Order Brusselator System
We consider the following fractional-order Brusselator system [20]: together with the initial conditions: In [20] approximate solutions of ( 17) are computed using the Variational Iteration Method (VIM) for the case  1 =  2 = 0.98.Also, a comparison with numerical solutions is presented for the particular case  1 =  2 = 1, illustrating the applicability of the method.
3.1.The Case  1 =  2 = 0.98.For the case  1 =  2 = 0.98, using PLSM with  = 3, we obtain the following approximate polynomial solutions: In Figures 1 and 2 we compare these approximations with the corresponding approximations of the same order computed by VIM (relations (15) in [20]), obtaining a good  agreement.In Figures 3 and 4 we compare the expressions of remainders (4) obtained by replacing the approximate solutions back in the equations.It is easy to observe that the errors obtained by using PLSM are smaller than the ones obtained by using VIM.

The Case
In this case both approximations (VIM and PLSM) consist of third-order polynomials.
We omitted the figures which compare our approximations with the ones given by VIM since they look almost the same as the corresponding ones from the case  1 =  2 = 0.98.
However, in the case  1 =  2 = 1 it is possible to compute the absolute error corresponding to an approximate solution Advances in Mathematical Physics as the difference in absolute value between the approximate solution and the numerical solution (in this case computed by using the Wolfram Mathematica software).Figures 5 and 6 present the comparison between the absolute errors corresponding to the approximate solutions from [20] obtained by VIM and the absolute errors corresponding to our approximate solutions.
Again, it is easy to observe that the errors obtained by using PLSM are smaller than the ones obtained by using VIM.

Conclusions
In this paper we present the Polynomial Least Squares Method, which is a relatively straightforward and efficient method to compute approximate solutions for the fractionalorder Brusselator system.
The comparison with previous results illustrates the accuracy of the method, since we were able to compute more precise approximations than the previously computed ones.
In closing we mention the fact that, due to the nature of the method, it is relatively easy to extend PLSM for the general case of fractional systems of  ≥ 3 nonlinear differential equations.

Figure 4 :Figure 5 :
Figure 4: The remainders corresponding to the second equation for the case  1 =  2 = 0.98.