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

Authors: Constantin Bota; Bogdan Căruntu
Journal: Advances in Mathematical Physics
 (2015)

---

## Abstract

The paper presents a new method, called the Polynomial Least Squares Method (PLSM).
PLSM allows us to compute approximate analytical solutions for the Brusselator system,
which is a fractional-order system of nonlinear differential equations.


---

## Body

## 1. Introduction

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.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.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:(i)
the Adomian Decomposition Method [7–9],(ii)
the Homotopy Analysis Method [10–12],(iii)
the Homotopy Perturbation Method [13, 14],(iv)
the Laplace Transform Method [15, 16],(v)
the Fourier Transform Method [17],(vi)
the Variational Iteration Method [18–20].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 [21–23] and can be expressed as follows.We consider the following Brusselator system:(1)Dtα1xt=a-μ+1·xt+xt2·yt,Dtα2yt=μ·xt-xt2·yt,together with the initial conditions:(2)x0=c1,y0=c2,where a>0, μ>0, 0<α1≤1, 0<α2≤1, c1,c2 are real constants, and Dtα denotes Caputo’s fractional derivative [15]:(3)Dtα=1Γ(1-α)·∫0t‍t-ζ-α·x′ζdζ,0<α≤1.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].


## 2. The Polynomial Least Squares Method

For problem ((1), (2)) we consider the remainder operators:(4)D1xt,yt=Dtα1x(t)-a-μ+1·xt+xt2·ytD2xt,yt=Dtα2xt-μ·xt-xt2·yt.We will find approximate polynomial solutionsx~(t),y~(t) of problem ((1), (2)) on the [0,b] interval, solutions which satisfy the following conditions:(5)Djx~t,y~t<ϵ,j=1,2,ϵ>0(6)x~0=c1,y~0=c2.Definition 1.
One calls anϵ-approximate polynomial solution of problem ((1), (2)) an approximate polynomial solution (x~(t),y~(t)) satisfying relations ((5), (6)).Definition 2.
One calls aweakδ-approximate polynomial solution of problem ((1), (2)) an approximate polynomial solution (x~(t),y~(t)) satisfying the relations: (7)∫0b‍Dj2x~t,y~tdt≤δ,j=1,2,together with initial conditions (6).Definition 3.
One considers the sequence of polynomials:Pmj(t)=a0j+a1jt+⋯+amjtm, aij∈R, i=0,1,…,m, j=1,2, satisfying the conditions:(8)Pm1(0)=c1,Pm2(0)=c2,m>1,m∈N.
One calls the sequence of polynomialsPmj(t) convergent to the solution of problem ((1), (2)) if limm→∞Dj(Pm1(t),Pm2(t))=0.We will find weakϵ-polynomial solutions of the type:(9)x~(t)=∑k=0m‍dk1·tk,y~(t)=∑k=0m‍dk2·tk,m>1,where the constants d0j,d1j,…,dmj, j=1,2 are calculated using the steps outlined as follows.(i)
We attach to problem ((1), (2)) the following real functional:(10)Jd21,…,dm1,d22,…,dm2=∑j=12‍∫0b‍Dj2x~t,y~tdt,whered01,d02 are computed as functions of d11,d21,…,dm1,d12,d22,…,dm2 by using initial conditions (6).(ii)
We computes the values ofd¯11,d¯21,…,d¯m1,d¯12,d¯22,…,d¯m2 as the values which give the minimum of functional (10) and the values of d¯01,d¯02 again as functions of d¯11,d¯21,…,d¯m1,d¯12,d¯22,…,d¯m2 by using the initial conditions.(iii)
Using the constantsd¯01,d¯11,…,d¯m1,d¯02,d¯12,…,d¯m2 thus determined, we consider the polynomials:(11)Tm1(t)=∑k=0m‍d¯k1·tk,Tm2(t)=∑k=0m‍d¯k2·tk,m>1.The following convergence theorem holds.Theorem 4.
The necessary condition for problem ((1), (2)) to admit sequences of polynomials Pmj(t) convergent to the solution of this problem is (12)limm→∞∫0b‍Dj2Tm1t,Tm2tdt=0.Moreover, ∀ϵ>0, ∃m0∈N, such that, ∀m∈N, m>m0, it follows that Tmj(t), j=1,2, are weak ϵ-approximate polynomial solutions of problem ((1), (2)).Proof.
Based on the way the coefficients of the polynomialsTmj(t) are computed and taking into account relations ((9)–(11)), the following inequality holds: (13)0≤∫0b‍Dj2Tm1t,Tm2tdt≤∫0b‍∑j=1n‍Dj2Pm1t,Pm2tdt,∀m∈N.It follows that (14)0≤limm→∞∫0b‍Dj2Tm1t,Tm2tdt≤limm→∞∫0b‍∑j=1n‍Dj2Pm1t,Pm2tdt=0,∀m∈N.We obtain (15)limm→∞∫0b‍Dj2Tm1t,Tm2tdt=0.From this limit we obtain that, ∀ϵ>0,∃m0∈N such that, ∀m∈N, m>m0, it follows that Tmj(t) are weak ϵ-approximate polynomial solutions of problem ((1), (2)), j=1,2.Remark 5.
Anyϵ-approximate polynomial solutions of problem ((1), (2)) are also weak ϵ2·b-approximate polynomial solutions, but the opposite is not always true. It follows that the set of weak approximate solutions of problem ((1), (2)) also contains the approximate solutions of the system.Taking into account the above remark, in order to findϵ-approximate polynomial solutions of problem ((1), (2)) by PLSM we will first determine weak approximate polynomial solutions, x~(t),y~(t). If Djx~t,y~t<ϵ, j=1,2, then x~(t),y~(t) are also ϵ-approximate polynomial solutions of the system.


## 3. Application: the Fractional-Order Brusselator System

We consider the following fractional-order Brusselator system [20]:(16)Dtα1xt=-2·xt+xt2·yt,Dtα2yt=xt-xt2·yt,together with the initial conditions:(17)x(0)=1,y(0)=1.In [20] approximate solutions of (17) are computed using the Variational Iteration Method (VIM) for the case α1=α2=0.98. Also, a comparison with numerical solutions is presented for the particular case α1=α2=1, illustrating the applicability of the method.


### 3.1. The Caseα1=α2=0.98

For the caseα1=α2=0.98, using PLSM with m=3, we obtain the following approximate polynomial solutions:(18)xPLSM(t)=0.0243682·t3+0.311138·t2-1.08655·t+1yPLSMt=-0.184414·t3+0.333424·t2+0.0349127·t+1.In Figures1 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.Figure 1
Approximations forx(t) in the case α1=α2=0.98.Figure 2
Approximations fory(t) in the case α1=α2=0.98.Figure 3
The remainders corresponding to the first equation for the caseα1=α2=0.98.Figure 4
The remainders corresponding to the second equation for the caseα1=α2=0.98.


### 3.2. The Caseα1=α2=1

For the caseα1=α2=1, using PLSM with m=3 we obtain the following approximate polynomial solutions:(19)xPLSM(t)=0.0750974·t3+0.201028·t2-1.02827·t+1yPLSMt=-0.180088·t3+0.334087·t2+0.0271107·t+1.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 caseα1=α2=0.98.However, in the caseα1=α2=1 it is possible to compute the absolute error corresponding to an approximate solution as the difference in absolute value between the approximate solution and the numerical solution (in this case computed by using the Wolfram Mathematica software).Figures5 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.Figure 5
The absolute errors corresponding tox(t) for the case α1=α2=1.Figure 6
The absolute errors corresponding toy(t) for the case α1=α2=1.Again, it is easy to observe that the errors obtained by using PLSM are smaller than the ones obtained by using VIM.


## 3.1. The Caseα1=α2=0.98

For the caseα1=α2=0.98, using PLSM with m=3, we obtain the following approximate polynomial solutions:(18)xPLSM(t)=0.0243682·t3+0.311138·t2-1.08655·t+1yPLSMt=-0.184414·t3+0.333424·t2+0.0349127·t+1.In Figures1 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.Figure 1
Approximations forx(t) in the case α1=α2=0.98.Figure 2
Approximations fory(t) in the case α1=α2=0.98.Figure 3
The remainders corresponding to the first equation for the caseα1=α2=0.98.Figure 4
The remainders corresponding to the second equation for the caseα1=α2=0.98.


## 3.2. The Caseα1=α2=1

For the caseα1=α2=1, using PLSM with m=3 we obtain the following approximate polynomial solutions:(19)xPLSM(t)=0.0750974·t3+0.201028·t2-1.02827·t+1yPLSMt=-0.180088·t3+0.334087·t2+0.0271107·t+1.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 caseα1=α2=0.98.However, in the caseα1=α2=1 it is possible to compute the absolute error corresponding to an approximate solution as the difference in absolute value between the approximate solution and the numerical solution (in this case computed by using the Wolfram Mathematica software).Figures5 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.Figure 5
The absolute errors corresponding tox(t) for the case α1=α2=1.Figure 6
The absolute errors corresponding toy(t) for the case α1=α2=1.Again, it is easy to observe that the errors obtained by using PLSM are smaller than the ones obtained by using VIM.


## 4. 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 fractional-order 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 ofn≥3 nonlinear differential equations.



---

*Source: 450235-2015-03-24.xml*