# New Exact  Solutions of the Brusselator Reaction Diffusion Model Using the Exp-Function Method

Authors: F. Khani; F. Samadi; S. Hamedi-Nezhad
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Firstly, using a series of transformations, the Brusselator reaction diffusion
model is reduced into a nonlinear reaction diffusion equation, and then through
using Exp-function method, more new exact solutions are found which contain
soliton solutions. The suggested algorithm is quite efficient and is practically
well suited for use in these problems. The results show the reliability and
efficiency of the proposed method.


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## Body

## 1. Introduction

The Brusselator reaction model plays an important role both in biology and in chemistry. Since the model was put forward by Prigogine and Lefever in 1968, much attention had been paid to the model and many properties of it had been researched by marly people via using different methods [1–5]. In this paper, we mainly consider the model improved by Lefever et al. in 1977 [1] . The model is described as follows:
(1.1)Kψxx-ψt+ψ2φ-Bψ=0,Kφxx-φt-ψ2φ+Bψ=0,
where B is a constant, and K is the diffusion coefficient; the functions ψ(x,t) and φ(x,t) denote the concentrations. System (1.1) describes a biochemical model. Recently, many approaches have been suggested to solve the nonlinear equations, such as the variational iteration method [6–8], the homotopy perturbation method [9–11], the tanh-method [12], the extended tanh-method [13], the sinh-method [14], the homogeneous balance method [15, 16], the F-expansion method [17], and the extended Fan's subequation method [18]. Recently, He and Wu [19] have proposed a straightforward method called Exp-function method to obtain the exact solutions of nonlinear evolution equations (NLEEs). It should be pointed out that the method is also valid for difference-different equations [20, 21]. The solution's procedure of this method is of utter simplicity, and this method has been successfully applied to many kinds of NLEEs [22–33]. The Exp-function method not only provides generalized solitonary solutions but also provides periodic solutions. Taking advantage of the generalized solitonary solutions, we can recover some known solutions obtained by the most existing methods such as decomposition method, tanh-function method, algebraic method, extended Jacobi elliptic function expansion method, F-expansion method, auxiliary equation method, and others [22–33].


## 2. Exp-Function Method and Exact Solutions

In this section we intend to find a solitary wave solution of (1.1). Therefore by using the following transformations:
(2.1)ψ=v,φ=-v,x=±y,t=t,
the system (1.1) is reduced to a nonlinear reaction diffusion equation with respect to v(y,t):(2.2)Kvyy-vt-Bv-v3=0.
After that we use the transformation
(2.3)v=ϕ(ξ),ξ=λ(y-wt+γ),
where λ and w are constants to be determined later, and γ is arbitrary constant. Therefore (2.2) converts to
(2.4)Kλ2d2ϕdξ2+λwdϕdξ-Bϕ-ϕ3=0.
By virtue of the Exp-function method [19], we assume that the solution of (2.4) is of the form(2.5)ϕ(ξ)=acexp(cξ)+⋯+a-dexp(-dξ)bfexp(fξ)+⋯+b-gexp(-gξ),
where c,d,f, and g are unknown positive integers and to be determined later; an and bm are constants.By balancing the highest order of linear termϕ′′ with the highest order of nonlinear term ϕ3, the values of c and f can be determined easily.Since(2.6)ϕ′′=c1exp[(c+3f)ξ]+⋯c2exp[4fξ]+⋯,ϕ3=c3exp[(3c+f)ξ]+⋯c4exp[4fξ]+⋯,
setting
(2.7)c+3f=3c+f
leads easily to c=f.Similarly, to determined and g, we balance the lowest-order linear term of Exp-function in (2.4):
(2.8)ϕ′′=⋯+d1exp[-(d+3g)ξ]⋯+d2exp[-3gξ],ϕ3=⋯+d3exp[-(3d+g)ξ]⋯+d4exp[-3gξ].
This requires
(2.9)-(d+3g)=-(3d+g),
which leads to g=d.We can freely choose the values ofc and d, but we will illustrate that the final solution does not strongly depend on the choice of the values of c and d [19, 28]. Choosing f=c=1 and d=g=1 for simplicity causes the trial function (2.5) to become
(2.10)ϕ(ξ)=a1exp(ξ)+a0+a-1exp(-ξ)b1exp(ξ)+b0+b-1exp(-ξ).
By substituting (2.10) into (2.4), we get
(2.11)1Q(k3exp(3ξ)+k2exp(2ξ)+k1exp(ξ)+k0+k-1exp(-ξ)+k-2exp(-2ξ)+k-3exp(-3ξ))=0,
where Q=(b1exp(ξ)+b0+b-1exp(-ξ))3; ki(i=-3,-2,…,2,3) are constants. If the coefficients of exp(nξ) are set to zero, we have
(2.12)k3=0,k2=0,k1=0,k0=0,k-3=0,k-2=0,k-1=0.
By solving this system of algebraic equations, we obtain the following sets of solutions.Case A.
(2.13)a1=0,a0=a0,a-1=0,λ2=BK,b1=b1,b0=b0,b-1=-a028Bb1,w=0.Case B.
(2.14)a1=0,a0=a0,a-1=Bb-1-B,λ2=-2BK,b1=0,b0=a0-B,b-1=b-1,w=0.Case C.
(2.15)a1=a1,a0=a0,a-1=Bb02+a024a1,λ2=-2BK,b1=a1-B,b0=b0,b-1=14Ba0b02+a03+b03(B2/-B)+b0(Ba02/-B)a1B(a0/-B-b0),w=0.Case D.
(2.16)a0=a0,a-1=0,a1=0λ2=-B2K,b0=a0-B,b-1=b-1,b1=0,w=3B-2B/K.Case 2 E.
(2.17)a0=a0,a-1=0,a1=a1,λ2=-B2K,b0=-a02+Ba1b-1/-BBa0/-B,b-1=b-1,b1=a1-B,w=3B-2B/K.Substituting (2.13)–(2.17) into (2.10) gives the generalized solitonary solution
(2.18)ϕ(ξ)=a0b1exp(ξ)-(a02/8Bb1)exp(-ξ),
where ξ=B/K(±x+γ) and the solutions
(2.19)ϕ(ξ)=a0+(Bb-1/-B)exp(-ξ)a0/-B+b-1exp(-ξ),
where ξ=-2B/K(±x+γ), and
(2.20)ϕ(ξ)=a1exp(ξ)+a0+(Bb02+a02/4a1)exp(-ξ)(a1/-B)exp(ξ)+b0+𝔖exp(-ξ),
where 𝔖 denotes 1/4((Ba0b02+a03+b03B2/-B+b0Ba02/-B)/(a1B(a0/-B-b0))), and ξ=-2B/K(±x+γ), and
(2.21)ϕ(ξ)=a0a0/-B+b-1exp(-ξ),
where ξ=-B/2K(±x-(3B/-2B/K)t+γ), and
(2.22)ϕ(ξ)=a0+a1exp(ξ)(a1/-B)exp(ξ)+(-a02+Ba1b-1/-B)/(Ba0/-B)+b-1exp(-ξ),
where ξ=-B/2K(±x-(3B/-2B/K)t+γ), respectively. The choice of a0=2-2B and b1=1 in our solution (2.18) gives the same bell solitary wave solution presented in [34] which was obtained on using the sine-cosine method
(2.23)v(x,t)=-2BsechBK(±x+γ).
Also if we set
(2.24)b1=±1,a0=22B,b1=-1,a0=4-B,
in (2.18), we obtain the new solutions
(2.25)v(x,t)=±2BcschBK(±x+γ),v(x,t)=4BcoshB/K(±x+γ)-3sinhB/K(±x+γ).
Also, setting a0=-B and b-1=±1 causes (2.19) to lead to the new kink solitary wave solution
(2.26)v(x,t)=-Bcoth-B2K(±x+γ),v(x,t)=-Btanh-B2K(±x+γ).
This solution is similar to the solution obtained in [34].By setting(2.27)b0=0,a1=-B,a0=±2i-B,
in (2.20), we will have the solitary wave solution
(2.28)v(x,t)=-B(tanh-2BK(±x+γ)±isech-2BK(±x+γ)),
where i=-1. Now if we set a0=-B and b-1=1 in (2.21), we will have another kink solitary wave solution
(2.29)v(x,t)=-B2+-B2tanh-B8K(±x-3B-2B/Kt+γ),
which is similar to the solution obtained in [34].If we also setb-1=1 and a0=--B in (2.21), we obtain the new kink solitary wave solution
(2.30)v(x,t)=-B2+-B2coth-B8K(±x-3B-2B/Kt+γ).
Also, setting b-1=a0 causes (2.21) to lead to the new soliton solution
(2.31)v(x,t)=1-(-B/B)+cosh-B/2K(±x-(3B/-2B/K)t+γ)-ℌ,
where ℌ denotes sinh-B/2K(±x-(3B/-2B/K)t+γ).By choosinga1=-B,a-1=1, and a0=i-B in (2.22), we can find solitary wave solution
(2.32)v(x,t)=-B2(1+tanh-B2K(±x-3B-2B/Kt+γ)+isech-B2K(±x-3B-2B/Kt+γ)),
where i=-1.


## 3. Conclusion

An investigation on the Brusselator reaction diffusion model was established by using the Exp-function method. During this procedure some new exact solitary wave solutions, mostly solitons and kinks solutions, were obtained as well as some special cases. In particular, Yan's solution [34] can be considered as a special case of our result, and our result can turn into kink, soliton, and bell solutions with a suitable choice of the parameters. The study reveals the power of the method.



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