Approximate Closed-form Solutions for a System of ...

ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.16(2013) No.4,pp.300-312

Approximate Closed-form Solutions for a System of Coupled Nonlinear Partial Differential Equations: Brusselator Model S.O. Ajadi, O.A. Adesina, O.A. Jejeniwa ∗ , Department of Mathematics, Obafemi Awolowo University,Ile-Ife, 220005, Nigeria. (Received 5 March 2013, accepted 17 September 2013)

Abstract: In this article, approximate analytical expressions for the solutions of a system of coupled nonlinear reaction-diffusion equations have been obtained using the the homotopy pertubation method(HPM) framework. The effectiveness of this method is elucidated by applying this procedure to the reaction diffusion Brusselator model in one and two space dimensions. For the cases considered, the study reveals that the HPM reduces computational work and converges rapidly to its closed-form solutions. Graphical demonstration of these solutions shed more lights on the behavior of the system. Keywords: HPM; Brusselator model; closed-form solutions; non-linear partial differential equation.

1

Introduction

The nonlinear partial differential equations arise in many areas of application in science and technology particularly in mechanics, chemical kinetics, and biological phenomena. The study of non-linear reaction-diffusion systems has been an active area of research and continues to be so [1,2 ,4-6, 10-15,22]. A typical example is the well known Brusselator model, which was developed in the 1970s in Brussels by Nobel laureate Ilya Prigogine among others. It is one of the simplest chemical models exhibiting a pattern forming instability called Turing instability. This class of reaction diffusion system includes some significant pattern formation equations arising from the modeling of kinetics of chemical or biochemical reactions and from the biological pattern formation theory. In other words, the Brusselator is a trimolecular chemical reaction model used to show cooperative phenomenon in chemically reacting systems. At the same time, it is one of the simplest (nonlinear) models of chemical systems for which the relative concentration of the constituents can vary either in space, or in time, or can even exhibit nonlinear traveling waves. Thus, the hypothetical reaction scheme described as the Brusselator model is a system whose chemical reactions scheme are given by[13,14,17,19,21-23]  A → U, R1 = k1    B + U → V + D, R2 = k2 (1) R3 = k3  2U + V → 3U,   U → E, R4 = k4 where U and V are spatially and temporally varying active centres(radical) concentrations, whereas the concentrations of the chemicals A and B are input chemicals, D and E are output chemicals, while Ri (ki ) are the reaction rates for each step(i). The first reaction is known as the radical initiation step(IS), the second and third are known as the radical propagation steps(PS), while the fourth one is regarded as the radical termination step(TS). Using the law of mass action [18,19 and 21], one can derive the mathematical equations corresponding to the reaction scheme defined by Eq. (1). Arising from this, we consider a general class of coupled nonlinear partial differential (also known as diffusion reaction system) equations of the form ∂U = d1 ∆U + F (U, V ), (2) ∂t ∂V = d2 ∆V + G(U, V ), (3) ∂t ∗ Corresponding

author.

E-mail address: [email protected] c Copyright⃝World Academic Press, World Academic Union IJNS.2013.12.30/772

S.O. Ajadi, O.A. Adesina et.al: Approximate Closed-form Solutions for a System of Coupled Nonlinear · · ·

301

where d1 , d2 are the diffusion coefficients, F (U, V ) and G(U, V ) are the reaction terms. Thus, the reaction terms in the Brusselator model in Eq. (1) are given as { F (U, V ) = k1 A − k2 BU + k3 U 2 V − k4 U, (4) G(U, V ) = k2 BU − k3 U 2 V By using the variables: U → u, V → v, A′ → Ak1 /k3 , B ′ → Bk2 /k3 , k4 → k3 , t′ → tk3 and after dropping primes, give rise to { F (u, v) = A − (B + 1)u + u2 v (5) G(u, v) = Bu − u2 v This reduces to a system of two nonlinearly coupled reaction–diffusion equations given by[13],[20] ∂u = d1 ∆u + A − (B + 1)u + u2 v, ∂t ∂v = d2 ∆v + Bu − u2 v, ∂t ICs : u(0, r) = u0 (r),

(t, r) ∈ (0, ∞) × Ω,

r = (x, y, z)

(t, r) ∈ (0, ∞) × Ω

(7)

v(0, r) = v0 (r)

BCs : u(t, r1 ) = u1 , u(t, r2 ) = u2 , v(t, r1 ) = v1 , v(t, r2 ) = v2 ,

(6)

(8) t > 0, ∂Ω.

(9)

The Brusselator model is still attracting more attentions from researchers because of its theoretical relevance[19] and relationship to many physical phenomenon Kuptsov et al.[23], Karafyllis et al.[22], Chowdhury et al.[20]. Manaa et al.[21] solved the Brusselator model numerically in one and two dimensions by using two finite differences methods. For the one-dimension, they showed that the Crank-Nicolson method is more accurate, though more rigorous than the explicit method, while for the two dimensions, the Alternating Direction Implicit(ADI) is more accurate than the Alternating Direction Explicit(ADE) method. Rauber and Raunger[12] considered the spatial discretization of nonlinear partial differential equations (PDEs) results in large systems of nonlinear ordinary differential equations (ODEs) using the Brusselator equation as a characteristic example. For the parallel numerical solution of the Brusselator equation, they used an iterated Runge-Kutta method. A theoretical analysis of the resulting speed up values shows that the efficiency cannot be improved considerably. Considering that another source of stiffness is the Brusselator system through the translation of diffusion terms by divided differences into a large system of ordinary differential equations in one spatial variable, Hairer and Warner[3] obtain numerical solutions by using the finite difference method. More recently, chowdhury et al[20] have obtained a 3–term HPM solutions for the reaction–diffusion Brusselator model in two dimension. However, they have applied the HPM procedure to an example which does not have known Exact or closed-form solutions, which makes it impossible to measure the relative accuracy of the HPM method. The motivation for this work is to obtain more accurate and efficient method for solving system of PDEs, which has long been an active research undertaking. The effectiveness of this method is implemented to solve some coupled system of nonlinear Brusselator equations and compare with existing closed-form or other approximate methods[Manaa et al.[21],Rauber and Runger [12]. It was observed that the 5–term HPM solutions obtained are very close to the closed-form solutions, where they exist. The paper has been organized as follows. To start with, the notations and basic definitions of the homotopy pertubation method has been introduced, then the effectiveness of this method has been carried out by applying to some systems of non-linear reaction diffusion equations. Finally the accuracy of this method has been elucidated by comparing with established solutions and displayed graphically and in tabular forms.

2

Homotopy perturbation method

Traditionally, solving a differential or system of differential equations simply means writing down formulae which satisfy the equations. The best form of solution is the closed-form(exact) solution or formula, which is a set of algebraic functions under the operation of integration. Due to the difficulty of obtaining closed-form solutions particularly for non-linear problems, less approximate solutions in the form of power series can often be found. Unfortunately, series constructed by such method does not converge [Rapp et al. [17]]. The homotopy perturbation method(HPM) is a novel and effective method for solving a wide range of problems whose mathematical models yield differential equation or system of differential equations. The solution obtained by this method is also expressed in series form and is evidently convergent[He[8 − 9]]. The HPM deforms a difficult problem into an infinite set of problems which are easier to solve without compromising

IJNS homepage: http://www.nonlinearscience.org.uk/

302

International Journal of Nonlinear Science, Vol.16(2013), No.4, pp. 300-312

the nonlinearity. In line with Ajadi[1] and Chowdhury et al. [20], we illustrate homotopy perturbation method for the nonlinear equation A(u) − f (r) = 0, r ∈ Ω (10) with the boundary conditions B(u,

∂u ) = 0, (r ∈ ∂Ω) ∂n

(11)

where A is a general differential operator, B is a boundary operator, f (r) is a known analytic function, and ∂Ω is the boundary of the domain Ω. The operator A are generally divided into two parts; L and N , where L and N are linear and nonlinear parts of A respectively. Therefore, Eqn. (10) may be written as L(u) + N (u) − f (r) = 0.

(12)

We construct a homotopy v(r, p) : Ω × [0, 1] → ℜ H(v, p) = [L(v) − L(u0 )] + pL(u0 ) + p[N (v) − f (r)] = 0,

(13)

H(v, p) = [L(v) − L(u0 )] + p[A(u) − f (r)] = 0,

(14)

or where p ∈ [0, 1] is called the homotopy parameter and u0 is an initial approximation of (10). At the two extremes p = 0 and p = 1, we have H(v, 0) = L(v) − L(u0 ) = 0 and H(v, 1) = A(u) − f (r) = 0. (15) In the interval 0 < p < 1, then the homotopy H(v, p) deforms from L(v) − L(u0 ) to A(u) − f (r). Thus, the solution of Eqns. (10) and (11) may be expressed as v = v0 + pv1 + p2 v2 + p3 v3 + · · ·

(16)

Eventually, at p = 1, the system takes the original form of the equation and the final stage of deformation gives the desired solution. Thus taking limit u = lim v = v0 + v1 + v2 + · · · (17) p→1

3

Homotopy perturbation method solutions

In order to solve Eqns. (6)-(9) and in line with Ajadi[1], we carefully choose an initial approximation and construct the homotopy[partial t-solution] as [ ] ∂u ∂u0 ∂u0 2 − = p d1 ∆u + A − (B + 1)u + u v − , (18) ∂t ∂t ∂t [ ] ∂v0 ∂v ∂v0 2 − = p d1 ∆v + Bu − u v − . ∂t ∂t ∂t Suppose that the solution of u(r, t) and v(r, t) are expressed in the form { u(r, t) = u0 (r) + pu1 (r, t) + p2 u2 (r, t) + p3 u3 (r, t) + p4 u4 (r, t) + p5 u5 (r, t) · · · , v(r, t) = v0 (r) + pv1 (r, t) + p2 v2 (r, t) + p3 v3 (r, t) + p4 v4 (r, t) + p5 v5 (r, t) · · · ,

(19)

(21)

where r = (x, y, z, · · · ). Substituting Eqn.(20) into Eqns.(18) and (19) and picking terms in p, we have  ∂u0 2 1 p : ∂u  ∂t = d1 ∆u0 + u0 v0 − (B + 1)u0 + A − ∂t   ∂u 2 2   p : ∂t2 = d1 ∆u1 + (2u0 u1 v0 + u0 v1 ) − (B + 1)u1 2 2 3 p3 : ∂u ∂t = d1 ∆u2 + (u1 v0 + 2u0 u2 v0 + u0 v2 + 2u0 u1 v1 ) − (B + 1)u2  ∂u4 4   p : ∂t = d1 ∆u3 + (2u1 u2 v0 + 2u0 u3 v0 + u21 v1 + 2u0 u2 v1 + 2u0 u1 v2 ) − (B + 1)u3   5 ∂u5 p : ∂t = d1 ∆u4 + H(u0 , u1 , u2 , u3 , u4 , u5 , v0 , v1 , v2 , v3 , v4 , v5 ) − (B + 1)u4

IJNS email for contribution: [email protected]

(22)


and

 ∂v0 2 1 p : ∂v  ∂t = d2 ∆v0 − u0 v0 + Bu0 + A − ∂t   2 ∂v2 2   p : ∂t = d2 ∆v1 − (2u0 u1 v0 + u0 v1 ) + Bu1 2 2 3 p3 : ∂v ∂t = d2 ∆v2 − (u1 v0 + 2u0 u2 v0 + u0 v2 + 2u0 u1 v1 ) + Bu2  ∂v4 4   p : ∂t = d2 ∆v3 − (2u1 u2 v0 + 2u0 u3 v0 + u21 v1 + 2u0 u2 v1 + 2u0 u1 v2 ) + Bu3   5 ∂v p : ∂t5 = d2 ∆v4 − H(u0 , u1 , u2 , u3 , u4 , u5 , v0 , v1 , v2 , v3 , v4 , v5 ) + Bu4

303

(23)

where H = u22 v0 + 2u1 u3 v0 + 2u0 u4 v0 + 2u1 u2 v1 + 2u0 u3 v1 + u21 v2 + 2u0 u2 v2 + 2u0 u1 v3 .The appropriate initial and boundary conditions of the partial differential equations in Eqns.(21) and (22) can be obtained by substituting Eqn. (20) into Eqns. (8) and (9). Hence { v0 (0, t) = v0 , vi (0, t) = 0, u0 (0, t) = u0 , ui (0, t) = 0 (24) v0 (x, 0) = v0 (x), vi (x, 0) = 0, u0 (x, 0) = u0 (x, 0), ui (x, 0) = 0, i = 1, 2 · · · If we start with ur and vr and all the linear equations above can be solved, then the solutions of (18) and (19) can be obtained by setting p = 1 in Eqn. (20). Thus { u(r, t) = u0 (r) + u1 (r, t) + u2 (r, t) + u3 (r, t) + u4 (r, t) + u5 (r, t) · · · (25) v(r, t) = v0 (r) + v1 (r, t) + v2 (r, t) + v3 (r, t) + v4 (r, t) + v5 (r, t) · · ·

4

Test examples

In order to test the validity and accuracy of this method, it is important to apply this method to some specific examples in literature with known exact or numerical solutions.

4.1

One-dimensional brusselator model

We start by applying the homotopy technique to an initial boundary value problem of the form ∂u ∂2u = d1 2 + A − (B + 1)u + u2 v, ∂t ∂x

(26)

∂v ∂2v = d2 2 + Bu − u2 v, ∂t ∂x

(27)

u(0, t) = u(1, t) = 1, v(0, t) = v(1, t) = 3, u(x, 0) = 1 + sin 2πx, v(x, 0) = 3.

(28)

subject to some initial boundary conditions: Case 1 [3]: A = 1, B = 3, d1 = d2 =

1 50 .

Case 2 [21]: A = 0.6, B = 0.2, d1 = d2 =

1 40 .

u(0, t) = u(1, t) = A, v(0, t) = v(1, t) = B A, u(x, 0) = A + x(1 − x), v(x, 0) =

(29) B A

+ x(1 − x).

In line with Eqns. (18) and (19), the homotopy is expressed as [ ] ∂u ∂u0 ∂2u ∂u0 − = p d1 2 + A − (B + 1)u + u2 v − , ∂t ∂t ∂x ∂t [ ] ∂v ∂v0 ∂2v ∂v0 − = p d2 2 + Bu − u2 v − . ∂t ∂t ∂x ∂t

(30) (31)

From Eqn. (16), the solutions of u(x, t) and v(x, t) are expressed in the form u(x, t) = u0 (x) + pu1 (x, t) + p2 u2 (x, t) + p3 u3 (x, t) + p4 u4 (x, t) + p5 u5 (x, t) · · · , v(x, t) = v0 (x) + pv1 (x, t) + p2 v2 (x, t) + p3 v3 (x, t) + p4 v4 (x, t) + p5 v5 (x, t) · · · .


(32)

304


Substituting Eqn. (20) into Eqns. (28) and (29), and the conditions ui = 0 and vi = 0(i ̸= 0) are determined by substituting Eqn. (30) into Eqns. (26) and (27) for cases I and II respectively. The analytical solutions can easily be obtained by successive integration. After each integration, we have ensured that the boundary conditions satisfy the solutions by an interactive procedure implemented on the Wolfram Mathematica 5.0 code(Appendix A). Using these schemes, the expressions for u1 , u2 , u3 , u4 , u5 ... and v1 , v2 , v3 , v4 , v5 ... are easily deduced. For simplicity and space economy, we have decided to present only the expressions for u1 , u2 ,v1 and v2 as follows Case 3 u1 = [(5 − B − 4d1 π 2 ) sin 2πx + 3 sin2 2πx]t, 1 u2 = − 16 [3(cos 8πx − 1) + (2B + 12) cos 6πx + (28B − 108 + 288d1 π 2 )(1 − cos 4πx) + (66B − 188 − 8B 2 + 320d1 π 2 − 64Bd1 π 2 − 128d21 π 4 ) sin 2πx]t2 , u3 = · · · u4 = · · · u5 = · · · ··· v1 = [(B − 6) sin 2πx − 3 sin2 2πx]t, 1 [3(cos 8πx − 1) + (2B + 12) cos 6πx + (28B − 120 + 96d1 π 2 + 192d2π2 )(1 − cos 4πx) v2 = − 16 +(74B − 228 − 8B 2 + 192d1 π 2 + 32Bd1 π 2 + 192d2 π 2 − 32Bd22 π 2 ) sin 2πx]t2 , v3 = · · · v4 = · · · v5 = · · · ···

(33)

u1 = At [A3 x(x − 1) − 2A2 x2 (x − 1)2 − Bx2 (x − 1)2 + Ax(x − 1)(x4 − 2x3 + x2 + B − 1)], t2 6 5 2 2 2 3 3 4 u2 = − 2A 2 [A x(x − 1) − 2A x (x − 1) − 2B x (x − 1) + A x(x − 1) 2 2 4 3 2 +AB(12d1 x(x − 1) + 3x (x − 1) (x − 2x + x + B − 1)) +A3 (24d1 x(x − 1) + x2 (x − 1)2 (2x4 − 4x3 + 2x2 + 3B − 4)) −A2 (2d1 (15x4 − 30x3 + 18x2 − 3x) + x(x − 1)(x8 − 4x7 + 6x6 − 4x5 + (6B − 2)x4 +6(1 − 2B)x3 + 3(2B − 1)x2 + (B − 1)2 ))], u3 = · · · u4 = · · · u5 = · · · ··· v1 = At [A3 x(x − 1) − (2A2 + B)x2 (x − 1)2 + Ax(x − 1)(x4 − 2x3 + x2 + B)], t2 6 5 2 2 2 3 3 v2 = − 2A 2 [A x(x − 1) − 2A x (x − 1) − 2B x (x − 1) 2 2 4 3 2 +AB(12d2 x(x − 1) + x (x − 1) (3x − 6x + 3x + 3B − 2)) −A2 (B 2 x(x − 1) + B(6x6 − 18x5 + 18x4 − 6x3 − x2 − x) + x(x − 1)(30d2 x(x − 1) +x2 (x − 1)2 (x4 − 2x3 + x2 − 2))) +A3 (24d2 x(x − 1) + x2 (x − 1)2 (2x4 − 4x3 + 2x2 + 3B − 2))], v3 = · · · v4 = · · · v5 = · · ·

(34)

Case 4

The final solutions are obtained by taking limit as p → 1, u(x, t) = u0 (x) + u1 (x, t) + u2 (x, t) + u3 (x, t) + u4 (x, t) + u5 (x, t) · · · , v(x, t) = v0 (x) + v1 (x, t) + v2 (x, t) + v3 (x, t) + v4 (x, t) + v5 (x, t) · · · .

4.2

Two-dimensional model

The procedure has been applied to some known examples in two-dimensions in the form ( 2 ) ∂ u ∂2u ∂u = d1 + + A − (B + 1)u + u2 v, ∂t ∂x2 ∂y 2


(35)


∂v = d2 ∂t

(

∂2v ∂2v + ∂x2 ∂y 2

305

) + Bu − u2 v,

(36)

subject to some initial boundary conditions:

Case 5 [12]: A = 1, B = 3.4, d1 = d2 = 2 × 10−3 ] ∂u ∂v ∂n = ∂n = 0, u(x, y, 0) = 0.5 + y, v(x, y, 0) = 1 + 5x.

(37)

Case 6 [21]: A = 0, B = 1, d1 = d2 = 14 .] ( ∂u

)

= (x + y)(− exp(−t/2 − x − y), exp(t/2 + x + y)), u(x, y, 0) = exp(−x − y), v(x, y, 0) = exp(x + y). ∂v ∂n , ∂n

(38)

In two dimensions, we define the homotopy in line with Eqns. (18) and (19) as [ ( 2 ) ] ∂u ∂u0 ∂u0 ∂ u ∂2u 2 − = p d1 + + A − (B + 1)u + u v − , ∂t ∂t ∂x2 ∂y 2 ∂t

(39)

[ ( 2 ) ] ∂ v ∂2v ∂v0 ∂v ∂v0 2 − = p d2 + 2 + Bu − u v − . ∂t ∂t ∂x2 ∂y ∂t

(40)

Suppose that the solutions of u(x, y, t) and v(x, y, t) are expressed in the form ∑∞ u(x, y, t) = u0 (x, y) + ∑ i=1 pi ui (x, y, t) = u0 (x, y) + pu1 (x, y, t) + p2 u2 (x, y, t) + · · · ∞ v(x, y, t) = v0 (x, y) + i=1 pi vi (x, y, t) = v0 (x, y) + pv1 (x, y, t) + p2 v2 (x, y, t) + · · · .

(41)

Substituting Eqn. (40) into Eqns. (38) and (39), we have (

)

∂u0 ∂ 2 u0 ∂ 2 u0 ∂u1 2 ∂t = d1 (∂x2 + ∂y 2 )+ u0 v0 − (B + 1)u0 + A − ∂t ∂ 2 u1 ∂ 2 u1 2 2 p2 : ∂u ∂t = d1 ( ∂x2 + ∂y 2 ) + (2u0 u1 v0 + u0 v1 ) − (B + 1)u1 2 2 ∂ u2 ∂ u2 2 2 3 p3 : ∂u ∂t = d1 ( ∂x2 + ∂y 2 ) + (u1 v0 + 2u0 u2 v0 + u0 v2 + 2u0 u1 v1 ) − (B + 1)u2 ∂ 2 u3 ∂ 2 u3 2 4 p4 : ∂u ∂t = d1 ( ∂x2 + ∂y 2 ) + (2u1 u2 v0 + 2u0 u3 v0 + u1 v1 + 2u0 u2 v1 + 2u0 u1 v2 ) − (B 2 2 ∂ u4 ∂ u4 5 p5 : ∂u + H(u0 , u1 , u2 , u3 , u4 , u5 , v0 , v1 , v2 , v3 , v4 , v5 ) − (B + 1)u4 ∂t = d1 ∂x2 + ∂y 2

p:

and

(

(42) + 1)u3

)

∂ 2 v0 ∂v0 ∂v1 ∂ 2 v0 2 ∂t = d2 (∂x2 + ∂y 2 )+ u0 v0 − Bu0 − ∂t ∂ 2 v1 ∂ 2 v1 2 2 p2 : ∂v ∂t = d2 ( ∂x2 + ∂y 2 ) + (2u0 u1 v0 + u0 v1 ) − Bu1 ∂ 2 v2 ∂ 2 v2 2 2 3 p3 : ∂v ∂t = d2 ( ∂x2 + ∂y 2 ) + (u1 v0 + 2u0 u2 v0 + u0 v2 + 2u0 u1 v1 ) − Bu2 2 2 ∂ v3 ∂ v3 2 4 p4 : ∂v ∂t = d2 ( ∂x2 + ∂y 2 ) + (2u1 u2 v0 + 2u0 u3 v0 + u1 v1 + 2u0 u2 v1 + 2u0 u1 v2 ) ∂ 2 v4 ∂ 2 v4 5 + H(u0 , u1 , u2 , u3 , u4 , u5 , v0 , v1 , v2 , v3 , v4 , v5 ) − Bu4 p5 : ∂v ∂t = d2 ∂x2 + ∂y 2

p:

(43) − Bu3

Based on the initial conditions(u0 and v0 ) and parameters in cases I and II, the solutions of Eqns. (41) and (42) can easily be obtained by successive integration using the initial-boundary conditions in Eqns. (36) and (37) and implemented on the Wolfram Mathematica 5.0 Code[Appendix A]:


306


Case 7

Case 8

u1 = [A − (1 + B)(0.5 + y) + (1 + 5x)(0.5 + y)2 ]t, 1 u2 = 2! {−2d1 (1 + 5x) + (0.5 + y)2 [B(0.5 + y) − (1 + 5x)(0.5 + y)2 ] −(1 + B)[A − (1 + B)(0.5 + y) + (1 + 5x)(0.5 + y)2 ] +2(1 + 5x)(0.5 + y)[A − (1 + B)(0.5 + y) + (1 + 5x)(0.5 + y)2 ]}t2 , u3 = · · · u4 = · · · u5 = · · · ··· v1 = [B(0.5 + y) − (1 + 5x)(0.5 + y)2 ]t, 1 v2 = 2! {−2d2 (1 + 5x) − (0.5 + y)2 [B(0.5 + y) − (1 + 5x)(0.5 + y)2 ] +B[A − (1 + B)(0.5 + y) + (1 + 5x)(0.5 + y)2 ] −2(1 + 5x)(0.5 + y)[A − (1 + B)(0.5 + y) + (1 + 5x)(0.5 + y)2 ]}t2 , v3 = · · · v4 = · · · v5 = · · · ···

(44)

u1 = (A − B − 2d1 − 1) exp(−x − y)t, 1 u2 = 2! {A(1 − B) + [2d21 − 3B + d1 + B 2 − 2d1 B + 2d2 ] exp(−x − y) +(B − 1) exp(−3x − 3y) + 2d1 d2 exp(x + y)}t2 , u3 = · · · u4 = · · · u5 = · · · (45) v1 = [(B − 1) exp(−x − y) + 2d2 exp(x + y)]t, 1 v2 = 2! {A(B − 2) + [2B − 4d1 − B 2 + 2d1 B − 4d2 + 2d2 B] exp(−x − y) +(1 − B) exp(−3x − 3y) + 4d22 exp(x + y)}t2 , v3 = · · · v4 = · · · v5 = · · ·

The final solutions are obtained by taking limit as p → 1, ∑∞ i u(x, y, t) = i=0 ui (x, y) ti! = u0 (x, y) + u1 (x, y)t + u2 (x, y)t2 /2! + u3 (x, y)t3 /3! + · · · ∑∞ i v(x, y, t) = i=0 vi (x, y) ti! = v0 (x, y) + v1 (x, y)t + v2 (x, y)t2 /2! + v3 (x, y)t3 /3! + · · ·

5

(44)

Conclusion

The choice of the non-negative initial data(u0 and v0 ) and the positive constants A, B, d1 and d2 in literature[3, 6] are for easy comparison. However, in applying the homotopy perturbation method, it is important to state that the rate of convergence of the homotopy perturbation method is greatly influenced by the choice of initial data[Liao [7]]; which is the power of the method. In all the cases considered, we observed that the HPM method provides a better and more accurate results than other approximate methods. For case 1, Figs. 1 − 14 shed more lights on the behaviour of the system. Figs. 1 − 3 and 4 − 6 show the variation and emergence of oscillatory pattern for u(x, t) and v(x, t) respectively with t for some fixed x, while Figs. 7 − 9 and 10 − 12 show variation and emergence of oscillatoty pattern for u(x, t) and v(x, t) respectively with t for some fixed x. In Figs. 1-3 and Figs. 4 − 6, it can be observed that oscillatory pattern starts in the vicinity of t = 1 and becomes more pronounced as t increases for u(x, t) and v(x, t) respectively. The 3D plots in Figs. 13 − 14 has not only confirm known results [3], they also confirm that the oscillatory behaviour of the system is noticed at t > 0.5. In case 2, Figs. 15 − 16 are the u(x, t) and v(x, t) profiles for t = 0.5. The 3D plots for case 3 are shown in Figs. 17 − 20. It can be observed that as t increases from t = 0.5 to t = 1.0, the frequency of oscillations of v(x, t) is more feasible than that of u(x, t). Case 4 is an example that provides us with the basis of comparing HPM solutions with Exact solutions [21]. Figs. 21 − 22 and Figs. 23 − 24 are representations of u(x, t) and v(x, t) profiles respectively. All these examples reveal that HPM is highly comparable with exact methods and a better improvement above other approximate methods.



UHtL

307

U 1

1

1

1 1

1 1

1

1 1

0.5

1.5

t

2

1

0.5

Figure 1: Plot of u(x, t) vs t for x = 0.

U

1.5

2

t

Figure 2: Plot of u(x, t) vs t for x = 0.5

VHx,tL 3 3

1 3 3

1

3 1

3 0.5

1

1.5

t

2

Figure 3: Plot of u(x, t) vs t for x = 1.0.

VHx,tL 3

1

0.5

1.5

2

t

Figure 4: Plot of v(x, t) vs t for x = 0.0.

VHx,tL 3

3

3

3

3

3 3

3

3

3 1

0.5

1.5

2

t


UHtL 2

1

0.5

1.5

2

t


UHtL 4

1.5

3 2

1

1 0.5 0.2 0.2

0.4

0.6

0.8

1

x

Figure 7: Plot of u(x, t) vs x for t = 0.0.

0.4

0.6

0.8

1

x

-1



308


UHtL 6

UHtL 150

5 100

4

50

3 2 0.2

0.4

0.6

0.8

1

x

1

-50

0.2


0.4

0.6

x

1

0.8

Figure 10: Plot of v(x, t) vs x for t = 0.0.

UHtL

UHtL

5.5 60

5 4.5

40

4

20

3.5 0.2

3

0.4

0.6

0.8

1

x

-20

2.5 0.2

0.4

0.6

0.8

x

1


10 UHx,tL 0 -10 -20 0

1 0.75 0.5x 0.25

0.25 0.5 t 0.75

-40

Figure 12: Plot of v(x, t) vs x for t = 1.0.

15 VHx,tL10 5 0 -5 0

1 0.75

0.5 t 0.75

10

Figure 13: 3D plot of u(x, t) vs x and t.

0.5x 0.25

0.25 10

Figure 14: 3D plot of v(x, t) vs x and t.



0.8 uHx,tL 0.7 0.6 0

1 0.75 0.5 0.5

0.25

1

0.5 vHx,tL 0.4

0.75

0

0.5 0.5

0.25

0.25

0.5 0.5 0.75

10

Figure 15: 3D plot of u(x, t) vs x and t.

Figure 16: 3D plot of v(x, t) vs x and t.

1 0.75 0.5y 0.25

0.25 0.5 x 0.75

6 vHx,tL 4 2 0 0

1 0.75

0.5 x 0.75 10

Figure 17: 3 D plot of u(x, t) for t = 0.5

1 0.75 0.5y 0.25

0.25 0.5 x 0.75

0.5y 0.25

0.25

10

6 uHx,tL 4 2 0 0

0.25

0.5 0.5 0.75

10

6 uHx,tL 4 2 0 0

309

Figure 18: 3 D plot of v(x, t) for t = 0.5.

6 vHx,tL 4 2 0 0

1 0.75

0.5 x 0.75

10

Figure 19: 3D plot of u(x, t) for t = 1.0.

0.5y 0.25

0.25 10

Figure 20: 3D plot of v(x, t) for t = 1.0.


310


0.8 uHx,yL 0.6 0.4

1 0.75

0

0.5 0.

0.25

0.6 uHx,yL 0.4 0.2 0 0

1 0.75

0.25

0.5 0. 0.75

0.5 0.

0.25

10

10

Figure 21: 3D plot of u(x, t) for t = 0.5[HPM].

8 vHx,yL 6 4 2 0 0

1 0.75 0.5 0.

0.25

0.25

0.5 0. 0.75

0.25

0.5 0. 0.75

Figure 22: 3D plot of u(x, t) for t = 0.5[EXACT].

8 vHx,yL 6 4 2 0 0

1 0.75 0.5 0.

0.25

10

Figure 23: 3D plot of v(x, t) for t = 0.5[HPM].

0.25

0.5 0. 0.75 10

Figure 24: 3D plot of v(x, t) for t = 0.5[EXACT].

References [1] S. O. Ajadi and M. Zuilino, Approximate analytical solutions of reaction-diffusion equations with exponential source term: Homotopy perturbation method(HPM), Applied Mathematics Letters 24(2011),1634-1639. [2] S. O. Ajadi and P. Vegulla, Instabilities In A Reaction Diffusion Model: Spartially Homogeneous And Distributed System, Applied Mathematics E-Notes, 10(2010),136-146. [3] E. Hairer and G. Wanner(1996): Solving Ordinary Differential Equation II. H. Stiff and Differential-Algebraic Problems. Springer Series in Comput. Math., vol. 14, 2nd edition. [4] S. L. Hollis, R. H. Martin, M. Pierre, Global existence and boundedness in reaction-diffusion systems, SIAM J, Math. Anal, 18(3),(1987),744-760. [5] D. Lesnic, The decomposition method for Cauchy reaction-diffusion problems, Applied Mathematics Letters 20(2007),412-418. [6] M. Ganjiani and H. Ganjiani, Solution of coupled system of non-linear differential equations using homotopy analysis method, Nonlinear Dynamics(2009)(56),159-167. [7] S. Liao, Comparison between homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computation 169(2005)1186-1194. [8] J. H. He, Homotopy perturbation method: A new nonlinear analytical technique, Applied Mathematics and Computation 135(2003),73-79. [9] J. H. He Homotopy perturbation method for solving boundary value problems, Physics Letters A 350(2006),87-88. [10] A. Yildirim, Application of He’s homotopy perturbation method for solving the Cauchy reaction-diffusion problem, Comput Math. Appl.57(2009),612-618. [11] A. Yildirim and H. Kocak, Homotopy perturbation method for solving space-time fractional advectiondispersion equation, Advances in Water Resources, 32(2009),1711-1716. [12] T. Rauber and G. Runger, Aspects of a distributed solution of the Brusselator equation, Proc. of the first Aizu



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International Symposium on Parallel Algorithms and Architecture Synthesis, (1995),114-120. [13] Yuncheng You, Global Dynamics of the Brusselator Equations, Dynamics of PDE, Vol.4, No.2, 167-196, 2007. [14] A. Yildirim and Y. Gulkanat, Analytical approach to fractional Zakharov-Kuzuetsov equations by He’s homotopy perturbation method, Communication in Theoretical Physics, 53(2010),1005-1010. [15] S. Momani and A. Yildirim, Analytical approximate solutions of tne fractional convection-diffusion equation with nonlinear source term by He’s homotopy perturbation method, International Journal of Computer Mathematics, 57(2010),1055-1065. [16] Y. Moriakawa, T. Yamaguchi and T. Amemiya, Analytical Approach for Oscillation Properties of Soft Materials, Forma, 15, 249256, 2000. [17] P. E. Rapp, T. I. Schmah and A. I. Mees, Models of knowing and the investigation of dynamical system, Physica D 132(1999), 133-149. [18] G. C. Wake, J. G. Graham-Eagle and B. F. Gray, Oscillating chemical reactions: the well stirred and spatially distributed cases, Lect. Appl. Math., 24(1986), 331-355. [19] M. P. McDowell and J. M. Powers, Mathematical Modeling of the Brusselator, Department of Aerospace and Mechanical Engineering, University of Notre Dame, Notre Dame, Indiana(2008) AME 36099-01. [20] M. S. H. Chowdhury, T. H. Hassan and S. Mawa, A New Application of Homotopy Perturbation Method to the reaction-diffusion Brusselator Model, Procedia Social and Behavioral Sciences 8(2010),648-653. [21] S. A. Manna, R. K. Saeed and F. H. Easif, Numerical Solution of Brusselator Model by finite Difference Method, Journal of Applied Sciences Research, 6(11) (2010), 1632-1646. [22] I. Karafyllis, P. D. Christofides and P. Daoutidis, Dynamics of a reaction-diffusion system with Brusselator kinetics under feedback control, PHYSICAL REVIEW E 59(1)(1999). [23] P. V. Kuptsov, S. P. Kuznetsov and E. Mosekilde, Particle in the Brusselator model with flow, Physica D 163(2002), 80-88.

Appendix A The Wolfram Mathematica program below provides the platform for obtaining the analytical solutions of the differential equations (24) and (25) subject to the initial-boundary conditions (26) and (27). By using the evaluated expression, then using the plot and plot3D graphic options, it is easy to obtain the 2 and 3 dimensional plots. A := A; B := B; d1 := d1 ; d2 := d2 ; x := x; t := t; y := y; u0 := u0 (x); v0 (x); u1 = Evaluate[d1 ∗ D[u0 , x, x] + u20 ∗ v0 − (B + 1) ∗ u0 + A] v1 = Evaluate[d2 ∗ D[v0 , x, x] − u20 ∗ v0 + B ∗ u0 ] u2 = Evaluate[d1 ∗ D[u1 , x, x] + (2u0 ∗ u1 ∗ v0 + u20 ∗ v1 − (B + 1) ∗ u1 ) − (B + 1) ∗ u1 ] v2 = Evaluate[d2 ∗ D[v1 , x, x] − (2u0 ∗ u1 ∗ v0 + u20 ∗ v1 ) + B ∗ u1 ] u3 = Evaluate[d1 ∗ D[u2 , x, x] + (2u21 ∗ v0 + 2u0 ∗ u2 ∗ v0 + u20 ∗ v2 + 4 ∗ u0 ∗ u1 ∗ v1 ) −(B + 1) ∗ u2 ] v3 = Evaluate[d2 ∗ D[v2 , x, x] − (2u21 ∗ v0 + 2u0 ∗ u2 ∗ v0 + u20 ∗ v2 + 4 ∗ u0 ∗ u1 ∗ v1 ) −B ∗ u2 ] u4 = Evaluate[d1 ∗ D[u3 , x, x] + (6 ∗ u1 ∗ u2 ∗ v0 + 2 ∗ v0 ∗ u3 ∗ v0 + 6 ∗ u21 + v1 +6 ∗ u0 ∗ u2 ∗ v1 + 6 ∗ u0 ∗ u1 ∗ v2 ) − (B + 1) ∗ u3 ] v4 = Evaluate[d2 ∗ D[v3 , x, x] − (6 ∗ u1 ∗ u2 ∗ v0 + 2 ∗ u0 ∗ u3 ∗ v0 + 6 ∗ u21 + v1 +6 ∗ u0 ∗ u2 ∗ v1 + 6 ∗ u0 ∗ u1 ∗ v2 ) − B ∗ u3 ] u5 = Evaluate[d1 ∗ D[u4 , x, x] + (6 ∗ u22 ∗ v0 + 8 ∗ u1 ∗ u3 ∗ v0 + 10 ∗ u0 ∗ u4 ∗ v0 +24 ∗ u1 ∗ u2 ∗ v1 + 8 ∗ u0 ∗ u3 ∗ v1 + 12 ∗ u21 ∗ v2 + 12 ∗ u0 ∗ u2 ∗ v2 + 8 ∗ u0 ∗ u1 ∗ v3 ) − (B + 1) ∗ u4 ] v5 = Evaluate[d2 ∗ D[v4 , x, x] + (6 ∗ u22 ∗ v0 + 8 ∗ u1 ∗ u3 ∗ v0 + 10 ∗ u0 ∗ u4 ∗ v0 +24 ∗ u1 ∗ u2 ∗ v1 + 8 ∗ u0 ∗ u3 ∗ v1 + 12 ∗ u21 ∗ v2 + 12 ∗ u0 ∗ u2 ∗ v2 + 8 ∗ u0 ∗ u1 ∗ v3 ) − B ∗ u4 ]

Appendix B The Wolfram Mathematica program below provides the platform for obtaining the analytical solutions of the differential equations (34) and (35) subject to the initial-boundary conditions (36) and (37). Taking advantage of the evaluated


312


expression and the plot3D graphic options on Mathematica, we obtain the 2 and 3 dimensonal plots. A := A; B := B; d1 := d1 ; d2 := d2 ; x := x; t := t; y := y; u0 := u0 (x); v0 (x); u1 = Evaluate[d1 ∗ D[u0 , x, x] + d1 ∗ D[u0 , y, y] + u20 ∗ v0 − (B + 1) ∗ u0 + A] v1 = Evaluate[d2 ∗ D[v0 , x, x] + d2 ∗ D[v0 , y, y] − u20 ∗ v0 + B ∗ u0 ] u2 = Evaluate[d1 ∗ D[u1 , x, x] + d1 ∗ D[v1 , y, y] + (2u0 ∗ u1 ∗ v0 + u20 ∗ v1 − (B + 1) ∗ u1 ) − (B + 1) ∗ u1 ] v2 = Evaluate[d2 ∗ D[v1 , x, x] + d2 ∗ D[v1 , y, y] − (2u0 ∗ u1 ∗ v0 + u20 ∗ v1 ) + B ∗ u1 ] u3 = Evaluate[d1 ∗ D[u2 , x, x] + d1 ∗ D[u2 , y, y] + (2u21 ∗ v0 + 2u0 ∗ u2 ∗ v0 + u20 ∗ v2 + 4 ∗ u0 ∗ u1 ∗ v1 ) −(B + 1) ∗ u2 ] v3 = Evaluate[d2 ∗ D[v2 , x, x] + d2 ∗ D[v2 , y, y] − (2u21 ∗ v0 + 2u0 ∗ u2 ∗ v0 + u20 ∗ v2 + 4 ∗ u0 ∗ u1 ∗ v1 ) −B ∗ u2 ] u4 = Evaluate[d1 ∗ D[u3 , x, x] + d1 ∗ D[u3 , y, y] + (6 ∗ u1 ∗ u2 ∗ v0 + 2 ∗ v0 ∗ u3 ∗ v0 + 6 ∗ u21 + v1 +6 ∗ u0 ∗ u2 ∗ v1 + 6 ∗ u0 ∗ u1 ∗ v2 ) − (B + 1) ∗ u3 ] v4 = Evaluate[d2 ∗ D[v3 , x, x] + d2 ∗ D[v3 , y, y] − (6 ∗ u1 ∗ u2 ∗ v0 + 2 ∗ u0 ∗ u3 ∗ v0 + 6 ∗ u21 + v1 +6 ∗ u0 ∗ u2 ∗ v1 + 6 ∗ u0 ∗ u1 ∗ v2 ) − B ∗ u3 ] u5 = Evaluate[d1 ∗ D[u4 , x, x] + d1 ∗ D[u4 , y, y] + (6 ∗ u22 ∗ v0 + 8 ∗ u1 ∗ u3 ∗ v0 + 10 ∗ u0 ∗ u4 ∗ v0 +24 ∗ u1 ∗ u2 ∗ v1 + 8 ∗ u0 ∗ u3 ∗ v1 + 12 ∗ u21 ∗ v2 + 12 ∗ u0 ∗ u2 ∗ v2 + 8 ∗ u0 ∗ u1 ∗ v3 ) − (B + 1) ∗ u4 ] v5 = Evaluate[d2 ∗ D[v4 , x, x] + d2 ∗ D[v4 , y, y] + (6 ∗ u22 ∗ v0 + 8 ∗ u1 ∗ u3 ∗ v0 + 10 ∗ u0 ∗ u4 ∗ v0 +24 ∗ u1 ∗ u2 ∗ v1 + 8 ∗ u0 ∗ u3 ∗ v1 + 12 ∗ u21 ∗ v2 + 12 ∗ u0 ∗ u2 ∗ v2 + 8 ∗ u0 ∗ u1 ∗ v3 ) − B ∗ u4 ]
