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SPATIO-TEMPORAL NUMERICAL MODELING OFAUTO-CATALYTIC BRUSSELATOR MODEL

NAUMAN AHMED1,5,a, MUHAMMAD RAFIQ2,b,DUMITRU BALEANU3,4,∗, MUHAMMAD AZIZ-UR REHMAN1,c

1Department of Mathematics, University of Management and Technology,Lahore, Pakistan

E-mail a: , E-mail c:2Faculty of Engineering, University of Central Punjab,

Lahore, PakistanE-mail b:

3Department of Mathematics, Cankaya University,Eskisehir Yolu 29. Km, Yukariyurtu Mahallesi Mimar Sinan Caddesi No:4 06790,

Etimesgut, Ankara, TurkeyCorresponding author∗:

4Institute of Space Sciences,RO-077125 Magurele-Bucharest, Romania

E-mail ∗:5Department of Mathematics and Statistics, University of Lahore,

Lahore, PakistanE-mail a:

Received June 5, 2019

Abstract. The main objective of this article is to propose a chaos free explicitfinite-difference (FD) scheme to find the numerical solution for the Brusselator reaction-diffusion model. The scheme is unconditionally stable and it is unconditionally dynam-ically consistent with the positivity property of continuous model as unknown quan-tities of auto-catalytic Brusselator system describe the concentrations of two reactantsubstances. Stability of the proposed FD method is showed with the help of Neumanncriteria of stability. Taylor series is used to validate the consistency of the proposed FDmethod. Forward Euler explicit FD approach and semi-implicit Crank-Nicolson FDscheme are also applied to solve the Brusselator reaction-diffusion system and to makethe comparison with the proposed FD scheme.

Key words: Auto-catalytic chemical reaction; structure preserving scheme;consistency; Von Neumann stability analysis; numericalsimulations.

1. INTRODUCTION

The dynamical systems described by ordinary and partial differential equationsare used very frequently in different disciplines of science, engineering, and technol-ogy. Nonlinearity factor describes the physical phenomena more accurately, but thenonlinear terms involved in the dynamical models make the solutions more difficult

Romanian Journal of Physics 64, 110 (2019)

Article no. 110 Nauman Ahmed et al. 2

to obtain and quite complex ones. Many important quantities such as concentra-tions, density, pressure, and population sizes, etc. depend on the time variable, hencethe structural properties like positivity, boundedness, and convergence towards sta-ble equilibrium point must be kept in mind in the analysis and the computationalschemes of such nonlinear dynamical systems. To find the analytical solutions of thedynamical models involving nonlinear evolution equations is either a very difficulttask or is not possible, therefore a plethora of numerical methods are frequently usedto find the solutions of ordinary differential equations (ODEs) and partial differentialequations (PDEs); see, for example, Refs. [1–7].

In this article, we consider a nonlinear autocatalytic Brusselator chemical re-action model, proposed in 1970 by Prigogine. This model is very useful in chemicalkinetics for the study of many physical processes. This type of Brusselator modelarises in autocatalytic reaction. This is a reaction with nonlinearity in which tworeactant species interact with each other to increase the production rate. The Brus-selator model also appears in the modeling of some chemical reaction-diffusion pro-cesses in various fields of physics such as plasma and laser physics. The Brusselatormodel consists of two nonlinear partial differential equations, therefore it attracts theresearchers working on numerical solutions of these types of problems. Mittal andRohila [8] have used the cubic B-spline differential quadrature technique for findingthe solution of one-dimensional reaction-diffusion systems. Ersoy and Dag [9] haveobtained the numerical solutions of the reaction-diffusion system by using exponen-tial cubic B-spline collocation algorithms. Hu et al. [10] have solved the Brusselatorand Gray-Scott models with moving finite element method. The homotopy pertur-bation method has been used to find the approximate closed form solution for Brus-selator system by Ajadi et al. [11]. Manna et al. [12] have used the forward Eulerexplicit and Crank-Nicolson implicit finite difference schemes to solve the Brusse-lator reaction-diffusion system. Twizell et al. [13] have proposed a second orderimplicit finite-difference (FD) scheme for the Brusselator reaction-diffusion system.Adomian [14] and Wazwaz [15] have solved this type of system by a decompositionmethod. Ang [16] has introduced the dual reciprocity boundary element method toobtain the numerical solution of the Brusselator model. Lin et al. [17] have proposedfinite volume method for pattern formation of inhomogeneous Brusselator cross dif-fusion system. Fernandes et al. [18] have introduced an extrapolated alternatingdirection implicit Crank-Nicolson orthogonal spline colocation method and have ap-plied it on chemical reaction-diffusion systems like Brusselator reaction-diffusionsystem, Grey-Scot reaction diffusion system, etc. In this work we propose an explicitchaos free FD scheme [19–23] that preserves the positivity property uncondition-ally. The proposed FD scheme, which is applied to find the numerical solution ofreaction-diffusion problems, preserves all important characteristics of the solution ofcontinuous systems.

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3 Numerical modeling of auto-catalytic Brusselator model Article no. 110

Let us consider the one-dimensional coupled reaction-diffusion Brusselatormodel

∂u

∂t= du

∂2u

∂x2+ρ2− (ρ1 + 1)u+ (u)2 v (1)

∂v

∂t= dv

∂2v

∂x2+ρ1u− (u)2 v (2)

with given initial and homogeneous Neumann boundary conditions. Here u= u(x,t)and v= v(x,t), u describes the concentration of first reactant species and v describesthe concentration of second reactant species, ρ1 and ρ2 represent the constant con-centrations during the reaction process and du,dv are the diffusion constants. Be-sides, ρ1,ρ2,du, and dv are positive. The equilibrium point of system (1)-(2) is(u∗,v∗) = (ρ2,ρ1/ρ2). The solution of the system (1)-(2) describes the positivityproperty as u,v are the concentrations of two species [24]. We recall that the non-standard finite difference (NSFD) method was developed by Mickens [25], whichgives the idea for designing of structure preserving FD schemes. This method is avery powerful technique to solve mathematical models containing systems of ordi-nary and partial differential equations, especially the models where positive solu-tions are necessary because NSFD schemes preserve the positivity property. Overthe years, many authors developed different NSFD schemes for different ODEs andPDEs models [26–31]. The Brusselator model contains a system of ODEs, solvednumerically by Zain et al. [24] with the help of a nonstandard FD method that pre-serves the positivity property. We also mention that Mickens [32] has constructed thepositivity-preserving nonstandard FD scheme for the Glycolysis Model.

This paper is organized as follows. In Sec. 2, we present two well-known me-thods and we introduce a positivity preserving method that is employed in this work.Also, we prove mathematically that the proposed method preserves the positive so-lution unconditionally. Section 3 is devoted to prove the unconditional stability ofthe proposed method by employing the Neumann stability criteria. The consistencyof the proposed method is presented in Sec. 4 by using the Taylor series method. InSec. 5, we provide numerical simulations in support of the convergent character ofour technique. Finally, in Sec. 6 we give our concluding remarks.

2. NUMERICAL TECHNIQUES

In this work, we use FD techniques for the solution of reaction-diffusion sys-tems. For the FD schemes, the domain in xt-plane is subdivided into aM×N matrixwith steps h= L

M and τ = TN . The grid points are

xi = ih, i= 1,2,3, · · ·,M (3)

tn = nτ, n= 1,2,3, · · ·,N (4)

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The FD approximations of uni and vni are denoted by u(ih,nτ) and v(ih,nτ), res-pectively. We get

(1 +λ1)un+1i − λ1

2(un+1i−1 +un+1

i+1 ) = (1−λ1)uni +λ1

2(uni−1 +uni+1)

+ τρ2− τ(ρ1 + 1)uni + τ(uni )2vni (5)

(1 +λ2)vn+1i − λ2

2(vn+1i−1 +vn+1

i+1 ) = (1−λ2)vni +λ2

2(vni−1 +vni+1)

+ τρ1uni − τ(uni )2vni (6)

where λ1 =du1τ

h2and λ2 =

du2τ

h2.

The numerical scheme is unconditionally stable for the system (1)-(2) [33].Now the forward Euler FD scheme for the system (1)-(2) is

un+1i =uni +λ1(uni−1−2uni +uni+1) + τρ2− τ(ρ1 + 1)uni + τ(uni )2vni (7)

vn+1i =vni +λ2(vni−1−2vni +vni+1) + τρ1u

ni − τ(uni )2vni (8)

The numerical scheme is conditionally stable for λ1 ≤ (2−τ(A+1))4 and λ2 ≤ 1/2

for the system (1)-(2) [33]. Now we design unconditionally positivity preservingexplicit FD scheme [19–21] with the help of rules proposed by Mickens for the NSFDschemes [25]. We thus get

un+1i =uni +λ1(uni−1 +uni+1)−2λ1u

n+1i

+τρ2− τ(ρ1 + 1)un+1i + τ(uni )2vni (9)

un+1i =

uni +λ1(uni−1 +uni+1) + τρ2 + τ(uni )2vni(1 + 2λ1 + τ(ρ1 + 1))

(10)

In a similar way, we have

vn+1i =vni +λ2(vni−1 +vni+1)−2λ2v

n+1i

+τρ1uni − τ(uni )2vn+1

i (11)

vn+1i =

vni +λ2(vni−1 +vni+1) + τρ1uni

(1 + 2λ2 + τ(uni )2)(12)

2.1. THEOREM

The numerical scheme demonstrates the positive solution under the suppositionof non-negative initial conditions if

uni ≥ 0,vni ≥ 0⇒ un+1i ≥ 0,vn+1

i ≥ 0 (13)

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2.1.1. Remark

The parameters ρ1,ρ2,du1 ,du2 involved in equations (10) and (12) are all pos-itive. If the initial conditions are non-negative then the equations (10) and (12) showpositivity unconditionally because the right hand sides of (10) and (12) have no neg-ative terms.

3. STABILITY OF THE PROPOSED TECHNIQUE

We present the Neumann stability technique to find the stability region of pro-posed FD scheme. To this aim, we linearize Eq. (9) and then substituting Φ(t)ei$x

for uni , we have

Φ(t+ ∆t)ei$x = Φ(t)ei$x+λ1(Φ(t)ei($(x−∆x)) + Φ(t)ei($(x+∆x)))

− 2λ1Φ(t+ ∆t)ei($x)− τ(ρ1 + 1)Φ(t+ ∆t)ei$x+ τab ,

where a and b are constants. We get

(1 + 2λ1 + τ(ρ1 + 1))Φ(t+ ∆t) = Φ(t) +λ1(2cos($∆x))Φ(t) + τab

Φ(t+ ∆t)

Φ(t)=

1 + τab+ 2λ1 cos($∆x)

1 + 2λ1 + τ(ρ1 + 1)=

1 + τab+ 2λ1(1−2sin2($∆x/2))

1 + 2λ1 + τ(ρ1 + 1)∣∣∣∣Φ(t+ ∆t)

Φ(t)

∣∣∣∣ =1 + τab+ 2λ1−4λ1 sin2($∆x/2)

1 + 2λ1 + τ(ρ1 + 1)≤ 1 + τab−2λ1

1 + 2λ1 + τ(ρ1 + 1)< 1

(14)

A similar procedure is used for (12), and we have∣∣∣∣Φ(t+ ∆t)

Φ(t)

∣∣∣∣ =1 + τρ1 + 2λ2−4λ2 sin2($∆x/2)

1 + 2λ2 + τ(a)2≤ 1 + τρ1−2λ2

1 + 2λ2 + τ(a)2< 1 (15)

From the above result, it is clear that the proposed FD technique is unconditionallystable.

4. CONSISTENCY OF THE PROPOSED TECHNIQUE

To check the consistency [19, 21] of the proposed FD scheme, the Taylor ex-pansion is used for un+1

i , uni−1, and uni+1:

un+1i = uni + τ

∂u

∂t+τ2

2!

∂2u

∂t2+τ3

3!

∂3u

∂t3+ · · ·

uni+1 = uni +h∂u

∂x+h2

2!

∂2u

∂x2+h3

3!

∂3u

∂x3+ · · ·

uni−1 = uni −h∂u

∂x+h2

2!

∂2u

∂x2− h

3

3!

∂3u

∂x3+ · · ·

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Considering the unconditionally positivity preserving scheme (10) and replacing thevalues of un+1

i ,uni+1, and uni−1 in it, we obtain the following(∂u

∂t+τ

2!

∂2u

∂t2+τ2

2!

∂3u

∂t3+ · · ·

)(1 + 2

duτ

h2+ τ(ρ1 + 1)

)= 2du

(1

2!

∂2u

∂x2+h2

4!

∂4u

∂x4+ · · ·

)+uni (−ρ1−1) +ρ2 + (uni )2vni

We replace τ = h3 and h→ 0 and we have

∂u

∂t= du

∂2u

∂x2+ρ2− (ρ1 + 1)u+ (u)2 v

(∂u1

∂t+τ

2!

∂2u1

∂t2+τ2

2!

∂3u1

∂t3+ · · ·

)(1 + 2

du1τ

h2+ τ(ρ1 + 1)

)= 2du1

(1

2!

∂2u1

∂x2+h2

4!

∂4u1

∂x4+ · · ·

)+un1i(−ρ1−1) +ρ2 + (un1i)

2un2i

We replace τ = h3 and h→ 0, and we have

∂u1

∂t=du1

∂2u1

∂x2+ρ2− (ρ1 + 1)u1 + (u1)2u2

In a similar way, the formulas for vn+1i , vni−1, and vni+1 are as follows

vn+1i =vni + τ

∂v

∂t+τ2

2!

∂2v

∂t2+τ3

3!

∂3v

∂t3+ · · · (16)

vni+1 =vni +h∂v

∂x+h2

2!

∂2v

∂x2+h3

3!

∂3v

∂x3+ · · · (17)

vni−1 =vni −h∂v

∂x+h2

2!

∂2v

∂x2− h

3

3!

∂3v

∂x3+ · · · (18)

We consider the proposed FD scheme (12) for equation (2)

vn+1i =vni +λ2(vni−1 +vni+1)−2λ2v

n+1i

+τρ1uni − τ(uni )2vn+1

i (19)

We put the values of vn+1i ,vni+1, and vni−1 in Eq. (19) and after some computations,

we have (∂v

∂t+τ

2!

∂2v

∂t2+τ2

3!

∂3v

∂t3+ · · ·

)(1 + 2

dvτ

h2+ τ(uni )2

)= 2dv

(1

2!

∂2v

∂x2+h2

4!

∂4v

∂x4+ · · ·

)+ρ1u

ni − (uni )2vni

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We replace τ = h3 and h→ 0, and we have

∂v

∂t=dv

∂2v

∂x2+ρ1u− (u)2 v

5. EXAMPLE

We consider the auto-catalytic chemical reaction Brusselator system in one di-mension [8] as

∂u

∂t= du

∂2u

∂x2+ρ2− (ρ1 + 1)u+ (u)2 v (20)

∂v

∂t= dv

∂2v

∂x2+ρ1u− (u)2 v, (21)

Fig. 1 – Mesh graph of u. Fig. 2 – Mesh graph of v.

Fig. 3 – Combined plot of u and v. Fig. 4 – Combined plot of u and v.

subject to the initial data

u(x,0) = 1/2, v(x,0) = 1 + 5x, (22)

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and the derivatives of the solution pair are prescribed on the boundary as follows

ux(0, t) = ux(1, t) = 0 vx(0, t) = vx(1, t) = 0. (23)

Now we will present the numerical simulations of all FD schemes at differentstep sizes to see to behavior of all these schemes under consideration.

(a) Mesh graph of u. (b) Mesh graph of v.

(c) Combined plot of u and v. (d) Combined plot of u and v.

Fig. 5 – Concentrations u and v using the Crank-Nicolson FD scheme.

In Figs. 1-3, we take the values h = 0.05, τ = 0.06, ρ1 = 1, ρ2 = 3.4,du = dv = 10−4, which implies that λ1 = λ2 = 0.0024. If we consider these val-ues, then the stability conditions of λ1 and λ2 for the forward Euler FD scheme areλ1 ≤ 0.434 and λ2 ≤ 0.5. This verifies that the values under consideration of thisnumerical experiment are within the stability criteria of forward Euler FD scheme.Now from Fig. 2 and Fig. 3 we see that the graph of v clearly shows that the for-ward Euler scheme violates the positivity property and gives negative values. Thenegative values for the concentration of reactant species are meaningless. Note that

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Twizell et al. [13] concluded that the system (1)-(2) is attracted to the equilibriumpoint (u∗,v∗) if 1−ρ1 +ρ2

2 ≥ 0. But the forward Euler FD scheme does not convergeto the equilibrium point and diverges at λ1 = λ2 = 0.0036 as shown in Fig. 4.

Figure 5 presents the solutions of concentrations u and v using the Crank-Nicolson FD scheme. We apply the semi-implicit Crank-Nicolson FD scheme thatis unconditionally stable. The graphs in Figs. 5 are plotted with the same val-ues as taken for the forward Euler FD scheme in Figs. 1-4. The behaviors of theCrank-Nicolson semi-implicit FD scheme and the forward Euler scheme are similar.The Crank-Nicolson scheme is also showing negative values of concentration. TheCrank-Nicolson FD scheme overflows and fails to show the convergence towards theequilibrium point (u∗,v∗).


(c) Combined plot of u and v.

Fig. 6 – Graphs of concentration u(x,t) and v(x,t).

Figure 6 shows the graphs of concentration u(x,t) and v(x,t) using the pro-posed FD scheme at h= 0.05, τ = 0.06, ρ1 = 1, ρ2 = 3.4, du = dv = 10−4, which

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implies that λ1 = λ2 = 0.0024. These values are the same as given for the for-ward Euler method and the Crank-Nicolson method. Figure 7 contains the graphsfor u and v of the proposed FD scheme at λ1 = λ2 = 0.0036. It can be observedthat the proposed scheme not only preserves the positivity but also converges to theequilibrium points of the Brusselator reaction-diffusion system under the condition1−ρ1 +ρ2

2 ≥ 0.


(c) Combined plot of u and v.

Fig. 7 – Graphs for u and v at λ1 = λ2 = 0.0036.

The properties possessed by the Brusselator one-dimensional reaction-diffusionsystem are preserved by the proposed FD scheme. Figures 8 and 9 demonstratethat the proposed method preserves the positive solution of the continuous Brusse-lator model even at large values λ1 = λ2 = 100. Also, the FD scheme converges tothe equilibrium point of the Brusselator one-dimensional reaction-diffusion system,which shows the novelty and efficacy of the proposed FD scheme.

As discussed above, the equilibrium point (u∗,v∗) of the system (1)-(2) is sta-

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ble for 1−ρ1 +ρ22 ≥ 0 and unstable for 1−ρ1 +ρ2

2 < 0. In Figs. 10-13, the valuesof ρ1 and ρ2 are taken in such a way that 1− ρ1 + ρ2

2 < 0. Here we take ρ1 = 3.4,ρ2 = 1, h= 0.1, λ1 = λ2 = 0.004. All the graphs in Figs. 10-13 validate the conclu-sion made by Twizzel et al. [13] and show that the equilibrium point (u∗,v∗) of thesystem (1)-(2) becomes unstable when 1−ρ1 +ρ2

2 < 0.

6. CONCLUSION

In this article, the auto-catalytic chemical reaction Brusselator system in one di-mension is solved by three different finite-difference numerical schemes, the forwardEuler scheme, the Crank-Nicolson semi-implicit scheme, and the proposed finite-difference scheme. Remember that for every numerical method, any property pos-sessed by a continuous system must be preserved by that numerical method. In non-linear problems, many well-known numerical methods fail to preserve the propertiespossessed by the continuous reaction-diffusion system. To overcome these issues,we have proposed a novel FD scheme that is designed for the numerical approxi-mation of reaction-diffusion problems. The proposed method preserves the positivesolution of a continuous system unconditionally. This novel method is an explicitFD scheme, which is computationally efficient. This numerical scheme is uncondi-tionally stable and unconditionally consistent with respect to the positivity property.These attributes are verified by numerical simulations. The graphs obtained usingthe proposed FD scheme show the convergence, consistency, and positivity of thenumerical scheme. On the other hand, the graphs show the divergence of the forwardEuler and Crank-Nicolson schemes for different values of the parameters. These twonumerical schemes also failed to preserve the positivity property for different valuesof step sizes.

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Fig. 12 – Plot graph of u. Fig. 13 – Plot graph of v.

REFERENCES

1. I. A. Cristescu, Rom. J. Phys. 63, 105 (2018).2. Y. H. Youssri, W. M. Abd-Elhameed, Rom. J. Phys. 63, 107 (2018).3. M. Oane, D. Ticos, C. Ristoscu, M. Badiceanu, A. Buca, I. N. Mihailescu, Rom. J. Phys. 63, 501

(2018).4. R. M. Hafez, E. H. Doha, A. H. Bhrawy, D. Baleanu, Rom. J. Phys. 62, 111 (2017).5. A. A. El-Kalaawy et al., Rom. Rep. Phys. 70, 109 (2018).6. H. O. Bakodah, A. Ebaid, Rom. Rep. Phys. 70, 115 (2018).7. M. Tlidi, K. Panajotov, Rom. Rep. Phys. 70, 406 (2018).8. R. C. Mittal, Rajni Rohila, Chaos, Solitons and Fractals 92, 9-19 (2016).9. O. Ersoy, I. Dag, Open Phys. 13(1), 414-427 (2015).

10. G. Hu, Z. Qiao, T. Tang, Adv. Appl. Math. Mech. 4(3), 365-381 (2012).11. S. O. Ajadi, O. A. Adesina, O. A. Jejeniwa, Int. J. Nonl. Sci. 16(4), 300-312 (2013).12. S. A. Manna, R. K. Saeed, F. H. Easif, Journal of Applied Sciences Research, 6(11), 1632-1646

(2010).13. E. H. Twizell, A. B. Gumel, Q. Cao, J. Math. Chem. 26, 297-316 (1999).14. G. Adomian, Comput. Math. Appl. 59, 1-3 (1995).

(c) RJP64(Nos. 7-8), ID 110-1 (2019) v.2.2r20190928 *2019.10.2#4239c10f


15. A. M. Wazwaz, Appl. Math. Comput. 110, 251-264 (2000).16. W. T. Ang, Eng. Anal. Bound Elem. 27, 897-903 (2003).17. Z. Lin, R. Ruiz-Baier, C. Tian, J. Comput. Phys. 256, 806-823 (2014).18. R. I. Fernandes, G. Fairweather, J. Comput. Phys. 231, 6248-6267 (2012).19. B. M. Chen-Charpentier, H. V. Kojouharov, Math. Comput. Modelling 57, 2177-1285 (2013).20. G. F. Sun, G. R. Liu, M. Li, Math. Prob. Eng. 2015, 708497 (2015).21. A. R. Appadu, Jean M-S. Lubuma, N. Mphephu, Prog. Comput. Fluid Dyn. 17, 114-129 (2017).22. N. Ahmed, M. Rafiq, M. A. Rehman, M. S. Iqbal, M. Ali, AIP Advances 9, 015205 (2019).23. N. Ahmed, M. Rafiq, M. A. Rehman, M. Ali, M. O. Ahmad, Communications in Mathematics and

Applications 9(3), 315-326 (2018).24. Z. A. Zafar, K. Rehan, M. Mushtaq, M. Rafiq, J. Diff. Eq. Appl. 23, 1-18 (2016).25. R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific,

1994.26. R. E. Mickens, Numer Methods Partial Diff. Eqs. 15, 201-214 (1999).27. M. Chapwanya, J. M.-S. Lubuma, R. E. Mickens, Comput. Math. Appl. 68(9), 1071-1082 (2014).28. R. E. Mickens, J. Diff. Eq. Appl. 10, 1307-1312 (2004).29. R. E. Mickens, Comput. Math. Appl. 45, 429-436 (2003).30. R. E. Mickens, Numer Methods Partial Diff. Eqs. 13, 51-55 (1997).31. R. Anguelov, J. M.-S. Lubuma, Numer Methods Partial Diff. Eqs. 17, 518-543 (2001).32. R. E. Mickens, Positivity preserving discrete model for the coupled ODE’s modeling glycolysis,

Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equa-tions, Wilmington, NC, USA, pp. 623-629, 2002.

33. F. H. Easif, R. K. Saeed, S. A. Manna, Al-Rafidain Journal of Computer Sciences and Mathematics8(2), 43-52 (2011).

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