An Efficient Method for Solving the Brusselator System



In this paper, a new efficient recurrence relation is constructed to solve a nonlinear Brusselator equation. The system, known as the reaction-diffusion Brusselator, arises in the modeling of certain diffusion processes. The Laplace transform method and the new homotopy perturbation method (NHPM) are used to solve these equations. Since mathematical modeling of numerous scientific and engineering experiment lead to the Brusselator equation, it is worthwhile to try new methods to solve this system. Comparison of the results with those of the homotopy perturbation method, the Adomian decomposition method and the dual-reciprocity boundary element method leads to significant consequences. The method is tested using various examples and the results show that the new method is more effective and convenient to use, and has an evident high accuracy rate.


Laplace transform method, new homotopy perturbation method (NHPM), Brusselator equation, reaction-diffusion systems

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M Ghergu and V Radulescu. Turing patterns in general reaction-diffusion systems of Brusselator type. Commun. Contemp. Math. 2010; 12, 661-79.

G Nicolis and I Prigogine. Self-Organization in Nonequilibrium Systems. John Wiley & Sons, New York, 1977, p. 1-152.

I Prigogine and R Lefever. Symmetry breaking Instabilities in Dissipative systems. J. Chem. Phys. 1968; 48, 1695-700.

J Tyson. Some further studies of nonlinear Oscillations in chemical systems. J. Chem. Phys. 1973; 58, 3919-30.

G Adomian. The diffusion-Brusselator equation. Comput. Math. Appl. 1995; 29, 1-3.

S Vandewalle and R Piessens. Numerical experiments with nonlinear multigrid waveform relaxation on a parallel processor. Appl. Numer. Math. 1991; 8, 149-61.

C Lubich and A Ostermann. Multigrid dynamic interaction for parabolic equations. BIT Numer. Math. 1987; 27, 216-34.

EH Twizell, AB Gumel and Q Cao. A second-order scheme for the Brusselator reaction-diffusion system. J. Math. Chem. 1999; 26, 297-316.

JH He. Homotopy perturbation technique. Comput. Meth. Appl. Mech. Eng. 1999; 178, 257-62.

JH He. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int. J. Nonlin. Mech. 2000; 35, 37-43.

JH He. New interpretation of homotopy perturbation method. Int. J. Mod. Phys. B 2006; 20, 2561-8.

JH He. Recent development of the homotopy perturbation method. Topol. Meth. Nonlinear Anal. 2008; 31, 205-9.

JH He. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl. Math. Comput. 2004; 151, 287-92.

JH He. Application of homotopy perturbation method to nonlinear wave equations. Chaos Soliton. Fract. 2005; 26, 695-700.

JH He. Limit cycle and bifurcation of nonlinear problems. Chaos Soliton. Fract. 2005; 26, 827-33.

A Rajabi and DD Ganji. Application of homotopy perturbation method in nonlinear heat conduction and convection equations. Phys. Lett. A 2007; 360, 570-3.

DD Ganji and A Sadighi. Application of homotopy perturbation and variational iteration methods to nonlinear heat transfer and porous media equations. J. Comput. Appl. Math. 2007; 207, 24-34.

DD Ganji. The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer. Phys. Lett. A 2006; 355, 337-41.

S Abbasbandy. A numerical solution of Blasius equation by Adomian’s decomposition method and comparison with homotopy perturbation method. Chaos Soliton. Fract. 2007; 31, 257-60.

J Biazar and H Ghazvini. Exact solutions for nonlinear Schrödinger equations by He’s homotopy perturbation method. Phys. Lett. A 2007; 366, 79-84.

S Abbasbandy. Numerical solutions of the integral equations: homotopy perturbation and Adomian’s decomposition method. Appl. Math. Comput. 2006; 173, 493-500.

JH He. Homotopy perturbation method for solving boundary value problems. Phys. Lett. A 2006; 350, 87-8.

Q Wang. Homotopy perturbation method for fractional KdV-Burgers equation. Chaos Soliton. Fract. 2008; 35, 843-50.

E Yusufoglu. Homotopy perturbation method for solving a nonlinear system of second order boundary value problems. Int. J. Nonlinear Sci. Numer. Simulat. 2007; 8, 353-8.

DD Ganji. A semi-Analytical technique for non-linear settling particle equation of Motion. J. Hydro Environ. Res. 2012; 6, 323-7.

M Jalaal, DD Ganji and G Ahmadi. Analytical investigation on acceleration motion of a vertically falling spherical particle in incompressible Newtonian media. Adv. Powder Tech. 2010; 21, 298-304.

M Jalaal and DD Ganji. On unsteady rolling motion of spheres in inclined tubes filled with incompressible Newtonian fluids. Adv. Powder Tech. 2011; 22, 58-67.

M Jalaal and DD Ganji. An analytical study on motion of a sphere rolling down an inclined plane submerged in a Newtonian fluid. Power Tech. 2010; 198, 82-92.

P Ashwin and Z Mei. Normal form for Hopf bifurcation of partial differential equations on the square. Nonlinearity 1995; 8, 715-34.

K J Brown and FA Davidson. Global bifurcation in the Brusselator system. Nonlin. Anal. 1995; 24, 1713-25.

T Erneux and E Reiss. Brusselator isolas. SIAM J. Appl. Math. 1983; 43, 1240-6.

M Ghergu. Non-constant steady states for Brusselator type systems. Nonlinearity 2008; 21, 2331-45.

H Kang. Dynamics of local map of a discrete Brusselator model: Eventually trapping regions and strange attractos. Discret. Contin. Dyn. Syst. 2008; 20, 939-59.

T Kolokolnikov, T Erneux and J Wei. Mesa-type patterns in the one-dimensional Brusselator and their stability. Phys. D 2006; 214, 63-77.

B Pena and C Perez-Garcia. Stability of turing patterns in the Brusselator model. Phys. Rev. E 2001; 64, 056213.

R Peng and M Wang. Pattern formation in the Brusselator system. J. Math. Anal. Appl. 2005; 309, 151-66.

Y You. Global dynamics of the Brusselator equations. Dynam. Part. Differ. Eq. 2007; 4, 167-96.

A Whye-Teong. The two-dimensional reaction-diffusion Brusselator system: A dual-reciprocity boundary element solution. Eng. Anal. Bound. Elem. 2003; 27, 897-903.

J Biazar and Z Ayati. A Numerical Solution of Reaction-Diffusion Brusselator System by ADM. In: MR Islam (ed.). Perspectives on Sustainable Technology. Nova Science Publishers Inc, United States, 2008, p. 191.

AM Wazwaz. The decomposition method applied to systems of partial differential equations and to the reaction-diffusion Brusselator model. Appl. Math. Comput. 2000; 110, 251-64.

AM Wazwaz. The variational iteration method for solving linear and nonlinear systems of PDEs. Comput. Math. Appl. 2007; 54, 895-902.

MSH Chowdhury, TH Hassan and S Mawa. A new application of homotopy perturbation method to the reaction-diffusion Brusselator model. Procedia Soc. Behav. Sci. 2010; 8, 648-53.

H Aminikhah and M Hemmatnezhad. An efficient method for quadratic Riccati differential equation. Commun. Nonlinear Sci. Numer. Simulat. 2010; 15, 835-9.

H Aminikhah. The combined Laplace transform and new homotopy perturbation methods for Stiff systems of ODEs. Appl. Math. Model. 2012; 36, 3638-44.


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Last updated: 12 August 2019