Semi-Analytical Solutions for the Brusselator Reaction–diffusion Model

Abstract

Semi-analytical solutions are derived for the Brusselator system in one- and two-dimensional domains. The Galerkin method is processed to approximate the governing partial differential equations via a system of ordinary differential equations. Both steady-state concentrations and transient solutions are obtained. Semi-analytical results for the stability of the model are presented for the identified critical parameter value at which a Hopf bifurcation occurs. The impact of the diffusion coefficients on the system is also considered. The results show that diffusion acts to stabilize the systems better than the equivalent nondiffusive systems with the increasing critical value of the Hopf bifurcation. Comparison between the semi-analytical and numerical solutions shows an excellent agreement with the steady-state transient solutions and the parameter values at which the Hopf bifurcations occur. Examples of stable and unstable limit cycles are given, and Hopf bifurcation points are shown to confirm the results previously calculated in the Hopf bifurcation map. The usefulness and accuracy of the semi-analytical results are confirmed by comparison with the numerical solutions of partial differential equations.

Semi-analytical solutions for the delayed and diffusive viral infection model with logistic growth

“…With the use of the Galerkin technique, the semi-analytical system for (2.1) has been derived in the 1-D geometrical domains, and a spatial form of the profile concentration in this technique is taken into consideration [2,4,17,18]. The Galerkin technique refers to an analytical method that applies the orthogonality of a set of rudimentary roles to substitute PDEs with an ODE system.…”

“…Semi-analytical solutions have been applied to solve several complications with reaction-diffusion systems, such as predator-prey model [8], pellet systems [18], BZ reactions [6], the Brusselator system [2], mixed quadratic-cubic autocatalytic reactions [7], and logistic equations [3,4]. All these models yielded accurate solutions compared to full numerical outcomes.…”

“…All these models yielded accurate solutions compared to full numerical outcomes. Recently, Alfifi [2] introduced a Brusselator model semi-analytical system in the 1-D and 2-D domains. For the ODE system in that study to be attained, the Galerkin technique was used, and the Hopf bifurcation map, stability analysis, and steadystate solutions of the Hopf bifurcation points were constructed.…”

Semi-analytical solutions for the diffusive logistic equation with mixed instantaneous and delayed density dependence

“…Reaction-diffusion phenomenon with time delays has been incorporated into many fields of biological applications. These applications have explained a number of practical applications in our everyday life by using partial differential equations (PDEs), for instance, in population ecology [15,17,20,30,31], animals [2,4,26], cell [5,19,22,25,33], chemicals [1,3,10], and heat and mass transfer [13,27]. This model can introduce instability, via a Hopf bifurcation, with the subsequent development of limit cycles.…”

“…Semi-analytical methods have been used to discuss many delay systems with reactiondiffusion phenomenon, such as delay logistic equations [6,7], predator-prey model [2], viral infection system [5], pellet systems [24], the Belousov-Zhabotinsky (BZ) reaction [9], the Brusselator model [3], the reversible Selkov model [1], the equation of Nicholsons blowflies [8], and the limited food model [4]. The outcomes for all papers that used this method revealed an excellent agreement between semi-analytical ODEs outcomes and the numerical solutions pertain to PDEs equations.…”