Non-smooth feedback control for Belousov–Zhabotinskii reaction–diffusion equations: semi-analytical solutions

Abstract

The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reaction-diffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full nonsmooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail. Abstract The Belousov-Zhabotinskii reaction is considered in one and two-dimensional reactiondiffusion cells. Feedback control is examined where the feedback mechanism involves varying the concentrations in the boundary reservoir, in response to the concentrations in the centre of the cell. Semi-analytical solutions are developed, via the Galerkin method, which assumes a spatial structure for the solution, and is used to approximate the governing delay partial differential equations by a system of delay ordinary differential equations. The form of feedback control considered, whilst physically realistic, is non-smooth as it has discontinuous derivatives. A stability analysis of the sets of smooth delay ordinary differential equations, which make up the full non-smooth system, allows a band of Hopf bifurcation parameter space to be obtained. It is found that Hopf bifurcations for the full non-smooth system fall within this band of parameter space. In the case of feedback with no delay a precise semi-analytical estimate for the stability of the full non-smooth system can be obtained, which corresponds well with numerical estimates. Examples of limit cycles and the transient evolution of solutions are also considered in detail.

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

“…The Runge-Kutta fourth-order technique [5,18] and a Crank-Nicholson finite-difference scheme [4,6] are introduced to the ODE and PDE numerical solutions and the spatial and temporal discretizations used in all the examples and figures are ∆x = 0.05 and ∆t = 5 × 10 −3 , respectively. The proportion of inaccuracy, which is considered to be the difference between the two-term semi-analytical solutions and the numerical solutions is defined in this paper as the unconditional value of the difference divided by the precise value times 100.…”

“…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.…”

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

“…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.…”

Semi-Analytical Solutions for the Brusselator Reaction–diffusion Model

“…Many chemical applications have been modelled and investigated using systems of ordinary differential equations (ODE) and partial differential equations (PDE) by both theoreticians and practical researchers for many decades. Previous work in this area has considered Belousov-Zhabotinsky (BZ) reactions [7], reversible Selkov models [4], cubic autocatalytic reactions [8,21] and pellet systems [23] (see also the references therein). These applications have described many oscillatory phenomena in daily life using continuous-flow well-stirred tank reactors (CSTRs).…”

“…A semi-analytical method for solving reaction-diffusion systems has been developed for various other problems, such as pellet systems [23], logistic equations [5], BZ reactions [7], feedback control for microwave heating [19], Nicholson's blowflies equation [6], the steady-state microwave heating of finite 1-D and twodimensional (2-D) slabs [22], the reversible Selkov model with feedback delay [4] and mixed quadratic-cubic autocatalytic reactions [8]. In 2002, Marchant [21] developed the Gray and Scott cubic autocatalytic system in a 1-D domain.…”