Traveling wave patterns in nonlinear reaction–diffusion equations

In this paper we study a reaction–diffusion model equation with general nonlinear diffusion and arbitrary kinetic orders in the reaction terms, which appears in the applied biochemical modeling. We carry both analytical and numerical studies of the model equation to show the existence of monotone and oscillatory waves. Our numerical computations are illustrated for a particular case of the equation by using different methods which lead to accurate wave profiles and confirm the analytical results.     

Optimal homotopy analysis method for the non-isothermal reaction–diffusion model equations in a spherical catalyst

We consider the Lane–Emden boundary value problems which appear in chemical applications, biochemical applications, and scientific disciplines. The Lane–Emden problem is transformed into an equivalent integral equation. The optimal homotopy analysis method is used to solve two specific models. The first problem models reaction–diffusion equation in a spherical catalyst, while the second problem models the reaction–diffusion process in a spherical biocatalyst. We obtain reliable analytical solutions of the concentrations and the effectiveness factors. Numerical results and graphs show the reliability and efficiency applicability of the employed method.     

A computationally efficient Galerkin technique for approximating transient diffusion—reaction equations in composite media

Non-linear transient two-point boundary value problems which describe the interaction between diffusion and chemical reaction in a catalyst pellet areThe technique is illustrated with a transient diffusion—reaction problem in poisoned automotive catalysts pellets.     

The wavelet methods to linear and nonlinear reaction–diffusion model arising in mathematical chemistry

In this paper, we have applied an accurate and efficient wavelet scheme (due to Legendre polynomial) to find the numerical solutions for a set of coupled reaction–diffusion equations. This technique provides the solutions in rapid convergence series with computable terms for the problems with high degree of non linear terms appearing in the governing differential equations. The highest derivative in the differential equation is expanded into wavelet series, this approximation is then integrated while the boundary conditions are applied by using integration constants. With the help of operational matrices, the nonlinear reaction–diffusion equations are converted into a system of algebraic equations. Finally, some numerical examples to demonstrate the validity and applicability of the method have been furnished. The use of Legendre wavelets is found to be accurate, efficient, simple, and computationally attractive. This wavelet method can be used for obtaining quick solution in many chemical Engineering problems.     

He–Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics

Journal of Mathematical Chemistry - In this article, we suggest an alternative approach for the study of some partial differential equations (PDEs) arising in physical phenomena such as chemical...     

Solving nonlinear reaction–diffusion problem in electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme systems using Taylor series method

A mathematical model of electrostatic interaction with reaction-generated pH change on the kinetics of immobilized enzyme is discussed. The model involves the coupled system of non-linear reaction–diffusion equations of substrate and hydrogen ion. The non-linear term in this model is related to the Michaelis–Menten reaction of the substrate and non-Michaelis–Menten kinetics of hydrogen ion. The approximate analytical expression of concentration of substrate and hydrogen ion has been derived by solving the non-linear reactions using Taylor’s series method. Reaction rate and effectiveness factor are also reported. A comparison between the analytical approximation and numerical solution is also presented. The effects of external mass transfer coefficient and the electrostatic potential on the overall reaction rate were also discussed.

     

Self-oscillating gels based on novel catalyst for the Belousov–Zhabotinsky reaction

We report on the synthesis of new Ru(bpy)₂(phen) catalyst for the oscillatory Belousov–Zhabotinsky chemical reaction and on the preparation of novel Ru(bpy)₂(phen)-based self-oscillating gels. The synthesized gels exhibit high-amplitude autonomous mechanical oscillations when the Belousov–Zhabotinsky reaction proceeds inside these gels     

Bifurcation analysis of reaction–diffusion Schnakenberg model

Bifurcations of spatially nonhomogeneous periodic orbits and steady state solutions are rigorously proved for a reaction–diffusion system modeling Schnakenberg chemical reaction. The existence of these patterned solutions shows the richness of the spatiotemporal dynamics such as oscillatory behavior and spatial patterns.     

Monte—Carlo model of the diffusion-limited reaction kinetics in microheterogeneous media. Comparison with the experimental plots of the forward and backward electron phototransfer kinetics on the pore diameter in silica gel

The kinetics of the diffusion-limited decay reaction A + B B was simulated by the Monte—Carlo method on a two-dimensional square lattice with defects presented by randomly distributed sites. The cases were considered where [B] [A] at the random initial distribution (quenching reaction) and [B] = [A] with the initial distribution of the A and B particles on neighboring sites (geminate recombination). The kinetic curves were approximated by the simplest analytical equation [A]/[A]₀ = (1 – )exp[–(kt)^1–h ] + (where k and are constants). The plots of the heterogeneity parameter (h) and time-averaged first-order rate constant vs. concentration of defects (p) or B particles (in the case of quenching) were obtained and compared with similar correlations obtained earlier by the experimental study of the kinetics of forward (quenching reaction) and backward (geminate recombination) electron phototransfer on the surface of different porous silica gels. The experimental plots of h vs. silica gel porosity are in satisfactory agreement with the plots of h vs. p in the model space, if the fraction of volume inaccessible for reactants, calculated from the free silica gel volume, is chosen as the p parameter for silica gel.Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 8, pp. 1536–1541, August, 2004.     

A new coupled wavelet-based method applied to the nonlinear reaction–diffusion equation arising in mathematical chemistry

In this paper, we have applied the wavelet-based coupled method for finding the numerical solution of Murray equation. To the best of our knowledge, until now there is no rigorous Legendre wavelets solution has been reported for the Murray equation. The highest derivative in the differential equation is expanded into Legendre series, this approximation is integrated while the boundary conditions are applied using integration constants. With the help of Legendre wavelets operational matrices, the Murray equation is converted into an algebraic system. Block pulse functions are used to investigate the Legendre wavelets coefficient vectors of nonlinear terms. The convergence of the proposed method is proved. Finally, we have given a numerical example to demonstrate the validity and applicability of the method. Moreover the use of proposed wavelet-based coupled method is found to be simple, efficient, less computation costs and computationally attractive.     

On the predictions and limitations of the Becker-Döring model for reaction kinetics in micellar surfactant solutions

We investigate the breakdown of a system of micellar aggregates in a surfactant solution following an order-one dilution. We derive a mathematical model based on the Becker-D?ring system of equations, using realistic expressions for the reaction constants fit to results from Molecular Dynamics simulations. We exploit the largeness of typical aggregation numbers to derive a continuum model, substituting a large system of ordinary differential equations for a partial differential equation in two independent variables: time and aggregate size. Numerical solutions demonstrate that re-equilibration occurs in two distinct stages over well-separated timescales, in agreement with experiment and with previous theories. We conclude by exposing a limitation in the Becker-D?ring theory for re-equilibration of surfactant solutions.     

On the kinetics of pozzolanic reaction in the system kaolin–lime–water

The kinetics of the pozzolanic reaction of enriched kaolin from the “Senovo” deposit (Bulgaria) with lime is the object of this article. The kaolin contains kaolinite as a major clay mineral as well as admixtures of quartz and illite. The experimental data of pozzolanic activity at temperatures of 100 and 23 °C are obtained for different reaction times. The reaction degrees of kaolinite and lime at 100 °C are determined from the pozzolanic activity data using a powder X-ray diffraction analysis. The kinetic analysis is performed by joint presentation of theoretical and experimental data in dimensionless coordinates having in mind the influence of particle size distribution on the reaction rate. It is found by the kinetic analysis that the rate of entire reaction is limited by the rate of chemical reaction on the reaction surface up to degree of reaction near to 0.4. The rate of penetration of the chemical reaction into the kaolinite particles for this area—from the beginning to degree of reaction 0.4, is determined to be equal to 2.10⁻¹¹ m/s.     

Weighted pseudo almost automorphic classical solutions and optimal mild solutions for fractional differential equations and application in fractional reaction–diffusion equations

In this paper, we are concerned with a class of fractional differential equations given by $$\begin{aligned} \hbox {D}_{t}^{\alpha }x(t)=Ax(t)+f(t,x(t)). \end{aligned}$$ Our main results concern the existence, uniqueness of weighted pseudo-almost automorphic classical solutions and optimal mild solutions. Moreover, as example and applications, we study the weighted pseudo-almost automorphic classical solutions and optimal mild solutions for a fractional reaction–diffusion equation to illustrate the practical usefulness of the analytical results that we establish in the paper.     

On the kinetics of pozzolanic reaction in metakaolin–lime–water system

The kinetics of pozzolanic reaction metakaolin–lime is studied in the present work. Metakaolin is prepared by calcination of enriched kaolin (deposit “Senovo”, Bulgaria) at temperature of 830 ± 10 °C in a labscale muffle oven. The reaction is performed in intensively stirred water suspension at different temperatures in the range 20–100 °C. The kinetics is analyzed by comparing the experimental data with theoretical curves, derived according to appropriate kinetic and diffusion models taking into account the grain size distribution of metakaolin. The macroscopic mechanism and activation energy of the reaction are determined. It is found, that the activation energy decreases gradually from 71 to 45 kJ/mol[Ca(OH)₂] with the increase of the reaction degree from 0.2 up to 0.6, respectively, which is a characteristic for transition regime reactions.     

A second-order scheme for the “Brusselator” reaction–diffusion system

A second-order method is developed for the numerical solution of the initial-value problems , , and , , , in which the functions and , where A and B are positive real constants, are the reaction terms arising from the mathematical modelling of chemical systems such as in enzymatic reactions and plasma and laser physics in multiple coupling between modes. The method is based on three first-order methods for solving u and v, respectively. In addition to being second-order accurate in space and time, the method is seen to converge to the correct fixed point ( , V* = A/B) provided . The approach adopted is extended to solve a class of non-linear reaction–diffusion equations in two-space dimensions known as the “Brusselator” system. The algorithm is implemented in parallel using two processors, each solving a linear algebraic system as opposed to solving non-linear systems, which is often required when integrating non-linear partial differential equations (PDEs). This revised version was published online in July 2006 with corrections to the Cover Date.     

Neural network devices based on reaction–duffusion media: an approach to artificial retina

Neural network architecture of devices based on reaction–diffusion media is discussed. Image processing operations performed by these media are similar to first steps of human vision mechanism.     

Oxygen reduction reaction on carbon supported Palladium–Nickel alloys in alkaline media

Carbon supported Palladium–Nickel alloys with various compositions (Pd–Ni/C) were synthesized by chemical reduction of the co-precipitated Pd and Ni hydroxides on carbon. The structure of these alloys was characterized using X-ray diffraction (XRD) analysis. The catalytic activity of Pd–Ni/C for oxygen reduction reaction (ORR) in alkaline media was studied using a glassy carbon rotating disk electrode (RDE). Pd/C showed ORR activity close to that of Pt/C. The activities of Pd–Ni (3:1)/C and Pd–Ni (1:1)/C were found unchanged compared with that of Pd/C. Ni/C showed about 175 mV lower onset potential than Pt/C, and the activity of Pd–Ni (1:3)/C was observed to be between that of Pd/C and Ni/C.