One-dimensional Reaction-diffusion Research Articles

Fast Propagation for Fractional KPP Equations with Slowly Decaying Initial Conditions

In this paper we study the large-time behavior of solutions of one-dimensional fractional Fisher-KPP reaction-diffusion equations, when the initial condition is asymptotically front-like and it decays at infinity more slowly than a power $x^{-b}$, where $b<2\alpha$ and $\alpha\in(0,1)$ is the order of the fractional Laplacian. We prove that the level sets of the solutions move exponentially fast as time goes to infinity. Moreover, a quantitative estimate of motion of the level sets is obtained in terms of the decay of the initial condition.

Weak Allee effect, grazing, and S-shaped bifurcation curves

We study a one-dimensional reaction-diffusion model arising in population dynamics where the growth rate is a weak Allee type. In particular, we consider the effects of grazing on the steady states and discuss the complete evolution of the bifurcation curve of positive solutions as the grazing parameter varies. We obtain our results via the quadrature method and Mathematica computations. We establish that the bifurcation curve is S-shaped for certain ranges of the grazing parameter. We also prove this occurrence of an S-shaped bifurcation curve analytically.

PARAMETER-UNIFORM FINITE ELEMENT METHOD FOR TWO-PARAMETER SINGULARLY PERTURBED PARABOLIC REACTION-DIFFUSION PROBLEMS

In this paper, parameter-uniform numerical methods for a class of singularly perturbed one-dimensional parabolic reaction-diffusion problems with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and finite element method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O(N-2( ln N)2 + Δt). Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.

Rigorous results for the minimal speed of Kolmogorov–Petrovskii–Piscounov monotonic fronts with a cutoff

We study the effect of a cutoff on the speed of pulled fronts of the one-dimensional reaction diffusion equation. To accomplish this, we first use variational techniques to prove the existence of a heteroclinic orbit in phase space for traveling wave solutions of the corresponding reaction diffusion equation under conditions that include discontinuous reaction profiles. This existence result allows us to prove rigorous upper and lower bounds on the minimal speed of monotonic fronts in terms of the cut-off parameter ɛ. From these bounds we estimate the range of validity of the Brunet–Derrida formula for a general class of reaction terms.

Experiments and Kinetic Modeling for CO2 Gasification of Indian Coal Chars in the Context of Underground Coal Gasification

Gasification of four Indian coals is carried out in a CO2 atmosphere, using a thermogravimetric analyzer (TGA) to determine the intrinsic kinetics over a temperature range of 800–1050 °C with different partial pressures of CO2. The applicability of three models, viz., the volumetric reaction model, the shrinking core model and the random pore model, is evaluated. Of these three models, the random pore model is found to be the most suitable for all the coals considered in the current study. The dependence of the reaction rate on the gas-phase partial pressures is explained by the Langmuir–Hinshelwood model, and the parameters for the inhibition due to CO and CO2 are determined by performing experiments at different partial pressures. In underground coal gasification, the reaction takes place on reasonably large sized coal particles, wherein diffusion effects are significant. A one-dimensional reaction diffusion model is therefore developed in order to determine the diffusional resistance in the coal partic...

Spreading speeds for one-dimensional monostable reaction-diffusion equations

We establish in this article spreading properties for the solutions of equations of the type ∂tu − a(x)∂xxu − q(x)∂xu = f(x, u), where a, q, f are only assumed to be uniformly continuous and bounded in x, the nonlinearity f is of monostable Kolmogorov, Petrovsky, and Piskunov type between two steady states 0 and 1 and the initial datum is compactly supported. Using homogenization techniques, we construct two speeds \documentclass[12pt]{minimal}\begin{document}$\underline{w}\le \overline{w}$\end{document}w̲≤w¯ such that \documentclass[12pt]{minimal}\begin{document}$\lim _{t\rightarrow +\infty }\sup _{0\le x\le wt} |u(t,x)-1| = 0$\end{document}limt→+∞sup0≤x≤wt|u(t,x)−1|=0 for all \documentclass[12pt]{minimal}\begin{document}$w\in (0,\underline{w})$\end{document}w∈(0,w̲) and \documentclass[12pt]{minimal}\begin{document}$\lim _{t\rightarrow +\infty } \sup _{x \ge wt} |u(t,x)| =0$\end{document}limt→+∞supx≥wt|u(t,x)|=0 for all \documentclass[12pt]{minimal}\begin{document}$w&amp;gt;\overline{w}$\end{document}w&amp;gt;w¯. These speeds are characterized in terms of two new notions of generalized principal eigenvalues for linear elliptic operators in unbounded domains. In particular, we derive the exact spreading speed when the coefficients are random stationary ergodic, almost periodic or asymptotically almost periodic (where \documentclass[12pt]{minimal}\begin{document}$\overline{w}=\underline{w}$\end{document}w¯=w̲).

Reduction waves in the two-variable Oregonator model for the BZ reaction

Numerical simulations of the two-variable Oregonator in a one-dimensional reaction–diffusion model are undertaken to show the formation of single reduction pulses. These are seen to exist over relatively narrow ranges of the (dimensionless) kinetic parameters ϵ and f arising in the derivation of the Oregonator model, though they are seen for all values of the diffusion coefficient ratio D considered. For the smaller values of D a direct transition from single reduction pulses to wave trains is found. For equal diffusion coefficients, D=1.0, this transition involves a sequence of complex spatio-temporal dynamics, including localized oscillatory behaviour and the successive spreading of a region in the reduced state. The present results are compared with results from the three-variable Oregonator and the Rovinsky–Zhabotinsky models for the BZ reaction, as well as with previous experimental observations.

Inside dynamics of pulled and pushed fronts

We investigate the inside structure of one-dimensional reaction–diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.

Spreading Properties and Complex Dynamics for Monostable Reaction–Diffusion Equations

This paper is concerned with the study of the large-time behavior of the solutions u of a class of one-dimensional reaction–diffusion equations with monostable reaction terms f, including in particular the classical Fisher-KPP nonlinearities. The nonnegative initial data u 0(x) are chiefly assumed to be exponentially bounded as x tends to + ∞ and separated away from the unstable steady state 0 as x tends to − ∞. On the one hand, we give some conditions on u 0 which guarantee that, for some λ > 0, the quantity c λ = λ +f′(0)/λ is the asymptotic spreading speed, in the sense that lim t→+∞ u(t, ct) = 1 (the stable steady state) if c < c λ and lim t→+∞ u(t, ct) = 0 if c > c λ. These conditions are fulfilled in particular when u 0(x) e λx is asymptotically periodic as x → + ∞. On the other hand, we also construct examples where the spreading speed is not uniquely determined. Namely, we show the existence of classes of initial conditions u 0 for which the ω-limit set of u(t, ct + x) as t tends to + ∞ is equal to the whole interval [0, 1] for all x ∈ ℝ and for all speeds c belonging to a given interval (γ1, γ2) with large enough γ1 < γ2 ≤ + ∞.

Phase Description of Stable Limit-cycle Solutions in Reaction-diffusion Systems

Phase reduction theory for stable limit-cycle solutions of one-dimensional reaction-diffusion systems is developed. By locally approximating the isochrons of the limit-cycle orbit, we derive the phase sensitivity function, which is a key quantity in the phase description of limit cycles. As an example, synchronization of traveling pulses in a pair of mutually interacting reaction-diffusion systems is analyzed. It is shown that the traveling pulses can exhibit multi–modal phase locking.

Single and Two-Scale Sharp-Interface Models for Concrete Carbonation---Asymptotics and Numerical Approximation

We investigate the fast-reaction asymptotics for a one-dimensional reaction-diffusion system describing the penetration of the carbonation reaction in concrete. The technique of matched-asymptotics is used to show that the reaction-diffusion system leads to two distinct classes of sharp-interface models. These correspond to different scalings of a small parameter $\epsilon$ representing the fast-reaction and defined here as the ratio between the characteristic scale of diffusion for the fastest species and the characteristic scale of the carbonation reaction. We explore three conceptually different diffusion regimes in terms of the behavior of the effective diffusivities for the driving chemical species. The limiting models include one-phase and two-phase generalized Stefan moving-boundary problems as well as a nonstandard two-scale (micro-macro) moving-boundary problem---the main result of the paper. Numerical results, supporting the asymptotics, illustrate the behavior of the concentration profiles for relevant parameter regimes.

Anomalous kinetics in diffusion limited reactions linked to non-Gaussian concentration probability distribution function

We investigate anomalous reaction kinetics related to segregation in the one-dimensional reaction-diffusion system A + B → C. It is well known that spatial fluctuations in the species concentrations cause a breakdown of the mean-field behavior at low concentration values. The scaling of the average concentration with time changes from the mean-field t(-1) to the anomalous t(-1/4) behavior. Using a stochastic modeling approach, the reaction-diffusion system can be fully characterized by the multi-point probability distribution function (PDF) of the species concentrations. Its evolution is governed by a Fokker-Planck equation with moving boundaries, which are determined by the positivity of the species concentrations. The concentration PDF is in general non-Gaussian. As long as the concentration fluctuations are small compared to the mean, the PDF can be approximated by a Gaussian distribution. This behavior breaks down in the fluctuation dominated regime, for which anomalous reaction kinetics are observed. We show that the transition from mean field to anomalous reaction kinetics is intimately linked to the evolution of the concentration PDF from a Gaussian to non-Gaussian shape. This establishes a direct relationship between anomalous reaction kinetics, incomplete mixing and the non-Gaussian nature of the concentration PDF.

Stability transitions and dynamics of mesa patterns near the shadow limit of reaction-diffusion systems in one space dimension

We consider a class of one-dimensional reaction-diffusion systems,\[\left\{\begin{array}[u_{t}=\varepsilon^{2}u_{xx}+f(u,w)\\\tau w_{t}=Dw_{xx}+g(u,w)\end{array}\right.\]with homogeneous Neumann boundary conditions on a one dimensional interval.Under some generic conditions on the nonlinearities $f,g$ and in the singularlimit $\varepsilon\rightarrow0,$ such a system admits a steady state for which$u$ consists of sharp back-to-back interfaces. For a sufficiently large $D$and for sufficiently small $\tau$, such a steady state is known to be stablein time. On the other hand, it is also known that in the so-called shadowlimit $D\rightarrow\infty,$ patterns having more than one interface areunstable. In this paper we analyse in detail the transition between the stablepatterns when $D=O(1)$ and the shadow system when $D\rightarrow\infty$. Weshow that this transition occurs when $D$ is exponentially large in $\varepsilon$ and we derive instability thresholds $D_{1}\gg D_{2}\ggD_{3}\gg\ldots$ such that a periodic pattern with $2K$ interfaces is stable if$D D_{K}$. We also study the dynamics of theinterfaces when $D$ is exponentially large; this allows us to describe indetail the mechanism leading to the instability. Direct numerical computationsof stability and dynamics are performed, and these results are in excellentagreement with corresponding results as predicted by the asymptotic theory.

Nonuniform autonomous one-dimensional exclusion nearest-neighbor reaction-diffusion models

In a recent article, the most general non-uniform reaction–diffusion models on a one-dimensional lattice with boundaries were considered, for which the time evolution equations of correlation functions are closed and the stationary profile can be obtained using a transfer-matrix method. Here, models are investigated for which the equation of relaxation toward the stationary profile could also be solved through a similar transfer-matrix method. A classification is given, and dynamical phase transitions are studied.

Uniqueness from pointwise observations in a multi-parameter inverse problem

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\varepsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases $N=2$ and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term.

A comparative study of Haar Wavelet Method and Homotopy Perturbation Method for solving one-dimensional Reaction-Diffusion Equations

Abstract—In this paper, we introduce a homotopy perturbation method to obtain exact solutions to some linear and nonlinear onedimensional reaction-diffusion equations. This method is a powerful device for solving a wide variety of problems. Using the homotopy perturbation method, it is possible to find the exact solution or an approximate solution of the problem. The power of this manageable method is confirmed. An attempt is made to combine the advantages of the Homotopy Perturbation Method (HPM) and Haar wavelets. Good agreement with the exact solution has been observed. The results reveal that the HPM and HWM are very effective, convenient and quite accurate to systems of partial differential equations. It is predicted that the HWM and HPM can be found widely applicable in engineering.

A viscosity solution method for the spreading speed formula in slowly varying media

In this paper, we consider reaction-diffusion-advection equations in slowly periodi-cally oscillating media. We prove the existence of and we give explicit expressions of the asymptotic spreading speeds of invasion of the unstable state 0 in any direction, when the period of the invaded medium becomes infinitely large. The limiting sprea-ding speeds involve families of 1-periodic Hamilton-Jacobi equations. In the case of one-dimensional reaction-diffusion equations, we analyze the relative effects of small perturbations of the diffusion and the reaction coefficients, and we compare the spreading speeds in slowly oscillating media to the homogenized spreading speeds in rapidly oscillating media.

Role of vacancy and metal doping on combustive oxidation of Zr/ZrO 2 core-shell particles

We studied self-propagated combustion synthesis of transition-metal-doped tetragonal ZrO 2 (t-ZrO 2) with first principles-based one-dimensional diffusion reaction model. The optimal reaction condition for the combustion process was investigated by calculating energetic stability and surface reactivity of oxygen vacancy defects on (101) surface termination of t-ZrO 2 using first-principles density functional methods. In the first-principles model, the surface was doped with 14 different metal impurities in the 4th and 5th row of the periodic table to examine the role of transition-metal doping on the combustion process. Results indicate that there are clear trends in the defect stability and reactivity depending upon the type of metal impurity and their relative location with respect to the oxygen vacancy. Surface density of states and charge density information also show that there is a trade-off between the vacancy stability and chemical activity of the surface defect states. Based on the thermodynamic information obtained from first principles, we analyze the combustion process of a Zr metal particle by using a one-dimensional diffusion-reaction model. The competition between the vacancy-assisted chemisorption and the vacancy diffusion results in an optimal point for rate of combustion reaction with respect to the vacancy stability. From this, we suggest a plausible screening strategy for metal-doping which can be applied at different temperatures and pressures, as well as with different particle sizes. Our analysis indicates that first-principles calculation provides key information that can be subsequently used for an optimization of the reaction rate for a self-sustained combustion process. An explicit inclusion of rates of defect and ionic transport will be introduced into our model in future work.

ON THE SUPERCRITICALITY OF THE FIRST HOPF BIFURCATION IN A DELAY BOUNDARY PROBLEM

The goal of this paper is to analyze the character of the first Hopf bifurcation (subcritical versus supercritical) that appears in a one-dimensional reaction–diffusion equation with nonlinear boundary conditions of logistic type with delay. We showed in the previous work [Arrieta et al., 2010] that if the delay is small, the unique non-negative equilibrium solution is asymptotically stable. We also showed that, as the delay increases and crosses certain critical value, this equilibrium becomes unstable and undergoes a Hopf bifurcation. This bifurcation is the first one of a cascade occurring as the delay goes to infinity. The structure of this cascade will depend on the parameters appearing in the equation. In this paper, we show that the first bifurcation that occurs is supercritical, that is, when the parameter is bigger than the delay bifurcation value, stable periodic orbits branch off from the constant equilibrium.

Singly diagonally implicit runge-kutta method for time-dependent reaction-diffusion equation

In this article, an efficient fourth-order accurate numerical method based on Padé approximation in space and singly diagonally implicit Runge-Kutta method in time is proposed to solve the time-dependent one-dimensional reaction-diffusion equation. In this scheme, we first approximate the spatial derivative using the second-order central finite difference then improve it to fourth-order by applying Padé approximation. A three stage fourth-order singly diagonally implicit Runge-Kutta method is then used to solve the resulting system of ordinary differential equations. It is also shown that the scheme is unconditionally stable, and is suitable for stiff problems. Several numerical examples are solved by the scheme and the efficiency and accuracy of the new scheme are compared with two widely used high-order compact finite difference methods. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 1423–1441, 2011