Simple Reaction-diffusion Equations Research Articles

Non-vanishing sharp-fronted travelling wave solutions of the Fisher-Kolmogorov model.

The Fisher-Kolmogorov-Petrovsky-Piskunov (KPP) model, and generalizations thereof, involves simple reaction-diffusion equations for biological invasion that assume individuals in the population undergo linear diffusion with diffusivity $D$, and logistic proliferation with rate $\lambda $. For the Fisher-KPP model, biologically relevant initial conditions lead to long-time travelling wave solutions that move with speed $c=2\sqrt {\lambda D}$. Despite these attractive features, there are several biological limitations of travelling wave solutions of the Fisher-KPP model. First, these travelling wave solutions do not predict a well-defined invasion front. Second, biologically relevant initial conditions lead to travelling waves that move with speed $c=2\sqrt {\lambda D}> 0$. This means that, for biologically relevant initial data, the Fisher-KPP model cannot be used to study invasion with $c \ne 2\sqrt {\lambda D}$, or retreating travelling waves with $c < 0$. Here, we reformulate the Fisher-KPP model as a moving boundary problem and show that this reformulated model alleviates the key limitations of the Fisher-KPP model. Travelling wave solutions of the moving boundary problem predict a well-defined front that can propagate with any wave speed, $-\infty < c < \infty $. Here, we establish these results using a combination of high-accuracy numerical simulations of the time-dependent partial differential equation, phase plane analysis and perturbation methods. All software required to replicate this work is available on GitHub.

Evolution of Microstructure and Creep Behavior in an Fe-Ni-Cr-Nb-C Alloy during Service in Hydrocarbon Cracker Tubes

Heat resistant, cast austenitic Fe-Ni-Cr-Nb-C alloy tubes are used by the petrochemical industry to transport hydrocarbons at temperatures between 850 and 1050 °C. Exposure to carbon from the process stream and nitrogen from the air causes severe carburization and nitridation in the tube walls, which influences creep resistance and degrades their ductility. A better understanding of these processes would enable their service life to be extended. To this end, we report the results of an experimental and computational study of the changes in microstructure and creep behavior in a representative alloy during service. The microstructure of the fresh material is compared to that of 8- and 10-year-old tubes. The results confirm the presence of M23C6, M7C3, and M2(CN) phases reported in prior studies, and also reveal the nucleation and growth of the carbo-nitride phase M6(CN). We show that the spatial variations of carbide and nitride particle volume fractions in aged tubes can be predicted by a simple reaction–diffusion equation. In addition, the uniaxial compressive creep response of the fresh and aged tubes was obtained as functions of temperature and strain rate. The materials deform by power-law creep, with strain rate versus stress relation as $$\dot{\varepsilon } = A\sigma^{n}$$. Aging reduces the stress exponent from n = 8 in fresh tubes to n = 5 in aged tubes, and also reduces the pre-exponential coefficient A. Tests on model alloys with composition identical to that of the matrix combined with finite element simulations of creep in model microstructures reveal that Cr depletion is responsible for the change in stress exponent, while the change in A is caused by the increase in particle volume fraction. By combining the microstructure-scale simulations of creep with the predictions of the reaction–diffusion model of microstructure evolution, the creep behavior of tubes left in service beyond 10 years is estimated.

Algorithm for the Time-Propagation of the Radial Diffusion Equation Based on a Gaussian Quadrature.

The numerical integration of the time-dependent spherically-symmetric radial diffusion equation from a point source is considered. The flux through the source can vary in time, possibly stochastically based on the concentration produced by the source itself. Fick’s one-dimensional diffusion equation is integrated over a time interval by considering a source term and a propagation term. The source term adds new particles during the time interval, while the propagation term diffuses the concentration profile of the previous time step. The integral in the propagation term is evaluated numerically using a combination of a new diffusion-specific Gaussian quadrature and interpolation on a diffusion-specific grid. This attempts to balance accuracy with the least number of points for both integration and interpolation. The theory can also be extended to include a simple reaction-diffusion equation in the limit of high buffer concentrations. The method is unconditionally stable. In fact, not only does it converge for any time step Δt, the method offers one advantage over other methods because Δt can be arbitrarily large; it is solely defined by the timescale on which the flux source turns on and off.

Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration

Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction–diffusion equation, we explain the observed phenomena of front acceleration (when the motility is unbounded) as well as other qualitative results, such as the existence of traveling waves and the selection of the most motile individuals (when the motility is bounded). The key argument for constructing and analysing the traveling waves is the derivation of a dispersion relation linking the wave speed and the spatial decay. When the motility is unbounded we show that the position of the front scales as t3/2. When the mutation rate is low we show that the canonical equation for the dynamics of the fittest trait should be stated as a PDE in our context. It turns out to be a type of Burgers equation with a source term.

Analysis of grain boundary sinks and interstitial diffusion in neutron-irradiated SiC

The widths of the interstitial loop denuded zone (DZ) along grain boundaries were examined for 3C-SiC irradiated at $1010--1380 \ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$ by transmission electron microscopy (TEM) in an effort to obtain the activation energy of interstitial migration. Denuded-zone widths as small as 17 nm were observed below $1130 \ifmmode^\circ\else\textdegree\fi{}\mathrm{C}$, indicating that a substantial population of ``TEM invisible'' voids of diameter $&lt;0.7$ significantly contribute to interstitial annihilation. By using the obtained loop DZ width and the matrix sink strength (including the invisible voids), the activation energy of interstitial diffusion was determined to be 1.5 eV for the slower moving Si interstitial of SiC by application of simple reaction-diffusion equations.

Producing swimmers by coupling reaction-diffusion equations to a chemically responsive material

We propose a mechanism for generating snakelike motion in a micrometer-scale, responsive, synthetic material, which thereby undergoes net movement in a fluid (i.e., swimming). By responsive material, we refer to a material that can expand or contract in response to a chemical concentration change. The concentrations of the chemical species are modeled by simple reaction-diffusion equations with suitably chosen source terms. Using linear stability analysis, we isolate the key properties of the material and reaction rate parameters.

Amplitude–shape approximation as an extension of separation of variables

AbstractSeparation of variables is a well‐known technique for solving differential equations. However, it is seldom used in practical applications since it is impossible to carry out a separation of variables in most cases. In this paper, we propose the amplitude–shape approximation (ASA) which may be considered as an extension of the separation of variables method for ordinary differential equations. The main idea of the ASA is to write the solution as a product of an amplitude function and a shape function, both depending on time, and may be viewed as an incomplete separation of variables. In fact, it will be seen that such a separation exists naturally when the method of lines is used to solve certain classes of coupled partial differential equations. We derive new conditions which may be used to solve the shape equations directly and present a numerical algorithm for solving the resulting system of ordinary differential equations for the amplitude functions. Alternatively, we propose a numerical method, similar to the well‐established exponential time differencing method, for solving the shape equations. We consider stability conditions for the specific case corresponding to the explicit Euler method. We also consider a generalization of the method for solving systems of coupled partial differential equations. Finally, we consider the simple reaction diffusion equation and a numerical example from chemical kinetics to demonstrate the effectiveness of the method. The ASA results in far superior numerical results when the relative errors are compared to the separation of variables method. Furthermore, the method leads to a reduction in CPU time as compared to using the Rosenbrock semi‐implicit method for solving a stiff system of ordinary differential equations resulting from a method of lines solution of a coupled pair of partial differential equations. The present amplitude–shape method is a simplified version of previous ones due to the use of a linear approximation to the time dependence of the shape function. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Formulation of gas diffusion dynamics for thin film semiconductor gas sensor based on simple reaction–diffusion equation

In response and recovery steps of a thin film semiconductor gas sensor, target gas molecules diffuse in and out of the thin film. The gas diffusion dynamics taking place in these steps have been formulated based on a simple reaction–diffusion equation assuming a first-order reaction of target gas. In order to facilitate mathematical treatments, the actual thin film device was replaced by an equivalent model, for which boundary conditions could be set properly. With this model, the reaction–diffusion equation could be solved by using the methods of Fourier expansion and separation of variables. The solutions given as a function of diffusion coefficient D, rate constant k, film thickness L, depth x and time t, are shown to express well how target gas concentration profile in the thin film develops or vanishes in the response or recovery step, respectively.

Development of experimental techniques and an analytical model for aluminum nitriding

The rates of diffusion of nitrogen into aluminum and the subsequent formation of aluminum nitride are unclear. By matching experimental data to analytical models one can gain a better understanding of the diffusion process for nitrogen into aluminum and formation of aluminum nitride. We combine novel experimental and modeling approaches to achieve this goal. We develop a technique for in situ measurement of nitrogen deposition using a microbalance and present a simple reaction diffusion equation to simulate the diffusion of nitrogen into aluminum and subsequent precipitation into aluminum nitride. A mixed boundary condition formulation is presented to account for the initial influx of nitrogen at the surface until a saturation level is reached. We curve fit the derived model equations to the experimental data and determine the system diffusion coefficient and reaction rate constant.

On Stochastic Reaction-Diffusion Equations with Singular Force Term

Differential equations may be ill behaved in the sense that they have no solutions, or nonunique solutions, or solutions which do not depend continuously on the given data. Surprisingly, by introducing some random perturbation we can often transform these equations into well-behaved stochastic differential equations. This phenomenon is well known for ordinary differential equations. In the present paper we are interested in this regularization effect of the noise on ill-posed partial differential equations (PDEs). In order to illustrate the result of the paper, let us consider the following examples of simple reaction-diffusion equations:

Solution of nonlinear equations and computation of multiple solutions of a simple reaction-diffusion equation

AbstractThe first part of this paper starts with a brief discussion of some methods for solution of nonlinear equations which have interested the first author over the last twenty years or so. In the second part we discuss a recent research involvement, the success of which relies heavily on the numerical solution of nonlinear equation systems. We briefly describe path-following methods and then present an application to a simple steady-state reaction-diffusion equation arising in combustion theory. Results for some regular geometric shapes are shown and compared with those from an approximate method.

On explicit solutions to stochastic differential equations

This note is concerned with the study of explicit solutions to stochastic differential equations. Previously, Doss and Sussman showed that the unique strong solution to the scalar Itô equation X can be represented as a function ρ of a Brownian motion and an auxiliary stochastic process Yt determined, for every path of by ordinary differential equation (ODE). ρ itself is determined by a second differential equation. Now, it will be shown that X can be solved explicitly as with f(.) being a continuous real valued function, provided solves a differential equation related to the one defining ρ as well as a simple reaction-diffusion equation strongly. In particular, for a given dispersion coefficient σ(.), there will be a class drift coefficients b(.) are provided. The corresponding explicit solution xt for any given dispersion σ is also supplied

Traveling Waves in Buffered Systems: Applications to Calcium Waves

Traveling waves of calcium are widely observed under conditions where the free calcium is heavily buffered. It is thus of considerable physiological interest to determine the precise effects that buffers have on the properties of traveling waves. Since calcium waves are widely believed to be the result of the reaction and diffusion of calcium, we study the properties of traveling waves in simple reaction-diffusion equations in which the diffusing species is buffered. For the buffered bistable equation we derive a constraint on the model parameters in order for a traveling wave of excitation to exist, and we show that stationary buffers cannot eliminate traveling waves. When the buffer is of low affinity, a constant effective diffusion coefficient may be defined, but no effective diffusion coefficient can be defined for high affinity buffers. We derive an approximate expression for the wave speed, show numerically that this approximation applies for both high and low affinity buffers, and hence derive an a...

On the Evolution of Periodic Plane Waves in Reaction-Diffusion Systems of $\lambda $–$\omega$ Type

$\lambda $–$\omega$ systems are a class of simple reaction-diffusion equations with a limit cycle in the reaction kinetics. The author considers the solution of the system given by $\lambda ( r ) = \lambda _0 - r^p $, $\omega ( r ) = \omega_0 - r^p $ on a semi-infinite spatial domain with initial data decaying exponentially across the domain. Numerical evidence is presented, showing that this initial condition induces a wave front moving across the domain, with periodic plane waves behind the front. These periodic waves can move in either direction, depending on the parameter values. The author uses intuitive criteria to derive an expression for the speed of the advancing front, and by reducing the system to ordinary differential equations for similarity solutions, the amplitude and speed of the periodic plane waves are determined. The speed of these periodic waves varies continuously with the initial data. Perturbation theory is then used to obtain an analytical approximation to the solutions. These anal...

Reaction-diffusion inequalities in cones

Reaction-diffusion equations are considered which are weakly coupled relative to an arbitrary cone. A result is proven on flow-invariance which is then utilized to obtain a useful comparison theorem and a theorem on differential inequalities. The results obtained are applied to simple reaction diffusion equations to derive positivity of solutions, upper and lower bounds and stability properties. Finally it is demonstrated by means of a simple example that working with a suitable cone other than R/sub +//sup N/ is more advantageous in the investigation of equations of reaction-diffusion. 15 references. (JFP)