Wave features of a hyperbolic prey–predator model

A hyperbolic predator–prey model is proposed within the context of extended thermodynamics. The nature of the steady state solutions for the uniform and non‐uniform perturbations are analyzed. The existence of smooth traveling wave‐like solutions, related to the invasion of the predator population into a prey‐only state is discussed. Validation of the model in point is also accomplished by searching for numerical solutions of the system, which also points out limit cycles in the populations. Copyright © 2010 John Wiley & Sons, Ltd.     

Numerical simulation of reaction–diffusion equations on spherical domains

The two-variable reaction diffusion equations on the spherical domain is considered and simulated, using the semi-implicit Euler finite difference method. It is shown that the method keeps the kinetics from overshooting the stable branches when a large time step is used in the simulation.     

Solvability of reaction–diffusion model with variable exponents

In this article, we study the existence and uniqueness of a weak solution of a degenerate reaction–diffusion parabolic system with variable exponents. This model describes the spread of epidemic diseases with a nonlinear diffusion operator. Copyright © 2013 John Wiley & Sons, Ltd.     

A family of numerical methods for diffusion and reaction–diffusion equations

A family of methods is developed for the numerical solution of second-order parabolic partial differential equations in one space dimension. The methods are second-, third-, or fourth-order accurate in time; five of them are seen to be L₀-stable in the sense of Gourlay and Morris, while the sixth is seen to be A₀-stable, The methods are tested on four problems from the literature, three diffusion problems and one reaction–diffusion problem.     

A hyperbolic model for the effects of urbanization on air pollution

A hyperbolic model to study effects of industrialization and urbanization on air pollution propagation is proposed.     

A reaction–diffusion problem with a reaction front

We state a 1D model with quasi-stationary gas flows approximation for a carbon reactivity test in the production of silicon. The mathematical problem we formulate is a non-linear boundary value problem for a third-order ordinary differential equation with non-linear boundary conditions, which are non-local in time. We prove existence and uniqueness of a classical solution and provide a numerical example. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Algebraic multigrid preconditioners for the bidomain reaction–diffusion system

The so-called bidomain system is possibly the most complete model for the cardiac bioelectric activity. It consists of a reaction–diffusion system, modeling the intra, extracellular and transmembrane potentials, coupled through a nonlinear reaction term with a stiff system of ordinary differential equations describing the ionic currents through the cellular membrane. In this paper we address the problem of efficiently solving the large linear system arising in the finite element discretization of the bidomain model, when a semiimplicit method in time is employed. We analyze the use of structured algebraic multigrid preconditioners on two major formulations of the model, and report on our numerical experience under different discretization parameters and various discontinuity properties of the conductivity tensors. Our numerical results show that the less exercised formulation provides the best overall performance on a typical simulation of the myocardium excitation process.     

Lyapunov functionals for reaction–diffusion equations with memory

We consider a reaction‐diffusion equation in which the usual diffusion term also depends on the past history of the diffusion itself. This equation has been analysed by several authors, with an emphasis on the longtime behaviour of the solutions. In this respect, the first results have been obtained by using the past history approach. They show that the equation, subject to a suitable boundary condition, defines a dissipative dynamical system which possesses a global attractor. A similar theorem has been recently proved by Chepyzhov and Miranville, using a different method based on the notion of trajectory attractors. In addition, those authors provide sufficient conditions that ensure the existence of a Lyapunov functional. Here we show that a similar result can be demonstrated within the past history approach, with less restrictive conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

Global a posteriori error estimates for convection–reaction–diffusion problems

In this note we propose a nonstandard technique for constructing global a posteriori error estimates for the stationary convection–reaction–diffusion equation. In order to estimate the approximation error in appropriate weighted energy norms, which measures the overall quality of the approximations, the underlying bilinear form is decomposed into several terms which can be directly computed or easily estimated from above using elementary tools of functional analysis. Several auxiliary parameters are introduced to construct such a splitting and tune the resulting upper error bound. It is demonstrated how these parameters can be chosen in some natural and convenient way for computations so that the weighted energy norm of the error is almost recovered, which shows that the estimates proposed are, in fact, quasi-sharp. The presented methodology is completely independent of numerical techniques used to compute approximate solutions. In particular, it is applicable to approximations which fail to satisfy the Galerkin orthogonality, e.g., due to an inconsistent stabilization, flux limiting, low-order quadrature rules, round-off and iteration errors etc. Moreover, the only constant that appears in the proposed error estimates is of global nature and comes from the Friedrichs–Poincaré inequality.     

Spike patterns in a reaction–diffusion ODE model with Turing instability

We explore a mechanism of pattern formation arising in processes described by a system of a single reaction–diffusion equation coupled with ordinary differential equations. Such systems of equations arise from the modeling of interactions between cellular processes and diffusing growth factors. We focus on the model of early carcinogenesis proposed by Marciniak‐Czochra and Kimmel, which is an example of a wider class of pattern formation models with an autocatalytic non‐diffusing component. We present a numerical study showing emergence of periodic and irregular spike patterns because of diffusion‐driven instability. To control the accuracy of simulations, we develop a numerical code on the basis of the finite‐element method and adaptive mesh grid. Simulations, supplemented by numerical analysis, indicate a novel pattern formation phenomenon on the basis of the emergence of nonstationary structures tending asymptotically to a sum of Dirac deltas. Copyright © 2013 John Wiley & Sons, Ltd.     

Solving the reaction–diffusion equations with nonlocal boundary conditions based on reproducing kernel space

The reaction–diffusion equations with initial condition and nonlocal boundary conditions are discussed in this article. A reproducing kernel space is constructed, in which an arbitrary function satisfies the initial condition and nonlocal boundary conditions of the reaction‐diffusion equations. Based on the reproducing kernel space, a new algorithm for solving the reaction–diffusion equations with initial condition and nonlocal boundary conditions is presented. Some examples are displayed to demonstrate the validity and applicability of the proposed method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Quenching for a reaction–diffusion system with logarithmic singularity

In this paper, we study the quenching phenomenon for a reaction–diffusion system with singular logarithmic source terms and positive Dirichlet boundary conditions. Some sufficient conditions for quenching of the solutions in finite time are obtained, and the blow-up of time-derivatives at the quenching point is verified. Furthermore, under appropriate hypotheses, the non-simultaneous quenching of the system is proved, and the estimates of quenching rate is given.     

Global existence to a reaction–diffusion equation through a self‐similar‐like upper solution

A demonstration method is presented, which will ensure the existence of positive global solutions in time to the reaction–diffusion equation ?u_t+Δu+u^p=0 in ?ⁿ×[0, ∞), for exponents p?3 and space dimensions n?3. This method does not require the initial value to have a specific uniform smallness condition, but rather to satisfy a bell‐like form. The method is based on a specific upper solution, which models the diffusion process of the heat equation. The upper solution is not self‐similar, but does have a self‐similar‐like form. After transforming the reaction–diffusion problem into an equivalent one, whose initial value is uniformly very small, a local solution is obtained in the time interval [0, 1] by the use of this upper solution. This local solution is then extended to [0, ∞) through an infinite sequence of extensions. At each step, an appropriate change of variables will transform the extension into a problem nearly identical to the local problem in [0, 1]. These transformations exploit the diffusive and self‐similar‐like nature of the upper solution. Copyright © 2009 John Wiley & Sons, Ltd.     

The decomposition method applied to systems of partial differential equations and to the reaction–diffusion Brusselator model

In this work, systems of linear and nonlinear partial differential equations and the reaction–diffusion Brusselator model are handled by applying the decomposition method. The advantage of this work is twofold. Firstly, the decomposition method reduces the computational work. Secondly, in comparison with existing techniques, the decomposition method is an improvement with regard to its accuracy and rapid convergence. The decomposition method has the advantage of being more concise for analytical and numerical purposes.     

A parameter-uniform Schwarz method for a coupled system of reaction–diffusion equations

We consider an arbitrarily sized coupled system of one-dimensional reaction–diffusion problems that are singularly perturbed in nature. We describe an algorithm that uses a discrete Schwarz method on three overlapping subdomains, extending the method in [H. MacMullen, J.J.H. Miller, E. O’Riordan, G.I. Shishkin, A second-order parameter-uniform overlapping Schwarz method for reaction-diffusion problems with boundary layers, J. Comput. Appl. Math. 130 (2001) 231–244] to a coupled system. On each subdomain we use a standard finite difference operator on a uniform mesh. We prove that when appropriate subdomains are used the method produces ε-uniform results. Furthermore we improve upon the analysis of the above-mentioned reference to show that, for small ε, just one iteration is required to achieve the expected accuracy.     

Blow‐up profiles of solutions for the exponential reaction‐diffusion equation

We consider the blow‐up of solutions for a semilinear reaction‐diffusion equation with exponential reaction term. It is known that certain solutions that can be continued beyond the blow‐up time possess a non‐constant self‐similar blow‐up profile. Our aim is to find the final time blow‐up profile for such solutions. The proof is based on general ideas using semigroup estimates. The same approach works also for the power nonlinearity. Copyright © 2011 John Wiley & Sons, Ltd.     

An ETD Crank‐Nicolson method for reaction‐diffusion systems

A novel Exponential Time Differencing Crank‐Nicolson method is developed which is stable, second‐order convergent, and highly efficient. We prove stability and convergence for semilinear parabolic problems with smooth data. In the nonsmooth data case, we employ a positivity‐preserving initial damping scheme to recover the full rate of convergence. Numerical experiments are presented for a wide variety of examples, including chemotaxis and exotic options with transaction cost. © 2011Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Mathematical modeling of time fractional reaction–diffusion systems

We study a fractional reaction–diffusion system with two types of variables: activator and inhibitor. The interactions between components are modeled by cubical nonlinearity. Linearization of the system around the homogeneous state provides information about the stability of the solutions which is quite different from linear stability analysis of the regular system with integer derivatives. It is shown that by combining the fractional derivatives index with the ratio of characteristic times, it is possible to find the marginal value of the index where the oscillatory instability arises. The increase of the value of fractional derivative index leads to the time periodic solutions. The domains of existing periodic solutions for different parameters of the problem are obtained. A computer simulation of the corresponding nonlinear fractional ordinary differential equations is presented. For the fractional reaction–diffusion systems it is established that there exists a set of stable spatio-temporal structures of the one-dimensional system under the Neumann and periodic boundary conditions. The characteristic features of these solutions consist of the transformation of the steady-state dissipative structures to homogeneous oscillations or space temporary structures at a certain value of fractional index and the ratio of characteristic times of system.     

Multilevel a posteriori error analysis for reaction–convection–diffusion problems

We present a new approach to the a posteriori error analysis of stable Galerkin approximations of reaction–convection–diffusion problems. It relies upon a non-standard variational formulation of the exact problem, based on the anisotropic wavelet decomposition of the equation residual into convection-dominated scales and diffusion-dominated scales. The associated norm, which is stronger than the standard energy norm, provides a robust (i.e., uniform in the convection limit) control over the streamline derivative of the solution. We propose an upper estimator and a lower estimator of the error, in this norm, between the exact solution and any finite dimensional approximation of it. We investigate the behaviour of such estimators, both theoretically and through numerical experiments. As an output of our analysis, we find that the lower estimator is quantitatively accurate and robust.