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.     

A variable nonlinear splitting algorithm for reaction diffusion systems with self‐ and cross‐ diffusion

Self‐ and cross‐diffusion are important nonlinear spatial derivative terms that are included into biological models of predator–prey interactions. Self‐diffusion models overcrowding effects, while cross‐diffusion incorporates the response of one species in light of the concentration of another. In this paper, a novel nonlinear operator splitting method is presented that directly incorporates both self‐ and cross‐diffusion into a computational efficient design. The numerical analysis guarantees the accuracy and demonstrates appropriate criteria for stability. Numerical experiments display its efficiency and accuracy.     

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.     

Higher-order compact finite difference method for systems of reaction–diffusion equations

This paper is concerned with a compact finite difference method for solving systems of two-dimensional reaction–diffusion equations. This method has the accuracy of fourth-order in both space and time. The existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. Three monotone iterative algorithms are provided for solving the resulting discrete system efficiently, and the sequences of iterations converge monotonically to a unique solution of the system. A theoretical comparison result for the various monotone sequences is given. The convergence of the finite difference solution to the continuous solution is proved, and Richardson extrapolation is used to achieve fourth-order accuracy in time. An application is given to an enzyme–substrate reaction–diffusion problem, and some numerical results are presented to demonstrate the high efficiency and advantages of this new approach.     

Global existence of solutions to reaction–diffusion systems without conditions on the nonlinearities growth

It is well known that, for reaction–diffusion systems, if the nonlinearities grow faster than a polynomial, nothing seems to be known for instance. The purpose of this paper is to give sufficient conditions guaranteeing global existence, uniqueness and uniform boundedness of solutions for coupled reaction–diffusion equations without condition growth on the reactions terms f and g in case f + g ≠ 0. These systems possess many and various applications in physics as the diffusion of the Phosphorus in the Silicone or some models describing some nuclear reactions; there have also been other applications in chemistry and biology. Our techniques are based on the Lyapunov functional methods. Copyright © 2010 John Wiley & Sons, Ltd.     

A weak solution approach to a reaction–diffusion system modeling pattern formation on seashells

We investigate a reaction–diffusion system proposed by H. Meinhardt as a model for pattern formation on seashells. We give a new proof for the existence of a local weak solution for general initial conditions and parameters upon using an iterative approach. Furthermore, the solution is shown to exist globally for suitable initial data. The behavior of the solution in time and space is illustrated through numerical simulations. Copyright © 2009 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     

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.     

Normal deviations from the averaged motion for some reaction–diffusion equations with fast oscillating perturbation

We study the normalized difference between the solution u of a reaction–diffusion equation in a bounded interval [0,L], perturbed by a fast oscillating term arising as the solution of a stochastic reaction–diffusion equation with a strong mixing behavior, and the solution of the corresponding averaged equation. We assume the smoothness of the reaction coefficient and we prove that a central limit type theorem holds. Namely, we show that the normalized difference converges weakly in C([0,T];L²(0,L)) to the solution of the linearized equation, where an extra Gaussian term appears. Such a term is explicitly given.     

A Lyapunov functional for a triangular reaction–diffusion system with nonlinearities of exponential growth

The aim of this work is to study the global existence of solutions to a triangular system of reaction–diffusion equations, which describes epidemiological or chemical situations. On the basis of the construction of a suitable Lyapunov functional, we show that for any initial data, classical global solutions exist even when the nonlinearities are of exponential growth. Copyright © 2012 John Wiley & Sons, Ltd.     

Traveling wave solutions of n‐dimensional delayed reaction–diffusion systems and application to four‐dimensional predator–prey systems

This paper deals with the existence of traveling wave solutions for n‐dimensional delayed reaction–diffusion systems. By using Schauder's fixed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper–lower solutions that are easy to construct. As an application, we apply our main results to a four‐dimensional delayed predator–prey system and obtain the existence of traveling wave solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Block-circulant preconditioners for systems arising from discretization of the three-dimensional convection–diffusion equation

We consider the system of equations arising from finite difference discretization of a three-dimensional convection–diffusion model problem. This system is typically nonsymmetric. The GMRES method with the Strang block-circulant preconditioner is proposed for solving this linear system. We show that our preconditioners are invertible and study the spectra of the preconditioned matrices. Numerical results are reported to illustrate the effectiveness of our methods.     

Asymptotic stability of traveling wave fronts in nonlocal reaction–diffusion equations with delay

This paper is concerned with the asymptotic stability of traveling wave fronts of a class of nonlocal reaction–diffusion equations with delay. Under monostable assumption, we prove that the traveling wave front is exponentially stable by means of the (technical) weighted energy method, when the initial perturbation around the wave is suitable small in a weighted norm. The exponential convergent rate is also obtained. Finally, we apply our results to some population models and obtain some new results, which recover, complement and/or improve a number of existing ones.     

Electro–Reaction–Diffusion Systems Including Cluster Reactions of Higher Order

In this paper we consider electro–reaction–diffusion systems modelling the transport of charged species in two–dimensional heterostructures. Our aim is to investigate the case that besides of reactions with source terms of at most second order so called cluster reactions of higher order are involved. We prove the unique solvability of the model equations and show the global boundedness and asymptotic properties of the solution. In order to get necessary a priori estimates we apply an anisotropic iteration scheme followed by usual Moser iterations. Then existence is obtained by cutting off the reaction terms.     

Transcendental Bernstein series for solving reaction–diffusion equations with nonlocal boundary conditions through the optimization technique

In this paper, we apply transcendental Bernstein series (TBS) for solving reaction–diffusion equations with nonlocal boundary conditions which is the novel approximation tool. To carry out the method, we firstly expand the solution of the system in the term of TBS through the operational matrix scheme. To determine the unknown free coefficients and control parameters appeared in TBS expansion, we define an optimization problem which combines the reaction–diffusion equation with its nonlocal boundary conditions. Then we use the Lagrange multipliers technique for converting the problem under study into a system of algebraic equations. High accuracy and simplicity in reducing the integral boundary conditions are some privileges of the proposed scheme. We emphasize that Bernstein polynomials is the particular case of transcendental Bernstein series. Theoretical discussion about convergence confirms the reliability of the proposed method. Some test problems are chosen to investigate the applicability and computational efficiency. The experimental results confirm that the obtained results are in good agreement with the exact solutions with high rate of convergence.     

Superconvergence of finite difference approximations for convection–diffusion problems

An Erratum has been published for this article in Numerical Linear Algebra with Applications 8 (4) 2001, iii–iv. We are concerned with numerical solutions of convection–diffusion equations. The convergence behaviour of numerical solutions is considered by using the finite difference approximation with respect to spatial variables and implicit method with respect to time variable. It is shown that superconvergence occurs near a part of the boundary which has Dirichlet's data. Copyright © 2001 John Wiley & Sons, Ltd.     

On the well‐posedness for Keller–Segel system with fractional diffusion

In this paper, we study the Cauchy problem for the Keller–Segel system with fractional diffusion generalizing the Keller–Segel model of chemotaxis for the initial data (u₀,v₀) in critical Fourier‐Herz spaces with q ∈ [2, ∞], where 1 < α ≤ 2. Making use of some estimates of the linear dissipative equation in the frame of mixed time‐space spaces, the Chemin ‘mono‐norm method’, the Fourier localization technique and the Littlewood–Paley theory, we get a local well‐posedness result and a global well‐posedness result with a small initial data. In addition, ill‐posedness for ‘doubly parabolic’ models is also studied. Copyright © 2011 John Wiley & Sons, Ltd.     

Stability and Hopf bifurcation of a delayed reaction–diffusion neural network

In this paper, a delayed reaction–diffusion neural network with Neumann boundary conditions is investigated. By analyzing the corresponding characteristic equations, the local stability of the trivial uniform steady state is discussed. The existence of Hopf bifurcation at the trivial steady state is established. Using the normal form theory and the center manifold reduction of partial function differential equations, explicit formulae are derived to determine the direction and stability of bifurcating periodic solutions. Numerical simulations are carried out to illustrate the main results. Copyright © 2011 John Wiley & Sons, Ltd.     

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.