GLOBAL STABILITY OF TRAVELING WAVEFRONTS FOR NONLOCAL REACTION-DIFFUSION EQUATIONS WITH TIME DELAY

This paper is concerned with the stability of traveling wavefronts for a population dynamics model with time delay. Combining the weighted energy method and the comparison principle, the global exponential stability of noncritical traveling wavefronts （waves with speeds c 〉 c＊, where c=c＊ is the minimal speed） is established, when the initial perturbations around the wavefront decays to zero exponentially in space as x → -∞, but it can be allowed arbitrary large in other locations, which improves the results in[9, 18, 21].

UNIFORM QUASI-DIFFERENTIABILITY OF SEMIGROUP TO NONLINEAR REACTION-DIFFUSION EQUATIONS WITH SUPERCRITI C AL EXPONENT

A new approach, is established to show that the semigroup {S（t）≥0 generated by a reaction-diffusion equation with supercritical exponent is uniformly quasi-differentiable in L^q（Ω） （2 ≤ q 〈 ∞） with respect to the initial value. As an application, this proves the upper-bound of fractal dimension for its global attractor in the corresponding space.

Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains

In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.

Traveling Wave Solution for Two Kinds of Reaction-Diffusion Equations

The generialized Kuramoto Sivashinski equation and Fisher equation in chemical reaction diffusion was studied in this paper. By introducing a new method, the anthors obtained the exact traveling wave solution for the two types of reaction diffusion equations.

A Split-Step Predictor-Corrector Method for Space-Fractional Reaction-Diffusion Equations with Nonhomogeneous Boundary Conditions

A split-step second-order predictor-corrector method for space-fractional reaction-diffusion equations with nonhomogeneous boundary conditions is presented and analyzed for the stability and convergence.The matrix transfer technique is used for spatial discretization of the problem.The method is shown to be unconditionally stable and second-order convergent.Numerical experiments are performed to confirm the stability and secondorder convergence of the method.The split-step predictor-corrector method is also compared with an IMEX predictor-corrector method which is found to incur oscillatory behavior for some time steps.Our method is seen to produce reliable and oscillatioresults for any time step when implemented on numerical examples with nonsmooth initial data.We also present a priori reliability constraint for the IMEX predictor-corrector method to avoid unwanted oscillations and show its validity numerically.

THE EXTINCTION BEHAVIOR OF THE SOLUTIONS FOR A CLASS OF REACTION-DIFFUSION EQUATIONS

The methods of L-p estimation are used to discuss the extinction phenomena of the solutions to the following reaction-diffusion equations with initial-boundary values partial derivativeu/partial derivativet = Deltau - lambda \u\(gamma-1) u - betau ((x, t) is an element of Omega x (0, + infinity)), u(x, t) \(partial derivativeOmegax (0, +infinity)) = 0, u(x, 0) = u(0) (x) is an element of H-0(1) (Omega) boolean AND L1+gamma(Omega) (x is an element of Omega). Sufficient and necessary conditions about the extinction of the solutions is given. Here lambda > 0, gamma > 0, beta > 0 are constants, Omega is an element of R-N is bounded with smooth boundary partial derivativeOmega. At last, it is simulated with a higher order equation by using the present methods.

Wave equations and reaction-diffusion equations with several nonlinear source terms

The initial boundary value problem of wave equations and reaction-diffusion equations with several nonlinear source terms in a bounded domain is studied by potential well method. The invarianee of some sets under the flow of these problems and the vacuum isolation of solutions are obtained by introducing a family of potential wells. Then the threshold result of global existence and nonexistence of solutions are given. Finally, the problem with critical initial conditions are discussed.

STABILITY OF TIME-PERIODIC TRAVELING FRONTS IN BISTABLE REACTION-ADVECTION-DIFFUSION EQUATIONS

This paper is concerned with the global exponential stability of time periodic traveling fronts of reaction-advection-diffusion equations with time periodic bistable nonlinearity in infinite cylinders. It is well known that such traveling fronts exist and are asymptotically stable. In this paper, we further show that such fronts are globally exponentially stable. The main difficulty is to construct appropriate supersolutions and subsolutions.

Exponential Time Differencing Method for a Reaction-Diffusion System with Free Boundary

For reaction-diffusion equations in irregular domains with moving boundaries,the numerical stability constraints from the reaction and diffusion terms often require very restricted time step sizes,while complex geometries may lead to difficulties in the accuracy when discretizing the high-order derivatives on grid points near the boundary.It is very challenging to design numerical methods that can efficiently and accurately handle both difficulties.Applying an implicit scheme may be able to remove the stability constraints on the time step,however,it usually requires solving a large global system of nonlinear equations for each time step,and the computational cost could be significant.Integration factor(IF)or exponential time differencing(ETD)methods are one of the popular methods for temporal partial differential equations(PDEs)among many other methods.In our paper,we couple ETD methods with an embedded boundary method to solve a system of reaction-diffusion equations with complex geometries.In particular,we rewrite all ETD schemes into a linear combination of specificФ-functions and apply one state-of-the-art algorithm to compute the matrix-vector multiplications,which offers significant computational advantages with adaptive Krylov subspaces.In addition,we extend this method by incorporating the level set method to solve the free boundary problem.The accuracy,stability,and efficiency of the developed method are demonstrated by numerical examples.

Relation between the complex Ginzburg-Landau equation and reaction-diffusion system

The complex Ginzburg-Landau equation （CGLE） has been used to describe the travelling wave behaviour in reaction-diffusion （RD） systems. We argue that this description is valid only when the RD system is close to the Hopf bifurcation, and is not valid when a RD system is away from the onset. To test this, we study spirals and anti-spirals in the chlorite-iodide-malonic acid （CIMA） reaction and the corresponding OGLE. Numerical simulations confirm that the OGLE can only be applied to the CIMA reaction when it is very near the Hopf onset.

NUMERICAL SOLUTION OF A NONLINEAR REACTION-DIFFUSION EQUATION

A nonlinear reaction-diffusion equation is studied numerically by a Petrov-Galerkin finite element method, which has been proved to be 2nd-order accurate in time and 4th-order in space. The comparison between the exact and numerical solutions of progressive waves shows that this numerical scheme is quite accurate, stable andefflcient. It is also shown that any local disturbance will spread, have a full growth and finally form two progressive waves propagating in both directions. The shape and the speed of the long term progressive waves are determined by the system itself, and do not depend on the details of the initial values.

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS OF THE REACTION DIFFUSION EQUATIONS IN A PART OF DOMAIN

A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.

A CLASS OF NONLINEAR SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

A class of nonlinear singularly perturbed problems for reaction diffusion equations are considered. Under suitable conditions, by using the theory of differential inequalities, the asymptotic behavior of solutions for the initial boundary value problems are studied, reduced problems of which possess two intersecting solutions.

Analytical Solutions of System of Non-Linear Differential Equations in the Single-Enzyme, Single-Substrate Reaction with Non-Mechanism-Based Enzyme Inactivation

A closed form of an analytical expression of concentration in the single-enzyme, single-substrate system for the full range of enzyme activities has been derived. The time dependent analytical solution for substrate, enzyme-substrate complex and product concentrations are presented by solving system of non-linear differential equation. We employ He’s Homotopy perturbation method to solve the coupled non-linear differential equations containing a non-linear term related to basic enzymatic reaction. The time dependent simple analytical expressions for substrate, enzyme-substrate and free enzyme concentrations have been derived in terms of dimensionless reaction diffusion parameters ε, λ1, λ2 and λ3 using perturbation method. The numerical solution of the problem is also reported using SCILAB software program. The analytical results are compared with our numerical results. An excellent agreement with simulation data is noted. The obtained results are valid for the whole solution domain.

SINGULAR PERTURBATION FOR REACTION DIFFUSION EQUATIONS

The singularly perturbed initial boundary value problems for reaction diffusion equations are considered.Under suitable conditions and by using the theory of differential inequality,the asymptotic behavior of solution for initial boundary value problems are studied,where the reduced problems possess two intersecting solutions.

THE ASYMPTOTIC BEHAVIOR OF SOLUTION FOR THE SINGULARLY PERTURBED INITIAL BOUNDARY VALUE PROBLEMS OF THE REACTION DIFFUSION EQUATIONS IN A PART OF DOMAIN

A class of singularly perturbed initial boundary value problems for the reaction diffusion equations in a part of domain are considered. Using the operator theory the asymptotic behavior of solution for the problems is studied.

Approximate Solution of Non-Linear Reaction Diffusion Equations in Homogeneous Processes Coupled to Electrode Reactions for CE Mechanism at a Spherical Electrode

A mathematical model of CE reaction schemes under first or pseudo-first order conditions with different diffusion coefficients at a spherical electrode under non-steady-state conditions is described. The model is based on non-stationary diffusion equation containing a non-linear reaction term. This paper presents the complex numerical method (Homotopy perturbation method) to solve the system of non-linear differential equation that describes the homogeneous processes coupled to electrode reaction. In this paper the approximate analytical expressions of the non-steady-state concentrations and current at spherical electrodes for homogeneous reactions mechanisms are derived for all values of the reaction diffusion parameters. These approximate results are compared with the available analytical results and are found to be in good agreement.

Inference of General Mass Action-Based State Equations for Oscillatory Biochemical Reaction Systems Using <i>k</i>-Step Genetic Programming

Systems biology requires the development of algorithms that use omics data to infer interaction networks among biomolecules working within an organism. One major type of evolutionary algorithm, genetic programming (GP), is useful for its high heuristic ability as a search method for obtaining suitable solutions expressed as tree structures. However, because GP determines the values of parameters such as coefficients by random values, it is difficult to apply in the inference of state equations that describe oscillatory biochemical reaction systems with high nonlinearity. Accordingly, in this study, we propose a new GP procedure called “k-step GP” intended for inferring the state equations of oscillatory biochemical reaction systems. The k-step GP procedure consists of two algorithms: 1) Parameter optimization using the modified Powell method—after genetic operations such as crossover and mutation, the values of parameters such as coefficients are optimized by applying the modified Powell method with secondary convergence. 2) GP using divided learning data—to improve the inference efficiency, imposes perturbations through the addition of learning data at various intervals and adaptations to these changes result in state equations with higher fitness. We are confident that k-step GP is an algorithm that is particularly well suited to inferring state equations for oscillatory biochemical reaction systems and contributes to solving inverse problems in systems biology.

A NEW APPROACH TO THE EQUATIONS FOR THE STEADY STATE CURRENTS AT MICROELECTRODES:EC'PROCESSES(SECOND ORDER REACTION)

With a new approach,the general current expressions of two typical second order catalytic reactions at microelectrodes are obtained.This allows the study of fast chemical reactions and systems where the reactants are present in similar concentrations.

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF REACTION DIFFUSION EQUATIONS

A class of initial boundary value problems for the reaction diffusion equations are considered.The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.