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.

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.

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.

Nonlinear Singularly Perturbed Problems for Reaction Diffusion Equations with Two Parameters and Boundary Perturbation

A class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with two parameters and boundary perturbation were considered.Under suitable conditions,the existence,uniqueness and asymptotic behavior of solutions for the initial boundary value problems were studied.An example was also given to illustrate our main results.

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.

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.

THE POINTWISE ESTIMATES OF SOLUTIONS FOR A NONLINEAR CONVECTION DIFFUSION REACTION EQUATION

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal Lp,1≤ p ≤ ＋∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.

A CLASS OF SINGULARLY PERTURBED INITIAL BOUNDARY PROBLEM FOR REACTION DIFFUSION EQUATION

The singularly perturbed initial boundary value problem for a class of reaction diffusion equation is considered. Under appropriate conditions, the existence-uniqueness and the asymptotic behavior of the solution are showed by using the fixed-point theorem.

Global existence and blow up of solutions for two classes of reaction diffusion systems with two nonlinear source terms in bounded domain

In this paper we deal with the initial boundary value problem for two classes of reaction diffusion systems with two source terms in bounded domain. Under some assumptions on the exponents and the initial data, applying the comparison principle, the maximum prin- ciple and the supersolution-subsolution method, we prove the global existence and blow up of solutions. We also establish some upper blow up rates.

STABILITY OF STATIONARY STATE SOLUTION FOR A REACTION DENSITY-DEPENDENT DIFFUSION EQUATION

In this paper we are interested in the large time behavior of the nonlinear diffusion equationWe consider functions which allow the equation to possess traveling wave solutions. We first present an existence and uniqueness as well as some comparison principle result of generalized solutions to the Cauchy problem. Then we give for some threshold results, from which we can see that u=a is stable, while u= 0 or u=1 is unstable under some assumptions, etc.

UPPER SEMICONTINUITY OF ATTRACTORS FORTHE REACTION DIFFUSION EQUATION

This paper deals with the reaction diffusion equation in domain, Omega = R or Omega = (-L, L) with L < infinity. Let A(L) and A be the global attractor of this equation corresponding to Omega = (-L,L) and Omega = R, respectively. It is showed that the global attractor A is upper semicontinuity at 0 with respect to the sets {A(L)} in some sense.

Lower Bounds of Blow Up Time for a Class of Slow Reaction Diffusion Equations with Inner Absorption Terms

In this paper,a class of slow reaction-diffusion equations with nonlocal source and inner absorption terms are studied.By using the technique of improved differential inequality,the lower bounds of blow up time for the system under either homogeneous Dirichlet or nonhomogeneous Neumann boundary conditions are obtained.

EXPONENTIAL TIME DIFFERENCING-PADE FINITE ELEMENT METHOD FOR NONLINEAR CONVECTION-DIFFUSION-REACTION EQUATIONS WITH TIME CONSTANT DELAY

In this paper,ETD3-Padéand ETD4-PadéGalerkin finite element methods are proposed and analyzed for nonlinear delayed convection-diffusion-reaction equations with Dirichlet boundary conditions.An ETD-based RK is used for time integration of the corresponding equation.To overcome a well-known difficulty of numerical instability associated with the computation of the exponential operator,the Padéapproach is used for such an exponential operator approximation,which in turn leads to the corresponding ETD-Padéschemes.An unconditional L^(2) numerical stability is proved for the proposed numerical schemes,under a global Lipshitz continuity assumption.In addition,optimal rate error estimates are provided,which gives the convergence order of O(k^(3)+h^(r))(ETD3-Padé)or O(k^(4)+h^(r))(ETD4-Padé)in the L^(2)norm,respectively.Numerical experiments are presented to demonstrate the robustness of the proposed numerical schemes.

ASYMPTOTIC PROPERTIES OF FUNCTIONAL DIFFERENTIAL EQUATIONS IN BANACH SPACES

The paper is devoted to the asymptotic properties of functional differential equations in Banach spaces.The criteria of the invariant and attracting sets are obtained.Particularly, the sufficient condition of asymptotic stability of the equilibrium point is given as the system has an equilibrium point.Several examples are also worked out to demonstrate the validity of the results.

SINGULAR PERTURBATION OF INITIAL BOUNDARY VALUE FROBLEMS OF REACTION DIFFUSlON EQUATION WITH DELAY

In this paper we consider the initial-boundary value problems for a class ofapplications, such as biomathematics and biochemistry.Applying the method ofcomposile expansion we construct the formally asymptotic solution of the problemdescribed. With the help of theory of upper and lower solutions we prove the uniformlyvalidity of the formal solution and the existence of solution of the original problem.

On the gradient tracking problem for a bilinear reaction–diffusion equation excited by distributed bounded controls

This paper addresses a gradient tracking problem of a bilinear reaction–diffusion equation evolvingin a spatial domainΩ ⊂ Rn, n ≤ 3. Such an equation is excited with distributed and boundedcontrols. The problem is formulated by the minimisation of a functional constituted of the deviationbetween the desired gradient and the current one all over a time interval and the energyterm. Then we prove the existence of an optimal control that we characterise by an optimalitysystem. Moreover, we discuss two sets of particular controls: the set of time dependent controlsand the space dependent ones. A computational approach and illustrative simulations are alsogiven.

ENERGY ESTIMATES FOR DELAY DIFFUSION-REACTION EQUATIONS

In this paper we consider nonlinear delay diffusion-reaction equations with initial and Dirichlet boundary conditions. The behaviour and the stability of the solution of such initial boundary value problems （IBVPs） are studied using the energy method. Simple numerical methods are considered for the computation of numerical approximations to the solution of the nonlinear IBVPs. Using the discrete energy method we study the stability and convergence of the numerical approximations. Numerical experiments are carried out to illustrate our theoretical results.

GENERALIZED SOLUTION OF SINGULARLY PERTURBED PROBLEMS FOR REACTION DIFFUSION EQUATIONS

The singularly perturbed initial boundary value problem for reaction diffusion equation is considered. Under suitable conditions the existence, uniqueness and asymptotic behavior of the generalized solution for the problems are studied.

LAGRANGE STABILITY IN MEAN SQUARE OF STOCHASTIC REACTION DIFFUSION EQUATIONS

This work is devoted to the discussion of stochastic reaction diffusion equations and some new theorems on Lagrange stability in mean square of the solution are established via Lyapunov method which is nothing to be done in the past.