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 KIND OF NONLOCAL PROBLEMS FOR SINGULARLY PERTURBED REACTION DIFFUSION SYSTEMS

The problems of the nonlocal boundary conditions for the singularly perturbed reaction diffusion systems are considered. Under suitable conditions, using the comparison theorem the asymptotic behavior of solution for the initial boundary value problems are studied.

A modified Tikhonov regularization method for a Cauchy problem of a time fractional diffusion equation

In this paper,we consider a Cauchy problem of the time fractional diffusion equation(TFDE)in x∈[0,L].This problem is ubiquitous in science and engineering applications.The illposedness of the Cauchy problem is explained by its solution in frequency domain.Furthermore,the problem is formulated into a minimization problem with a modified Tikhonov regularization method.The gradient of the regularization functional based on an adjoint problem is deduced and the standard conjugate gradient method is presented for solving the minimization problem.The error estimates for the regularized solutions are obtained under Hp norm priori bound assumptions.Finally,numerical examples illustrate the effectiveness of the proposed method.

Modeling of Diffusion Transport through Oral Biofilms with the Inverse Problem Method

Aim The purpose of this study was to develop a mathe-matical model to quantitatively describe the passive trans-port of macromolecules within dental biofilms. Methodology Fluorescently labeled dextrans with different molecular mass （3 kD,10 kD,40 kD,70 kD,2 000 kD） were used as a series of diffusion probes. Streptococcus mutans,Streptococcus sanguinis,Actinomyces naeslundii and Fusobacterium nucleatum were used as inocula for biofilm formation. The diffusion processes of different probes through the in vitro biofilm were recorded with a confocal laser microscope. Results Mathematical function of biofilm penetration was constructed on the basis of the inverse problem method. Based on this function,not only the relationship between average concentration of steady-state and molecule weights can be analyzed,but also that between penetrative time and molecule weights. Conclusion This can be used to predict the effective concentration and the penetrative time of anti-biofilm medicines that can diffuse through oral biofilm. Further-more,an improved model for large molecule is proposed by considering the exchange time at the upper boundary of the dental biofilm.

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 new streamline diffusion finite element method for the generalized Oseen problem

This paper aims to present a new streamline diffusion method with low order rectangular Bernardi-Raugel elements to solve the generalized Oseen equations. With the help of the Bramble-Hilbert lemma, the optimal errors of the velocity and pressure are estimated, which are independent of the considered parameter e. With an interpolation postprocessing approach, the superconvergent error of the pressure is obtained. Finally, a numerical experiment is carried out to confirm the theoretical results.

Free boundary value problems for a class of generalized diffusion equation

The transport behavior of free boundary value problems for a class ofgeneralized diffusion equations was studied. Suitable similarity transformations were used toconvert the problems into a class of singular nonlinear two-point boundary value problems andsimilarity solutions were numerical presented for different representations of heat conductionfunction, convection function, heat flux function, and power law parameters by utilizing theshooting technique. The results revealed the flux transfer mechanism and the character as well asthe effects of parameters on the solutions.

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.

On the Robustness of the xy-Zebra-Gauss-Seidel Smoother on an Anisotropic Diffusion Problem

Studies of problems involving physical anisotropy are applied in sciences and engineering,for instance,when the thermal conductivity depends on the direction.In this study,the multigrid method was used in order to accelerate the convergence of the iterative methods used to solve this type of problem.The asymptotic convergence factor of the multigrid was determined empirically(computer aided)and also by employing local Fourier analysis(LFA).The mathematical model studied was the 2D anisotropic diffusion equation,in whichε>0 was the coefficient of a nisotropy.The equation was discretized by the Finite Difference Method(FDM)and Central Differencing Scheme(CDS).Correction Scheme(CS),pointwise Gauss-Seidel smoothers(Lexicographic and Red-Black ordering),and line Gauss-Seidel smoothers(Lexicographic and Zebra ordering)in x and y directions were used for building the multigrid.The best asymptotic convergence factor was obtained by the Gauss-Seidel method in the direction x for 0<ε<<1 and in the direction y forε>>1.In this sense,an xy-zebra-GS smoother was proposed,which proved to be efficient and robust for the different anisotropy coefficients.Moreover,the convergence factors calculated empirically and by LFA are in agreement.

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.

Improving the Stability Problem of the Finite Difference Scheme for Reaction-diffusion Equation

这份报纸处理特殊非线性的反应散开 equation.The 有接近到微分方程( 2.1 )的增长 unknowns 的有限差别计划借助于介绍增长 unknowns methods.Through 被建立为计划分析的稳定性，与稳定性相比，相应经典差别的条件什么时候策划，被显示出有增长 unknowns 的有限差别计划的稳定性条件极大地被改进。

A NONLINEAR TRANSFORMATION AND A BOUNDARY-INITIAL VALUE PROBLEM FOR A CLASS OF NONLINEAR CONVECTION-DIFFUSION EQUATIONS

With the aid of a nonlinear transformation, a class of nonlinear convection-diffusion PDE in one space dimension is converted into a linear one, the unique solution of a nonlinear boundary-initial value problem for the nonlinear PDE can be exactly expressed by the nonlinear transformation, and several illustrative examples are given.

Streamline upwind finite element method using 6-node triangular element with adaptive remeshing technique for convective-diffusion problems

A streamline upwind finite element method using 6-node triangular element is presented. The method is applied to the convection term of the governing transport equation directly along local streamlines. Several convective-diffusion examples are used to evaluate efficiency of the method. Results show that the method is monotonic and does not produce any oscillation. In addition, an adaptive meshing technique is combined with the method to further increase accuracy of the solution, and at the same time, to minimize computational time and computer memory requirement.

THE CAUCHY PROBLEM FOR A NONLINEAR FAST DIFFUSION EQUATION

In this paper, the existence, uniqueness and continuous dependence on initial data for the Cauchy problem of fast diffusion equation is discussed.

Asymptotic of the Solutions to the Initial Boundary Value Problem for the Diffusion Equations for Semiconductors

In this paper, we study the asymptotic behavior of the solutions to the initial boundary value problem for unipolar drift diffusion equations for semiconductors. Under the proper assumptions on doping profile and initial value, we prove that the smooth solutions to these evolutionary problems tend to the unique stationary solution exponentially as time tends to infinity.

A VARIATIONAL PROBLEM ARISING IN REGISTRATION OF DIFFUSION TENSOR IMAGES

The existence of a global minimizer for a variational problem arising in registration of diffusion tensor images is proved, which ensures that there is a regular spatial transformation for the registration of diffusion tensor images.

ANALYSIS OF BOUNDARY LAYER SINGULARITYIN A NONLINEAR DIFFUSION PROBLEM

In this paper, the author analyzes the singularity of a boundary layer in a nonlinear diffusion problem. Results show when the limiting solution satisfies the boundary condition, there is no boundary singularity. Otherwise, the boundary layer exists, and its thickness is proportional to epsilon(1/2), here epsilon is a small positive real parameter.

The Singularly Perturbed Problem for Nonlinear Nonlocal Reaction Diffusion Systems

The singularly perturbed initial boudary value problem for the nonlocal reaction diffusion systems was considered. Using iteration method the comparison theorem was obtained. Introducing stretched variable the formal asymptotic solution was constructed. And the existence and its asymptotic behavior of solution for the problem were studied by using the method of the upper and lower solution.

MOVING BOUNDARY PROBLEM FOR DIFFUSION RELEASE OF DRUG FROM A CYLINDER POLYMERIC MATRIX

An approximate analytical solution of moving boundary problem for diffusion release of drug from a cylinder polymeric matrix was obtained by use of refined integral method. The release kinetics has been analyzed for non-erodible matrices with perfect sink condition. The formulas of the moving boundary and the fractional drug release were given. The moving boundary and the fractional drug release have been calculated at various drug loading levels, mid the calculated results were in good agreement with those of experiments. The comparison of the moving boundary in spherical, cylinder, planar matrices has been completed. An approximate formula for estimating the available release time was presented. These results are useful for the clinic experiments. This investigation provides a new theoretical tool for studying the diffusion release of drug from a cylinder polymeric matrix and designing the controlled released drug.

Finite volume element method for analysis of unsteady reaction-diffusion problems

A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the high-gradient boundary layers.