一个奇摄动微分方程非线性混合边值问题

研究了一个三阶半线性微分方程的奇摄动非线性混合边值问题.利用边界层函数法构造了该问题的形式渐近解,并采用微分不等式理论证明了解的存在性,给出了渐近解的误差估计,最后得出了边界层函数指数型衰减的结论.     

The interior layer phenomena for a class of singularly perturbed delay-differential equations

In this article, we study a kind of vector singularly perturbed delay-differential equation. Using boundary layer function method and geometric analysis skill, the asymptotic expression of the system is constructed and the uniform validity of asymptotic solution is also proved.     

A higher order difference method for singularly perturbed parabolic partial differential equations

This paper studies a higher order numerical method for the singularly perturbed parabolic convection-diffusion problems where the diffusion term is multiplied by a small perturbation parameter. In general, the solutions of these type of problems have a boundary layer. Here, we generate a spatial adaptive mesh based on the equidistribution of a positive monitor function. Implicit Euler method is used to discretize the time variable and an upwind scheme is considered in space direction. A higher order convergent solution with respect to space and time is obtained using the postprocessing based extrapolation approach. It is observed that the convergence is independent of perturbation parameter. This technique enhances the order of accuracy from first order uniform convergence to second order uniform convergence in space as well as in time. Comparative study with the existed meshes show the highly effective behavior of the present method.     

A computational method for singularly perturbed nonlinear differential-difference equations with small shift

This paper is devoted to the numerical study of the boundary value problems for nonlinear singularly perturbed differential-difference equations with small delay. Quasilinearization process is used to linearize the nonlinear differential equation. After applying the quasilinearization process to the nonlinear problem, a sequence of linearized problems is obtained. To obtain parameter-uniform convergence, a piecewise-uniform mesh is used, which is dense in the boundary layer region and coarse in the outer region. The parameter-uniform convergence analysis of the method has been discussed. The method has shown to have almost second-order parameter-uniform convergence. The effect of small shift on the boundary layer(s) has also been discussed. To demonstrate the performance of the proposed scheme two examples have been carried out. The maximum absolute errors and uniform rates of convergence have been presented in the form of the tables.     

Uniformly convergent difference schemes for a singularly perturbed third order boundary value problem

In this paper we consider a numerical approximation of a third order singularly perturbed boundary value problem by an upwind finite difference scheme on a Shishkin mesh. The behavior of the solution, and the stability of the continuous problem are discussed. The proof of the uniform convergence of the proposed numerical method is based on the strongly uniform stability and a weak consistency property of the discrete problem. Numerical experiments verify our theoretical results.     

Exponential bounds for discrete-time singularly perturbed Markov chains

This paper develops exponential type upper bounds for scaled occupation measures of singularly perturbed Markov chains in discrete time. By considering two-time scale in the Markov chains, asymptotic analysis is carried out. The cases of the fast changing transition probability matrix is irreducible and that are divisible into l ergodic classes are examined first; the upper bounds of a sequence of scaled occupation measures are derived. Then extensions to Markov chains involving transient states and/or nonhomogeneous transition probabilities are dealt with. The results enable us to further our understanding of the underlying Markov chains and related dynamic systems, which is essential for solving many control and optimization problems.     

The singularly perturbed problem for combustion reaction diffusion

1. IntroductionThe anchor discussed a class of problems for the nonlinear ordinary differential equations(of. [1-3]) and the reaction dimsion equations (of. [4-7]).Now we consider a model of the combustion reaction decision problem as follows:where y is a density of combustion flame, s denotes the rate of diffusion effect for reactionspeed, whiCh is a small positive parameter, t is a position of combustion, functions y - t andy t are the positions of the matter for fuel and chide respectivel…     

Numerical analysis of singularly perturbed delay differential turning point problem

In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper.     

High order parameter-robust numerical method for singularly perturbed reaction-diffusion problems

We present a high order parameter-robust finite difference method for singularly perturbed reaction-diffusion problems. The problem is discretized using a suitable combination of fourth order compact difference scheme and central difference scheme on generalized Shishkin mesh. The convergence analysis is given and the method is proved to be almost fourth order uniformly convergent in maximum norm with respect to singular perturbation parameter ε. Numerical experiments are conducted to demonstrate the theoretical results.     

Higher order global solution and normalized flux for singularly perturbed reaction-diffusion problems

In this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction-diffusion two-point boundary-value problems. The HOC (higher order compact) finite difference scheme developed in Gracia et al. (2001) [4] and which is constructed on an appropriate piecewise uniform Shishkin mesh, has been considered to find an almost fourth order convergent solution at mesh points. Using these values, piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined. It has been shown that the global solution and the global normalized flux are also uniformly convergent. Moreover, for the global solution, the order of uniform convergence in the whole domain is optimal, i.e., it is the same as this one obtained at mesh points, whereas, for the global normalized flux, the uniform convergence is almost third order, except at midpoints of the mesh, where it is also almost fourth order. Theoretical error bounds have been provided along with some numerical examples, which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux.     

A higher order uniformly convergent method for singularly perturbed parabolic turning point problems

In this article, we study numerical approximation for a class of singularly perturbed parabolic (SPP) convection-diffusion turning point problems. The considered SPP problem exhibits a parabolic boundary layer in the neighborhood of one of the sides of the domain. Some a priori bounds are given on the exact solution and its derivatives, which are necessary for the error analysis. A numerical scheme comprising of implicit finite difference method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh is proposed. Then Richardson extrapolation method is applied to increase the order of convergence in time direction. The resulting scheme has second-order convergence up to a logarithmic factor in space and second-order convergence in time. Numerical experiments are conducted to demonstrate the theoretical results and the comparative study is done with the existing schemes in literature to show better accuracy of the proposed schemes.     

A brief survey on numerical methods for solving singularly perturbed problems

In the present paper, a brief survey on computational techniques for the different classes of singularly perturbed problems is given. This survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000-2009. However some older important relevant papers are also included in this survey. We also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper, see [Appl. Math. Comput. 30 (1989) 223-259, 130 (2002) 457-510, 134 (2003) 371-429] for details. Thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems.     

Concentration on Hopf-fibres for singularly perturbed elliptic equations

Singularly perturbed elliptic equations with superlinear nonlinearities of polynomial type are considered on an annulus in Rⁿ${\mathbb{R}}^n$, n≥4$n\ge 4$. It is shown that for small parameters there exist solutions which concentrate on manifolds of dimensions one, three and seven, which are given as Hopf-fibres.     

A class of singularly perturbed semilinear differential equations with interior layers

In this paper singularly perturbed semilinear differential equations with a discontinuous source term are examined. A numerical method is constructed for these problems which involves an appropriate piecewise-uniform mesh. The method is shown to be uniformly convergent with respect to the singular perturbation parameter. Numerical results are presented that validate the theoretical results.

     

奇摄动高阶椭圆型方程解的多重边界层现象

研究了一类奇摄动2m阶椭圆型方程解的多重边层现象．利用比较定理得到解的一致有效的渐近展开式．     

Numerical simulation and convergence analysis for a system of nonlinear singularly perturbed differential equations arising in population dynamics

In this article, we consider a system of nonlinear singularly perturbed differential equations with two different parameters. To solve this system, we develop a weighted monotone hybrid scheme on a nonuniform mesh. The proposed scheme is a combination of the midpoint scheme and the upwind scheme involving the weight parameters. The weight parameters enable the method to switch automatically from the midpoint scheme to the upwind scheme as the nodal points start moving from the inner region to the outer region. The nonuniform mesh in particular the adaptive grid is constructed using the idea of equidistributing a positive monitor function involving the solution gradient. The method is shown to be second order convergent with respect to the small parameters. Numerical experiments are presented to show the robustness of the proposed scheme and indicate that the estimate is optimal.     

Numerical solution of singularly perturbed convection-diffusion problem using parameter uniform B-spline collocation method

This paper is concerned with a numerical scheme to solve a singularly perturbed convection-diffusion problem. The solution of this problem exhibits the boundary layer on the right-hand side of the domain due to the presence of singular perturbation parameter ε. The scheme involves B-spline collocation method and appropriate piecewise-uniform Shishkin mesh. Bounds are established for the derivative of the analytical solution. Moreover, the present method is boundary layer resolving as well as second-order uniformly convergent in the maximum norm. A comprehensive analysis has been given to prove the uniform convergence with respect to singular perturbation parameter. Several numerical examples are also given to demonstrate the efficiency of B-spline collocation method and to validate the theoretical aspects.