A coupling difference scheme for the numerical solution of a singular perturbation problem

In this paper, we consider a singularly perturbed problem without turning points. On a special diseretization mesh, a coupling difference scheme, resulting from central difference scheme and Abrahamsson-Keller-Kreiss box scheme, is proposed and the second order convergence, uniform in the small parameter, is proved. Finally, numerical resulls are provided.     

THE NUMERICAL SOLUTION OF A SINGULARLY PERTURBED PROBLEM FOR SEMILINEAR PARABOLIC DIFFERENTIAL EQUATION

The numerical solution of a singularly perturbed problem for the semilinear parabolicdifferential equation with parabolic boundary layers is discussed.A nonlinear two-leveldifference scheme is constructed on the special non-uniform grids.The uniform convergenceof this scheme is proved and some numerical examples are given.     

The numerical solution of a singularly perturbed problem for quasilinear parabolic differential equation

We consider the numerical solution of a singularly perturbed problem for the quasilinear parabolic differential equation, and construct a linear three-level finite difference scheme on a nonuniform grid. The uniform convergence in the sense of discrete L~2 norm is proved and numerical examples are presented.     

A UNIFORMLY CONVERGENT DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF A HIGH ORDER ELLIPTIC DIFFERENTIAL EQUATION

ＡＵＮＩＦＯＲＭＬＹＣＯＮＶＥＲＧＥＮＴＤＩＦＦＥＲＥＮＣＥＳＣＨＥＭＥＦＯＲＴＨＥＳＩＮＧＵＬＡＲＰＥＲＴＵＲＢＡＴＩＯＮＰＲＯＢＬＥＭＯＦＡＨＩＧＨＯＲＤＥＲＥＬＬＩＰＴＩＣＤＩＦＦＥＲＥＮＴＩＡＬＥＱＵＡＴＩＯＮ（刘国庆）（苏煜城）ＡＵＮＩＦＯ...     

The boundary layer scheme for a singularly perturbed problem for the second order elliptic equation in the rectangle

Using singularly perturbation theory is constructed the boundary layer scheme for a Dirichlet problem for the second order singularly perturbed equation of elliptic type in the rectangle. The error estimate is given.     

Step-like contrast structure for class of nonlinear singularly perturbed optimal control problem

The step-like contrast structure for a class of nonlinear singularly perturbed optimal control problems is considered. The existence of the step-like contrast structure for the singularly perturbed optimal control problem is proved by equivalence, which is based on the necessary conditions. The authors not only give the conditions under which there exists a step-like contrast structure, but also determine where the internal transition time is. Meanwhile, the uniformly valid asymptotic expansion of the step-like contrast structure solution is constructed by the direct scheme method. Finally, an example is presented to show the result.     

A uniformly convergent second order difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form

In this paper, based on the idea of El-Mistikawy and Werle⁽¹⁾ we construct a difference scheme for a singularly perturbed self-adjoint ordinary differential equation in conservation form. We prove that it is a uniformly convergent second order scheme.     

Numerical solution of singular perturbation problems for the fourth-order elliptic differential equations

In this paper we construct the finite-difference scheme for the singularly perturbedboundary value problem for the fourth-order elliptic differential equation on the basis ofpaper[1],and prove the uniform convergence of this scheme with respect to the smallparameterεin the discrete energy norm,Finally,we give a numerical example.     

Numerical solutions for singularly perturbed semi-linear parabolic equation

In this paper,we discuss singularly perturbed semi-linear parabolic equations for one dimension and two dimension,we find numerical solutions by using both the line-method and the exact difference scheme on a special non-uniform discretization mesh.The uniform convergence in e of the first order accuracy is obtained.     

High order multiplication perturbation method for singular perturbation problems

This paper presents a high order multiplication perturbation method for sin- gularly perturbed two-point boundary value problems with the boundary layer at one end. By the theory of singular perturbations, the singularly perturbed two-point boundary value problems are first transformed into the singularly perturbed initial value problems. With the variable coefficient dimensional expanding, the non-homogeneous ordinary dif- ferential equations （ODEs） are transformed into the homogeneous ODEs, which are then solved by the high order multiplication perturbation method. Some linear and nonlinear numerical examples show that the proposed method has high precision.     

Uniform difference scheme for a singularly perturbed linear 2nd order hyperbolic problem with zeroth order reduced equation

In this paper a singularly perturbed linear second order hyperbolic problem with zeroth order reduced equation is discussed. Firstly, an energy inequality of the solution and an estimate of the remainder term of the asymptotic solution are given. Then an exponentially fitted difference scheme is developed in an equidistant mesh. Finally, uniform convergence in small parameter is proved in the sense of discrete energy norm.     

Uniformly higher order accurate extrapolations to solutions of uniformly convergent discretization methods for singularly perturbed problems

We discuss the uniformly higher order accurate extrapolations, which are based on the uniform expansion for global error, to solutions of uniformly convergent discretization methods for singularly perturbed problems. By applying the approach to the Il'in-Allen-Southwell scheme for a non-self-adjoint problem, we obtain an extrapolation solution which is uniformly convergent with order two. We confirm the result by numerical calculations.Supported by the National Natural Science Foundation of China.     

A class of singular perturbation solutions to semilinear equations of fourth order

A class of singularly perturbed boundary value problems for semilinear equations of fourth order with two parameters are considered. Under suitable conditions, using the method of lower and upper solutions, the existence and the asymptotic behavior of the solution to the boundary value problem are studied, In the present paper, the solution to the original singularly perturbed problem with two parameters has only one boundary layer.     

A singular perturbation analysis with applications to delay differential equations

We prove an approximation result for the solutions of a singularly perturbed, nonautonomous ordinary differential equation which has interesting applications to problems in higher dimensions. Here our result is applied to a singularly perturbed, delay differential equation with state dependent time-lags (i.e., aninfinite dimensional problem). We find a new dynamical system (also in infinite dimensions), which describes, in a certain sense, the dynamics of our delay equations for very small values of the singular parameter.     

Existence of Multi-Pulses of the Regularized Short-Pulse and Ostrovsky Equations

The existence of multi-pulse solutions near orbit-flip bifurcations of a primary single-humped pulse is shown in reversible, conservative, singularly perturbed vector fields. Similar to the non-singular case, the sign of a geometric condition that involves the first integral decides whether multi-pulses exist or not. The proof utilizes a combination of geometric singular perturbation theory and Lyapunov–Schmidt reduction through Lin’s method. The motivation for considering orbit flips in singularly perturbed systems comes from the regularized short-pulse equation and the Ostrovsky equation, which both fit into this framework and are shown here to support multi-pulses.     

Singularly Perturbed System of Differential Equations with Regular Singularity

We construct asymptotic solutions of a singularly perturbed system of differential equations with regular singularity.     

Construction of an asymptotic solution of one optimal control problem

We construct an asymptotic solution of a singularly perturbed optimal control problem in the case of a regular pencil of boundary matrices.     

On Asymptotic Decomposition of Solutions of Systems of Differential Equations with Slowly Varying Coefficients

We propose an algorithm for the reduction of a singularly perturbed system of differential equations whose characteristic equation has multiple roots to a system with simple roots.     

On the asymptotic integration of a singularly perturbed system of linear differential equations with deviating argument

We prove the existence and infinite differentiability of a solution of a singularly perturbed linear system of differential equations with deviating arguments and a turning point.