Application of singular perturbation method in analyzing traffic density waves

Car following model is one of microscopic models for describing traffic flow. Through linear stability analysis, the neutral stability lines and the critical points are obtained for the different types of car following models and two modified models. The singular perturbation method has been used to derive various nonlinear wave equations, such as the Kortewegde-Vries （KdV） equation and the modified Korteweg-de-Vries （mKdV） equation, which could describe different density waves occurring in traffic flows under certain conditions. These density waves are mainly employed to depict the formation of traffic jams in the congested traffic flow. The general soliton solutions are given for the different types of car following models, and the results have been used to the modified models efficiently.

ASYMPTOTIC METHOD FOR SINGULAR PERTURBATION PROBLEM OF ORDINARY DIFFERENCE EQUATIONS

This paper is taken up for the following difference equation problem (Pe);where e is a small parameter, c1, c2,constants and functions of k and e . Firstly, the case with constant coefficients is considered. Secondly, a general method based on extended transformation is given to handle (P.) where the coefficients may be variable and uniform asymptotic expansions are obtained. Finally, a numerical example is provided to illustrate the proposed method.

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.

INTERPOLATION PERTURBATION METHOD FORSOLVING NONLINEAR PROBLEMS

In this paper, using the interpolation perturbation method. the author seeks tosolve several nonlinear problems. Numerical examples show that the method Df thispaper has good accuracy.

INTERPOLATION PERTURBATION METHOD FOR SOLVING THE BOUNDARY LAYER TYPE PROBLEMS

In this paper,on the basis of Ref.[1],the author studies the boundary value problems of the second-order differential equations,the highest order derivatives of which contain the small parameters.The numerical examples show that the calculating process of this method is quite simple and its accuracy is even higher than that of the multiple scales method.

DIAGONALIZATION METHOD FOR A SINGULARLY PERTURBED VECTOR ROBIN PROBLEM

In this paper the method and technique of the diagonalization are employed to transform a vector second-order nonlinear system into two first-order approximate diagonalized systems. The existence and the asymptotic behavior of the solutions are obtained for a vector second-order nonlinear Robin problem of singular perturbation type.

SECOND-ORDER ACCURATE DIFFERENCE METHOD FOR THE SINGULARLY PERTURBED PROBLEM OF FOURTH-ORDER ORDINARY DIFFERENTIAL EQUATIONS

In this paper, we construct a uniform second-order difference scheme for a class of boundary value problems of fourth-order ordinary differential equations. Finally, a numerical example is given.

Perturbation solutions for asymmetric laminar flow in porous channel with expanding and contracting walls

The cases of large Reynolds number and small expansion ratio for the asym- metric laminar flow through a two-dimensional porous expanding channel are considered. The Navier-Stokes equations are reduced to a nonlinear fourth-order ordinary differential equation by introducing a time and space similar transformation. A singular perturbation method is used for the large suction Reynolds case to obtain an asymptotic solution by matching outer and inner solutions. For the case of small expansion ratios, we are able to obtain asymptotic solutions by double parameter expansion in either a small Reynolds number or a small asymmetric parameter. The asymptotic solutions indicate that the Reynolds number and expansion ratio play an important role in the flow behavior. Nu- merical methods are also designed to confirm the correctness of the present asymptotic solutions.

Existence of multiple solutions for the laminar flow in a porous channel with suction at both slowly expanding and contracting walls

The asymptotic behavior of solutions of a similarity equation for the laminar flow in a porous channel with suction at both expanding and contracting walls has been obtained by using a singular perturbation method.However,in the matching process,this solution neglects exponentially small terms.To take into account these exponentially small terms,a method involving the inclusion of exponentially small terms in a perturbation series was used to find two of the solutions analytically.The series involving the exponentially small terms and expansion ratio predicts dual solutions.Furthermore,the result indicates that the expansion ratio has much important influence on the solutions.

AN APPROXIMATE ANALYTICAL SOLUTION OF THE CYLINDRICAL DETONATION WAVE GENERATED BY THE LINEAR EXPLOSION

The solution of the cylindrical detonation wave generated by the linear explosion was obtained by numerical method in ref. [1.].In this paper, when the ratio of specific heat by using the enlargement coordinate method, the first-order analytical solutions are obtained. The perturbation parameter is The correction of these solutions is checked at the end of this paper.

LOCAL ANALYSIS OF THE FULLY DISCRETE LOCAL DISCONTINUOUS GALERKIN METHOD FOR THE TIME-DEPENDENT SINGULARLY PERTURBED PROBLEM

In this paper we consider the fully discrete local discontinuous Galerkin method, where the third order explicit Runge-Kutta time marching is coupled. For the one-dimensional time-dependent singularly perturbed problem with a boundary layer, we shall prove that the resulted scheme is not only of good behavior at the local stability, but also has the double-optimal local error estimate. It is to say, the convergence rate is optimal in both space and time, and the width of the cut-off subdomain is also nearly optimal, if the boundary condition at each intermediate stage is given in a proper way. Numerical experiments are also given.

ε-Dependent Controllability for Two Time-Scale Systems

This paper investigates the controllability of two time-scale systems using both the time-scale separation model and the slow-fast order reduction model. This work considers the effect of a singular perturbation parameter on the model transformations to improve the criterion precision. The Maclaurin expansion method and homotopy arithmetic are introduced to obtain t-dependent controllability criteria. Examples indicate that the s-dependent controllability criteria are more accurate and that the controllability of two time-scale systems does not change during model transformations with these more accurate forms.