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High accuracy non-equidistant method for singular perturbation reaction-diffusion problem

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Applied Mathematics and Mechanics(English Edition) 2009年第2期30卷 175-182页

作者：蔡新蔡丹琳吴瑞潜谢康和 School of Sciences Zhejiang University of Science and Technology Institute of Geotechnical Engineering Zhejiang University School of Sciences Quanzhou Normal University

singular perturbation reaction-diffusion problem with Dirichlet boundary condition is considered. It is a multi-scale problem. Presence of small parameter leads to boundary layer phenomena in both sides of the region. A non-equidistant finite difference method is presented according to the property of boundary layer. The region is divided into an inner boundary layer region and an outer boundary layer region according to transition point of Shishkin. The steps sizes are equidistant in the outer boundary layer region. The step sizes are gradually increased in the inner boundary layer region such that half of the step sizes are different from each other. Truncation error is estimated. The proposed method is stable and uniformly convergent with the order higher than 2. Numerical results are given, which are in agreement with the theoretical result.

关键词： singular perturbation reaction-diffusion uniform convergence high accuracy non-equidistant

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

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Applied Mathematics and Mechanics(English Edition) 2009年第11期30卷 1455-1460页

作者：莫嘉琪 Department of Mathematics Anhui Normal University Division of Computational Science E-Institute of Shanghai Universities at SJTU

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.

关键词： nonlinear two parameters singular perturbation lower and upper solutions

Multiscale Basis Functions for singular perturbation on Adaptively Graded Meshes

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Advances in Applied Mathematics and Mechanics 2014年第5期6卷 604-614页

作者： Mei-Ling Sun Shan Jiang College of Mathematics Science Yangzhou UniversityYangzhou 225002China Education and Technology Center Nantong Vocational CollegeNantong 226007China Department of Mathematics Texas A&M UniversityCollege StationTX 77843USA

We apply the multiscale basis functions for the singularly perturbed reaction-diffusion problem on adaptively graded meshes,which can provide a good balance between the numerical accuracy and computational *** multiscale space is built through standard finite element basis functions enriched with multiscale basis *** multiscale basis functions have abilities to capture originally perturbed information in the local problem,as a result our method is capable of reducing the boundary layer errors remarkably on graded meshes,where the layer-adapted meshes are generated by a given *** numerical experiments we demonstrate that the multiscale method can acquire second order convergence in the L^(2)norm and first order convergence in the energy norm on graded meshes,which is independent ofε.In contrast with the conventional methods,our method is much more accurate and effective.

关键词： Multiscale basis functions singular perturbation boundary layer adaptively graded meshes.

ON singular perturbation FOR A NONLINEAR INITIAL-BOUNDARY VALUE PROBLEM (Ⅱ)

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Applied Mathematics and Mechanics(English Edition) 1992年第2期13卷 149-157页

作者：康连城 Jiangsu Institute of Chemical Technology Changzhou

In this paper, we consider a singularly perturbed problem of a kind of quasilinear hyperbolic-parabolic equations, subject to initial-boundary value conditions with moving boundary:When certain assumptions are satisfied and e is sufficiently small, the solution of this problem has a generalized asymptotic expansion (in the Van der Corput sense), which takes the sufficiently smooth solution of the reduced problem as the first term, and is uniformly valid in domain Q where the sufficiently smooth solution exists. The layer exists in the neighborhood of t = 0. This paper is the development of references [3 - 5].

关键词： singular perturbation moving boundary asymptotic expansion

Output regulation of nonaffine nonlinear systems using singular perturbation theory

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控制理论与应用（英文版） 2009年第2期7卷 181-184页

作者： Bo MENG Yuanwei JING, Xiaoping LIU College of Information Science and Engineering Northeastern University Shenyang Liaoning 110004 China

A control synthesis method for output regulation based on singular perturbation theory combined with inverting design is considered for a class of nonaffine nonlinear systems. The resulting control signal is defined as a solution to ＂fast＂ dynamics which inverts a series error model, whose state is exponentially stable. It is shown that, under sufficient conditions being consistent with the assumptions of singular perturbation theory, this problem is solvable with （ε） tracking error if and only if a set of first-order nonlinear partial differential equations are solvable. The control law can be easily constructed and the simulations show the feasibility and effectiveness of the controller.

关键词： Output regulation singular perturbation Nonaffine nonlinear system Dynamic inversion

singular perturbation APPROACH TO MOVING MASS CONTROL OF BUOYANCY-DRIVEN AIRSHIP IN 3-D SPACE

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Transactions of Nanjing University of Aeronautics and Astronautics 2011年第4期28卷 343-352页

作者：吴小涛 Claude H Moog 胡跃明华南理工大学自动化学院法国南特通信与控制研究所

The attitude control problem and the guidance problem are solved in 3-D for a buoyancy-driven airship actuated by the combined effects of an internal air bladder which modulates the airshiprs net weight and of two moving masses which modulate its center of mass. A simple and clear modeling is introduced to derive the 8 degree of freedom （DOF） mathematical model. Nonlinear control loops are derived through maximal feedback linearization with internal stability for both dynamics in the longitudinal plane and in the lateral plane. Based on a singular perturbation approach, the superposition of these two control actions in the longitudinal plane and in the lateral plane is shown to achieve the control of the dynamics in 3-D space. The simulations of the airship tracking specified attitude, moving direction and speed in 3-D space are presented.

关键词： buoyancy-driven airship nonlinear control 3-D singular perturbation

Finite-Difference Methods for a Class of Strongly Nonlinear singular perturbation Problems

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Numerical Mathematics(Theory,Methods and Applications) 2008年第2期1卷 235-244页

作者： Relja Vulanovi Department of Mathematical Sciences Kent State University Stark Campus

The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one *** problems are solved numerically by finite- difference schemes on special meshes which are dense in the boundary *** Bakhvalov mesh and a special piecewise equidistant mesh are *** the central scheme,error estimates are derived in a discrete L^1 *** are of second order and decrease together with the perturbation parameterε.The fourth-order Numerov scheme and the Shishkin mesh are also tested *** results showε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes.

关键词： Boundary-value problem singular perturbation finite differences Bakhvalov and piecewise equidistant meshes L^1 stability

singular perturbation OF THIRD ORDER NONLINEAR BOUNDARY VALUE PROBLEM WITH TWO TURNING POINTS

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Annals of Differential Equations 1997年第1期13卷 77-84页

作者：蔡建平漳州师范学院 363000

In this paper, by the theorem of differential inequalities, we prove the existence of the solution of a singularly perturbed boundary value problem of thirdorder nonlinear differential equation with two turning points.

关键词： turning point nonlinear boundary value prblem singular perturbation

singular　perturbation　OF　TWO─POINT　BOUNDARY　VALUE　PROBLEM　FOR　NONLINEAR　SYSTEM

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Annals of Differential Equations 1995年第1期11卷 25-36页

作者：陈育森福建师范大学福清分校

In this paper we study the singular perturbation of boundary value problem for second order nonlinear system by the method and the technique of diagonalization. Under the appropriate assumptions we prove the existenc... 详细信息

关键词： Nonlinear system the method and the technique of diagonalization singular perturbation