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Alternative Methods of Regular and Singular Perturbation Problems
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作者 Boampong Asare Manohar Sah Ram Krishna Hona 《Applied Mathematics》 2024年第10期687-708,共22页
Making exact approximations to solve equations distinguishes applied mathematicians from pure mathematicians, physicists, and engineers. Perturbation problems, both regular and singular, are pervasive in diverse field... Making exact approximations to solve equations distinguishes applied mathematicians from pure mathematicians, physicists, and engineers. Perturbation problems, both regular and singular, are pervasive in diverse fields of applied mathematics and engineering. This research paper provides a comprehensive overview of algebraic methods for solving perturbation problems, featuring a comparative analysis of their strengths and limitations. Serving as a valuable resource for researchers and practitioners, it offers insights and guidance for tackling perturbation problems in various disciplines, facilitating the advancement of applied mathematics and engineering. 展开更多
关键词 perturbation Regular perturbation singular perturbation Asymptotic Expansion Matched Asymptotic Strained Coordinates Multiple Scales
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SINGULAR PERTURBATION APPROACH TO MOVING MASS CONTROL OF BUOYANCY-DRIVEN AIRSHIP IN 3-D SPACE 被引量:1
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作者 吴小涛 Claude H Moog 胡跃明 《Transactions of Nanjing University of Aeronautics and Astronautics》 EI 2011年第4期343-352,共10页
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 mov... 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. 展开更多
关键词 buoyancy-driven AIRSHIP nonlinear control 3-D singular perturbation
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High accuracy non-equidistant method for singular perturbation reaction-diffusion problem 被引量:5
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作者 蔡新 蔡丹琳 +1 位作者 吴瑞潜 谢康和 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第2期175-182,共8页
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.... 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. 展开更多
关键词 singular perturbation REACTION-DIFFUSION uniform convergence high accuracy non-equidistant
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Singular perturbation for the weakly nonlinear reaction diffusion equation with boundary perturbation 被引量:6
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作者 MO Jia-qi 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2008年第8期1105-1110,共6页
In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the re... In this paper, a class of nonlinear singularly perturbed initial boundary value problems for reaction diffusion equations with boundary perturbation are considered under suitable conditions. Firstly, by dint of the regular perturbation method, the outer solution of the original problem is obtained. Secondly, by using the stretched variable and the expansion theory of power series the initial layer of the solution is constructed. And then, by using the theory of differential inequalities, the asymptotic behavior of the solution for the initial boundary value problems is studied. Finally, using some relational inequalities the existence and uniqueness of solution for the original problem and the uniformly valid asymptotic estimation are discussed. 展开更多
关键词 NONLINEAR reaction diffusion singular perturbation boundary perturbation
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A Singular Perturbation Method for Parametric Investigation on J-lay Installation of Deepwater Pipelines 被引量:3
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作者 王琴 段梦兰 +1 位作者 李海明 张庆元 《China Ocean Engineering》 SCIE EI CSCD 2013年第6期751-766,共16页
The maximum bending moment or curvature in the neighborhood of the touch down point (TDP) and the maximum tension at the top are two key parameters to be controlled during deepwater J-lay installation in order to en... The maximum bending moment or curvature in the neighborhood of the touch down point (TDP) and the maximum tension at the top are two key parameters to be controlled during deepwater J-lay installation in order to ensure the safety of the pipe-laying operation and the normal operation of the pipelines. In this paper, the non-linear governing differential equation for getting the two parameters during J-lay installation is proposed and solved by use of singular perturbation technique, from which the asymptotic expression of stiffened catenary is obtained and the theoretical expression of its static geometric configuration as well as axial tension and bending moment is derived. Finite element results are applied to verify this method. Parametric investigation is conducted to analyze the influences of the seabed slope, unit weight, flexural stiffness, water depth, and the pipe-laying tower angle on the maximum tension and moment of pipeline by this method, and the results show how to control the installation process by changing individual parameters. 展开更多
关键词 deepwater touch down point singular perturbation technique non-linear governing differential equation J-lay of pipeline geometric nonlinearity stiffened catenary
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ON THE SINGULAR PERTURBATION OF A NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH TWO PARAMETERS 被引量:3
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作者 张汉林 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1989年第5期471-480,共10页
In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and t... In this paper,the method of differential inequalities has been applied to study theboundary value problems of nonlinear ordinary differential equation with two parameters.The asymptotic solutions have been found and the remainders have been estimated. 展开更多
关键词 REAL ON THE singular perturbation OF A NONLINEAR ORDINARY DIFFERENTIAL EQUATION WITH TWO PARAMETERS exp
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SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEM FOR QUASILINEAR THIRD-ORDER ORDINARY DIFFERENTIAL EQUATIONS INVOLVING TWO SMALL PARAMETERS 被引量:2
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作者 林苏榕 田根宝 林宗池 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2001年第2期229-236,共8页
The singularly perturbed boundary value problem for quasilinear third-order ordinary differential equation involving two small parameters has been considered. For the three cases epsilon/mu (2) --> 0(mu --> 0), ... The singularly perturbed boundary value problem for quasilinear third-order ordinary differential equation involving two small parameters has been considered. For the three cases epsilon/mu (2) --> 0(mu --> 0), mu (2)/epsilon --> 0(epsilon --> 0) and epsilon = mu (2), the formal asymptotic solutions are constructed by the method of two steps expansions and the existences of solution are proved by using the differential inequality method. In addition, the uniformly valid estimations of the remainder term are given as well. 展开更多
关键词 two-parameters singular perturbation boundary value problem asymptotic expansion
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Prediction of velocity distribution in straight open-channel flow with partial vegetation by singular perturbation method 被引量:2
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作者 Wenxin HUAI Suwen SONG +1 位作者 Jie HAN Yuhong ZENG 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI CSCD 2016年第10期1315-1324,共10页
A numerical analysis model based on two-dimensional shallow water differential equations is presented for straight open-channel flow with partial vegetation across the channel. Both the drag force acting on vegetation... A numerical analysis model based on two-dimensional shallow water differential equations is presented for straight open-channel flow with partial vegetation across the channel. Both the drag force acting on vegetation and the momentum exchange between the vegetation and non-vegetation zones are considered. The depth-averaged streamwise velocity is solved by the singular perturbation method, while the Reynolds stress is calculated based on the results of the streamwise velocity. Comparisons with the experimental data indicate that the accuracy of prediction is significantly improved by introducing a term for the secondary current in the model. A sensitivity analysis shows that a sound choice of the secondary current intensity coefficient is important for an accurate prediction of the depth-averaged streamwise velocity near the vegetation and non-vegetation interfaces, and the drag force coefficient is crucial for predictions in the vegetation zone. 展开更多
关键词 depth-averaged velocity singular perturbation secondary flow
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Singular Perturbations for a Class of Fourth-order Nonlinear Boundary Value Problem 被引量:2
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作者 LI Xu WANG Shi-peng 《Chinese Quarterly Journal of Mathematics》 CSCD 2011年第2期217-222,共6页
This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y(4) =f(t,y',y'',y'''),0t1,0ε1 with the nonlinear boundary conditions y(0)=y'(1)=0,p... This paper is devoted to study the following the singularly perturbed fourth-order ordinary differential equation ∈y(4) =f(t,y',y'',y'''),0t1,0ε1 with the nonlinear boundary conditions y(0)=y'(1)=0,p(y''(0),y'''(0))=0,q(y''(1),y'''(1))=0 where f:[0,1]×R3→R is continuous,p,q:R2→R are continuous.Under certain conditions,by introducing an appropriate stretching transformation and constructing boundary layer corrective terms,an asymptotic expansion for the solution of the problem is obtained.And then the uniformly validity of solution is proved by using the differential inequalities. 展开更多
关键词 singular perturbation nonlinear boundary value problem differential inequality asymptotic expansion
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Precise integration method for solving singular perturbation problems 被引量:1
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作者 富明慧 张文志 S.V.SHESHENIN 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2010年第11期1463-1472,共10页
This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matr... This paper presents a precise method for solving singularly perturbed boundary-value problems with the boundary layer at one end. The method divides the interval evenly and gives a set of algebraic equations in a matrix form by the precise integration relationship of each segment. Substituting the boundary conditions into the algebraic equations, the coefficient matrix can be transformed to the block tridiagonal matrix. Considering the nature of the problem, an efficient reduction method is given for solving singular perturbation problems. Since the precise integration relationship introduces no discrete error in the discrete process, the present method has high precision. Numerical examples show the validity of the present method. 展开更多
关键词 singular perturbation problem first-order ordinary differential equation two-point boundary-value problem precise integration method reduction method
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SINGULAR PERTURBATION FOR REACTION DIFFUSION EQUATIONS 被引量:1
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作者 MoJiaqi WangHui ZhuJiang 《Applied Mathematics(A Journal of Chinese Universities)》 SCIE CSCD 2003年第3期251-257,共7页
The singularly perturbed initial boundary value problems for reaction diffusion equations are considered.Under suitable conditions and by using the theory of differential inequality,the asymptotic behavior of solution... The singularly perturbed initial boundary value problems for reaction diffusion equations are considered.Under suitable conditions and by using the theory of differential inequality,the asymptotic behavior of solution for initial boundary value problems are studied,where the reduced problems possess two intersecting solutions. 展开更多
关键词 NONLINEAR reaction diffusion equation singular perturbation
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ASYMPTOTIC SOLUTION OF SINGULAR PERTURBATION PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS 被引量:1
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作者 苏煜城 刘国庆 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1990年第7期637-650,共14页
In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal... In this paper we consider the singularly perturbed boundary value problem for the fourth-order elliptic differential equation, establish the energy estimates of the solutionand its derivatives and construct the formal asymptotic solution by Lyuternik- Vishik 's method. Finally, by means of the energy estimates we obtain the bound of the remainder of the asymptotic solution. 展开更多
关键词 ASYMPTOTIC SOLUTION OF singular perturbation PROBLEMS FOR THE FOURTH-ORDER ELLIPTIC DIFFERENTIAL EQUATIONS
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Singular perturbation composite control of a free-floating flexible dual-arm space robot
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作者 罗战武 王从庆 《Journal of Pharmaceutical Analysis》 SCIE CAS 2008年第1期43-47,70,共6页
The Free-floating Flexible Dual-arm Space Robot is a highly nonlinear and coupled dynamics system. In this paper, the dynamic model is derived of a Free-floating Flexible Dual-arm Space Robot holding a rigid payload. ... The Free-floating Flexible Dual-arm Space Robot is a highly nonlinear and coupled dynamics system. In this paper, the dynamic model is derived of a Free-floating Flexible Dual-arm Space Robot holding a rigid payload. Furthermore, according to the singular perturbation method, the system is separated into a slow subsystem representing rigid body motion of the robot and a fast subsystem representing the flexible link dynamics. For the slow subsystem, based on the second method of Lyapunov, using simple quantitative bounds on the model uncertainties, a robust tracking controller design is used during the trajectory tracking phase. The optimal control method is designed in the fast subsystem to guarantee the exponential stability. With the combination of the two above, the system can track the expected trajectory accurately, even though with uncertainty in model parameters, and its flexible vibration gets suppressed, too. Finally, some simulation tests have been conducted to verify the effectiveness of the proposed methods. 展开更多
关键词 free-floating Flexible Dual-arm Space Robot singular perturbation robust tracking control vibration suppression
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A SINGULAR PERTURBATION PROBLEM FOR PERIODIC BOUNDARY PARTIAL DIFFERENTIAL EQUATION
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作者 林鹏程 江本铦 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1991年第3期281-290,共10页
In this paper, we consider a singular perturbation elliptic-parabolic partial differential equation for periodic boundary value problem, and construct a difference scheme. Using the method of decomposing the singular ... In this paper, we consider a singular perturbation elliptic-parabolic partial differential equation for periodic boundary value problem, and construct a difference scheme. Using the method of decomposing the singular term from its solution and combining an asymptotic expansion of the equation, we prove that the scheme constructed by this paper converges uniformly to the solution of its original problem with O(r+h2). 展开更多
关键词 elliptic-parabolic partial differential equation singular perturbation problem periodic boundary difference scheme uniform convergence
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SINGULAR PERTURBATION OF BOUNDARY VALUE PROBLEMS FOR SECOND ORDER NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS ON INFINITE INTERVAL(Ⅱ)
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作者 Zhao Wei-li 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1991年第11期1105-1116,共12页
In this paper the existence of solutions of the singularly perturbed boundary value problems on infinite interval for the second order nonlinear equation containing a small parameterε>0,εy'=f(x,y,y'),y... In this paper the existence of solutions of the singularly perturbed boundary value problems on infinite interval for the second order nonlinear equation containing a small parameterε>0,εy'=f(x,y,y'),y'(0)=a,y(∞)=βis examined,where are constants,and i=0,1.Moreover,asymptotic estimates of the solutions for the above problems are given. 展开更多
关键词 singular perturbations NONLINEAR boundary value problems on infinite interval existence of solutions asymptotic estimates.
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A class of singular perturbation solutions to semilinear equations of fourth order
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作者 莫嘉琪 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 2009年第11期1455-1460,共6页
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 existe... 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. 展开更多
关键词 NONLINEAR two parameters singular perturbation lower and upper solutions
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SINGULAR PERTURBATIONS OF A HIGHER-ORDER SCALAR NONLINEAR BOUNDARY VALUE PROBLEM
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作者 史玉明 高灿柱 《Acta Mathematica Scientia》 SCIE CSCD 1997年第2期167-179,共13页
In the present paper, the singular perturbations for the higher-order scalar nonlinear boundary value problem epsilon(2)y(n)=f(t,epsilon y,y',...,y((n-2)),epsilon y((n-1)), t is an element of[0,1] H1(y(0,epsilon),... In the present paper, the singular perturbations for the higher-order scalar nonlinear boundary value problem epsilon(2)y(n)=f(t,epsilon y,y',...,y((n-2)),epsilon y((n-1)), t is an element of[0,1] H1(y(0,epsilon),...,y((n-3))(0,epsilon),epsilon y((n-2))(0,epsilon),epsilon y((n-1))(0,epsilon),epsilon)=0, H2(y(0,epsilon),y((n-1))(0,epsilon),y(1,epsilon)...,y((n-1))(1,epsilon),epsilon=0 are studied, where epsilon > 0 is a small parameter, n greater than or equal to 2. Under some mild assumptions, we prove the existence and local uniqueness of the perturbed solution and give out the uniformly valid asymptotic expansions up to its nth-order derivative function by employing the Banach/Picard fixed-point theorem. Then the existing results are extended and improved. 展开更多
关键词 singular perturbation uniformly valid asymptotic expansion Green function Banach/Picard fixed-point theorem
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SINGULAR PERTURBATION OF GENERAL BOUNDARY VALUE PROBLEM FOR NONLINEAR DIFFERENTIAL EQUATION SYSTEM
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作者 周雅丽 游哲丰 林宗池 《Applied Mathematics and Mechanics(English Edition)》 SCIE EI 1998年第6期531-538,共8页
In this paper, the singular perturbation of nonlinear differential equation system with nonlinear boundary conditions is discussed. Under suitable assumptions, with the asymptotic method of Lyusternik-Vishik([1]) and ... In this paper, the singular perturbation of nonlinear differential equation system with nonlinear boundary conditions is discussed. Under suitable assumptions, with the asymptotic method of Lyusternik-Vishik([1]) and fixed point theory, the existence of the solution of the perturbation problem is proved and its uniformly valid asymptotic expansion of higher order is derived. 展开更多
关键词 nonlinear system nonlinear boundary condition singular perturbation asymptotic expansion
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Finite-Difference Methods for a Class of Strongly Nonlinear Singular Perturbation Problems
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作者 Relja Vulanovi 《Numerical Mathematics(Theory,Methods and Applications)》 SCIE 2008年第2期235-244,共10页
The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one dimension.The problems are solved numerically by finite- difference schemes on special meshes which are dense in the... The paper is concerned with strongly nonlinear singularly perturbed bound- ary value problems in one dimension.The problems are solved numerically by finite- difference schemes on special meshes which are dense in the boundary layers.The Bakhvalov mesh and a special piecewise equidistant mesh are analyzed.For the central scheme,error estimates are derived in a discrete L^1 norm.They are of second order and decrease together with the perturbation parameterε.The fourth-order Numerov scheme and the Shishkin mesh are also tested numerically.Numerical results showε-uniform pointwise convergence on the Bakhvalov and Shishkin meshes. 展开更多
关键词 Boundary-value problem singular perturbation finite differences Bakhvalov and piecewise equidistant meshes L^1 stability
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Output regulation of nonaffine nonlinear systems using singular perturbation theory
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作者 Bo MENG Yuanwei JING, Xiaoping LIU 《控制理论与应用(英文版)》 EI 2009年第2期181-184,共4页
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 a... 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. 展开更多
关键词 Output regulation singular perturbation Nonaffine nonlinear system Dynamic inversion
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