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
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
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
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
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
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
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
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
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
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
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
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 parameteris examined, where are constants, a...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 parameteris examined, where are constants, and i=0,1 . Moreover, asymptotic estimates of the solutions for the above problems are given.展开更多
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
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
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
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
Based on the precise integration method (PIM), a coupling technique of the high order multiplication perturbation method (HOMPM) and the reduction method is proposed to solve variable coefficient singularly pertur...Based on the precise integration method (PIM), a coupling technique of the high order multiplication perturbation method (HOMPM) and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems (TPBVPs) with one boundary layer. First, the inhomogeneous ordinary differential equations (ODEs) are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.展开更多
The singular perturbation of boundary value problem of second-order nonlinear system of differential equations with integral operators and boundary perturbation is discussed. Under the suitable assumed conditions, by ...The singular perturbation of boundary value problem of second-order nonlinear system of differential equations with integral operators and boundary perturbation is discussed. Under the suitable assumed conditions, by the technique of diagonalization, the existence of the solutions is proved and its remainder term is estimated.展开更多
This paper considers the singular perturbation of a fourth order elliptic equation when the limit equation is elliptic-parabolic. The equation involves a positive parameter, a positive real number, a Laplacian operato...This paper considers the singular perturbation of a fourth order elliptic equation when the limit equation is elliptic-parabolic. The equation involves a positive parameter, a positive real number, a Laplacian operator, and sufficient smoothness. Under appropriate condition the sufficient condition of solvability is derived, the existence of solution is proved and a uniformly valid asymptotic solution of arbitrary order is given.展开更多
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 followin...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.展开更多
基金supported by the Educational Department Foundation of Fujian Province of China(Nos. JA08140 and A0610025)the Scientific Research Foundation of Zhejiang University of Scienceand Technology (No. 2008050)the National Natural Science Foundation of China (No. 50679074)
文摘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.
基金Project supported by the National Natural Science Foundation of China (Nos. 40676016, 10471039), the National Key Basic Research Special Foundation of China (No. 2004CB418304), the Key Basic Research Foundation of the Chinese Academy of Sciences (No. KZCX3-SW-221) and in part by EInstitutes of Shanghai Municipal Education Commission (No. E03004)
文摘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.
基金financially supported by the National Basic Research Program of China(Grant No.2011CB013702)the National Natural Science Foundation of China(Grant No.50979113).1
文摘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.
基金Project Supported by the Science Fund of the Chinese Academy of Sciences
文摘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.
文摘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.
基金Project supported by the National Natural Science Foundation of China(Nos.51439007 and11372232)the Specialized Research Fund for the Doctoral Program of Higher Education(No.20130141110016)
文摘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.
基金Project supported by the National Natural Science Foundation of China(No.10672194)the China-Russia Cooperative Project(the National Natural Science Foundation of China and the Russian Foundation for Basic Research)(No.10811120012)
文摘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.
基金Supported by the National Natural Scince Foundation of China( 1 0 0 71 0 4 8) ,and the"Hundred TalentsProject"of Chinese Academy of Sciences
文摘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.
文摘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.
基金This work was supported by the application foundation for basic research of Jiangsu(No.BJ98057)the innovation foundation for the scientific research of Nanjing University of Aeronautics and Astronautics(No.Y0487-031)
文摘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.
基金This work is supported by the National Fujian Province Nature Science Research Funds
文摘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).
文摘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 parameteris examined, where are constants, and i=0,1 . Moreover, asymptotic estimates of the solutions for the above problems are given.
基金supported by the National Natural Science Foundation of China (Nos.40676016 and 40876010)the Knowledge Innovation Program of Chinese Academy of Sciences (No.KZCX2-YW-Q03-08)the LASG State Key Laboratory Special Fund,and the E-Institute of Shanghai Municipal Education Commission (No.E03004)
文摘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.
文摘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.
文摘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.
基金supported by the National Natural Science Foundation of China (No.60274009)Specialized Research Fund for the Doctoral Program of Higher Education (No.20020145007)
文摘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.
基金Project supported by the National Natural Science Foundation of China(Key Program)(Nos.11132004 and 51078145)
文摘Based on the precise integration method (PIM), a coupling technique of the high order multiplication perturbation method (HOMPM) and the reduction method is proposed to solve variable coefficient singularly perturbed two-point boundary value prob lems (TPBVPs) with one boundary layer. First, the inhomogeneous ordinary differential equations (ODEs) are transformed into the homogeneous ODEs by variable coefficient dimensional expansion. Then, the whole interval is divided evenly, and the transfer ma trix in each sub-interval is worked out through the HOMPM. Finally, a group of algebraic equations are given based on the relationship between the neighboring sub-intervals, which are solved by the reduction method. Numerical results show that the present method is highly efficient.
文摘The singular perturbation of boundary value problem of second-order nonlinear system of differential equations with integral operators and boundary perturbation is discussed. Under the suitable assumed conditions, by the technique of diagonalization, the existence of the solutions is proved and its remainder term is estimated.
文摘This paper considers the singular perturbation of a fourth order elliptic equation when the limit equation is elliptic-parabolic. The equation involves a positive parameter, a positive real number, a Laplacian operator, and sufficient smoothness. Under appropriate condition the sufficient condition of solvability is derived, the existence of solution is proved and a uniformly valid asymptotic solution of arbitrary order is given.
基金supported by the National Basic Research Program of China (Grant No.2006CB705500)the National Natural Science Foundation of China (Grant Nos.10532060, 10602025, 10802042)the Natural Science Foundation of Ningbo (Grant No.2007A610050)
文摘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.