This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2...This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.展开更多
This note is concerned with an iterative method for the solution of singular boundary value problems. It can be considered as a predictor-corrector method. Sufficient conditions for the convergence of the method are i...This note is concerned with an iterative method for the solution of singular boundary value problems. It can be considered as a predictor-corrector method. Sufficient conditions for the convergence of the method are introduced. A number of numerical examples are used to study the applicability of the method.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 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
In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positiv...In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly increasing.Finally we give an example.展开更多
In this paper,we study a singular third-order three-point boundary value problem. By a fixed point theorem of cone expansion-compression type due to Krasnosel’skii,we obtain various new results on the existence of tw...In this paper,we study a singular third-order three-point boundary value problem. By a fixed point theorem of cone expansion-compression type due to Krasnosel’skii,we obtain various new results on the existence of two positive solutions to the problem,whose coefficient is allowed to have suitable singularities. Finally,we give an example to verify our results.展开更多
In this paper, we study a singular third-order three-point boundary value problem. By using a fixed-point theorem of cone expansion-compression type,we establish results on the existence of at least one, at least two,...In this paper, we study a singular third-order three-point boundary value problem. By using a fixed-point theorem of cone expansion-compression type,we establish results on the existence of at least one, at least two, and n positive solutions to the boundary value problem. Finally we give an example.展开更多
In this paper, three robust zero-watermark algorithms named Direct Current coefficient RElationship (DC-RE), CUmulant combined Singular Value Decomposition (CU-SVD), and CUmulant combined Singular Value Decomposition ...In this paper, three robust zero-watermark algorithms named Direct Current coefficient RElationship (DC-RE), CUmulant combined Singular Value Decomposition (CU-SVD), and CUmulant combined Singular Value Decomposition RElationship (CU-SVD-RE) are proposed. The algorithm DC-RE gets the feature vector from the relationship of DC coefficients between adjacent blocks, CU-SVD gets the feature vector from the singular value of third-order cumulants, while CU-SVD-RE combines the essence of the first two algorithms. Specially, CU-SVD-RE gets the feature vector from the relationship between singular values of third-order cumulants. Being a cross-over studying field of watermarking and cryptography, the zero-watermark algorithms are robust without modifying the carrier. Numerical simulation obviously shows that, under geometric attacks, the performance of CU-SVD-RE and DC-RE algorithm are better and all three proposed algorithms are robust to various attacks, such as median filter, salt and pepper noise, and Gaussian low-pass filter attacks.展开更多
The paper aims to obtain existence and uniqueness of the solution as well as asymptotic estimate of the solution for singularly perturbed nonlinear thirdorder Robin boundary value problem with a turning point. In orde...The paper aims to obtain existence and uniqueness of the solution as well as asymptotic estimate of the solution for singularly perturbed nonlinear thirdorder Robin boundary value problem with a turning point. In order to achieve this aim, existence and uniqueness of the solution for third-order nonlinear Robin boundary value problem is derived first based on the upper and lower solutions method under relatively weaker conditions. In this manner, the goal of this paper is gained by applying the existence and uniqueness results mentioned above.展开更多
In this paper, we first obtain the existence of solution to some n-point boundary value problem for third-order differential equations using upper and lower solutions method. Based on the results, we explore singular ...In this paper, we first obtain the existence of solution to some n-point boundary value problem for third-order differential equations using upper and lower solutions method. Based on the results, we explore singular perturbation of another n-point boundary value problem for third-order differential equations with a small positive parameter. Finally, a uniformly valid asymptotic solution is constructed and the error estimation is given.展开更多
文摘This paper deals with the singular nonlinear third-order periodic boundary value problem u'' + p(3)u = f (t, u), 0 less than or equal to t less than or equal to 2pi, with u((i)) (0) = u((i)) (2pi), i = 0, 1, 2, where p is an element of (Graphics) and f is singular at t = 0, t = 1 and u = 0. Under suitable weaker conditions than those of [1], it is proved by constructing a special cone in C[0, 2pi] and employing the fixed point index theory that the problem has at least one or at least two positive solutions.
文摘This note is concerned with an iterative method for the solution of singular boundary value problems. It can be considered as a predictor-corrector method. Sufficient conditions for the convergence of the method are introduced. A number of numerical examples are used to study the applicability of the method.
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.
文摘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.
基金Supported by the National Natural Science Foundation of China (Grant No. 10871160)
文摘In this paper,we study the existence of positive solutions for the nonlinear singular third-order three-point boundary value problemu （t） = λa（t）f（t,u（t））,u（0） = u （1） = u （η） = 0,where λ is a positive parameter and 0 ≤ η 1 2 .By using the classical Krasnosel’skii’s fixed point theorem in cone,we obtain various new results on the existence of positive solution,and the solution is strictly increasing.Finally we give an example.
基金the National Natural Science Foundation of China (10871160)
文摘In this paper,we study a singular third-order three-point boundary value problem. By a fixed point theorem of cone expansion-compression type due to Krasnosel’skii,we obtain various new results on the existence of two positive solutions to the problem,whose coefficient is allowed to have suitable singularities. Finally,we give an example to verify our results.
基金the National Natural Science Foundation of China(11261053)The Natural Science Foundation of Gansu Province(1308RJZA125)
文摘In this paper, we study a singular third-order three-point boundary value problem. By using a fixed-point theorem of cone expansion-compression type,we establish results on the existence of at least one, at least two, and n positive solutions to the boundary value problem. Finally we give an example.
基金Supported by the National Natural Science Foundation of China (No. 60672095, 60972165, and 61071111)the National High Technology Project of China (No. 2007AA-11Z210)+2 种基金the Doctoral Fund of Ministry of Education of China (No. 20100092120012 and 20070286004)the Foundation of High Technology Project in Jiangsu Provincethe Natural Science Foundation of Jiangsu Province (No.BK2010240)
文摘In this paper, three robust zero-watermark algorithms named Direct Current coefficient RElationship (DC-RE), CUmulant combined Singular Value Decomposition (CU-SVD), and CUmulant combined Singular Value Decomposition RElationship (CU-SVD-RE) are proposed. The algorithm DC-RE gets the feature vector from the relationship of DC coefficients between adjacent blocks, CU-SVD gets the feature vector from the singular value of third-order cumulants, while CU-SVD-RE combines the essence of the first two algorithms. Specially, CU-SVD-RE gets the feature vector from the relationship between singular values of third-order cumulants. Being a cross-over studying field of watermarking and cryptography, the zero-watermark algorithms are robust without modifying the carrier. Numerical simulation obviously shows that, under geometric attacks, the performance of CU-SVD-RE and DC-RE algorithm are better and all three proposed algorithms are robust to various attacks, such as median filter, salt and pepper noise, and Gaussian low-pass filter attacks.
基金Natural Science Foundation of Fujian Province under grant No.S0650010the Foundation of the Education Department of Fujian Province (JB06098).
文摘The paper aims to obtain existence and uniqueness of the solution as well as asymptotic estimate of the solution for singularly perturbed nonlinear thirdorder Robin boundary value problem with a turning point. In order to achieve this aim, existence and uniqueness of the solution for third-order nonlinear Robin boundary value problem is derived first based on the upper and lower solutions method under relatively weaker conditions. In this manner, the goal of this paper is gained by applying the existence and uniqueness results mentioned above.
基金the Natural Science Foundation of Fujian Province (S0650010)Fujian Provincial Department of Sci.& Tech.(2005K028)Department of Education of FuJian Province (JB06098)
文摘In this paper, we first obtain the existence of solution to some n-point boundary value problem for third-order differential equations using upper and lower solutions method. Based on the results, we explore singular perturbation of another n-point boundary value problem for third-order differential equations with a small positive parameter. Finally, a uniformly valid asymptotic solution is constructed and the error estimation is given.
