Some sufficient conditions are obtained for the oscillation of first order neutral differential_difference equations with positive and negative periodic coefficients.
The scattering theory in periodic structures has many applications in modern microoptics and industry. Periodic structures are often referred to as diffraction gratings. In this paper we consider a planar dielectric-l...The scattering theory in periodic structures has many applications in modern microoptics and industry. Periodic structures are often referred to as diffraction gratings. In this paper we consider a planar dielectric-layer modulated grating problem. The diffraction problem may be modeled by a Helmholtz equation with periodic coefficients. Results on existence and uniqueness of the solution for the diffraction problem are obtained by variational method and integral equation method, respectively. At the end of the paper, we also discuss the Born approximation to the solution of an equivalent integral equation.展开更多
In this paper, we study the large time behavior of solutions of the parabolic semilinear equation δtu-div（a（x）△↓u） = -｜u｜^αu in （0,∞） × R^N, where α 〉 0 is constant and a∈ Cb^1（R^N） is a symmetr...In this paper, we study the large time behavior of solutions of the parabolic semilinear equation δtu-div（a（x）△↓u） = -｜u｜^αu in （0,∞） × R^N, where α 〉 0 is constant and a∈ Cb^1（R^N） is a symmetric periodic matrix satisfying some ellipticity assumptions.Considering an integrable initial data u0 and α ∈ （2/N, 3/N）, we prove that the large time behavior of solutions is given by the solution U（t, x） of the homogenized linear problem δtU-div（a^h△↓U）=0,U（0） = C, where a^h is the homogenized matrix of a（x）, C is a positive constant and δ is the Dirac measure at 0.展开更多
Presents a study which proposed a dual approximate expression of the exact solution for mixed boundary value problems of second order elliptic partial differential equation with small periodic coefficients. Dual appro...Presents a study which proposed a dual approximate expression of the exact solution for mixed boundary value problems of second order elliptic partial differential equation with small periodic coefficients. Dual approximate error estimates; Main results.展开更多
In this paper, the concept of generalized ω-periodic solution is given for Riccati's equationy'=a(t)y^2+b(t)y+c(t)with perriodic coefficients, the relation between generalized ω-periodicsolutions and the cha...In this paper, the concept of generalized ω-periodic solution is given for Riccati's equationy'=a(t)y^2+b(t)y+c(t)with perriodic coefficients, the relation between generalized ω-periodicsolutions and the characteristic numbers of system x'_1=c(t)x_2, x'_2=-a(t)x_1-b(t)x_2 is indicated, andseveral necessary and sufficient conditions are given using the coefficients. Moreover, in the case of a(t)without zero, the relation between the number of continuous ω-periodic solutions of y'=a(t)y^2+b(t)y+c(t)+δand the parameter δ is given; thus the problem on the existence of continuous ω-periodic.solutions is basically solved.展开更多
In this paper, we prove that, under appropriate hypotheses, the odd-order nonlinear neutral delay differential equation has the same oscillatory character as the associated linear equation with periodic coefficients O...In this paper, we prove that, under appropriate hypotheses, the odd-order nonlinear neutral delay differential equation has the same oscillatory character as the associated linear equation with periodic coefficients Our method is based on the establishment of a new comparison theorem for the oscillation of Eq.(E). Hence we prove that the parameter set such that Eq.(E) has a nonoscillatory solution is closed in certain metric space, and avoid the difficulty to set the necessary and sufficient condition for the oscillation of Eq.(E).展开更多
This paper is concerned with the second order elliptic problems with small periodic coefficients on a bounded domain with a curved boundary. A two-scale curved element method which couples linear elements and isoparam...This paper is concerned with the second order elliptic problems with small periodic coefficients on a bounded domain with a curved boundary. A two-scale curved element method which couples linear elements and isoparametric elements is proposed. The error estimate is obtained over the given smooth domain. Furthermore an additive Schwarz method is provided for the isoparametric element method.展开更多
This article investigates the property of linearly dependence of solutions f（z） and f（z ＋ 2πi） for higher order linear differential equations with entire periodic coefficients.
A thorough investigation of the systemd^2y(x):dx^2+p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤x<x_0(2π>0> -η, x_0≤x<2π(η>p(x)=p(x+2π),-∞<x<∞is given, and the method can be appl...A thorough investigation of the systemd^2y(x):dx^2+p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤x<x_0(2π>0> -η, x_0≤x<2π(η>p(x)=p(x+2π),-∞<x<∞is given, and the method can be applied to one with other periodic impulse coefficients.展开更多
In this paper the homogenization method is improved to develop one kind of dual coupled approximate method, which reflects both the macro-scope properties of whole structure and its loadings, and micro-scope configura...In this paper the homogenization method is improved to develop one kind of dual coupled approximate method, which reflects both the macro-scope properties of whole structure and its loadings, and micro-scope configuration properties of composite materials. The boundary value problem of woven membrane is considered, the dual asymptotic expression of the exact solution is given, and its approximation and error estimation are discussed. Finally the numerical example shows the effectiveness of this dual coupled method.展开更多
We study the multiscale finite element method for solving multiscale elliptic problems with highly oscillating coefficients,which is designed to accurately capture the large scale behaviors of the solution without res...We study the multiscale finite element method for solving multiscale elliptic problems with highly oscillating coefficients,which is designed to accurately capture the large scale behaviors of the solution without resolving the small scale characters.The key idea is to construct the multiscale base functions in the local partial differential equation with proper boundary conditions.The boundary conditions are chosen to extract more accurate boundary information in the local problem.We consider periodic and non-periodic coefficients with linear and oscillatory boundary conditions for the base functions.Numerical examples will be provided to demonstrate the effectiveness of the proposed multiscale finite element method.展开更多
New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F...New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F（z）, in negative feedback with a matrix of periodic/aperiodic gains A（k）, k = 0,1, 2,... and a vector of certain classes of non-monotone/monotone nonlinearities φp（-）, without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features： i） They involve the positive definiteness of the real part （as evaluated on ｜z｜ = 1） of the product of Г （z） and a matrix multiplier function of z. ii） For periodic A（k）, one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A（k）, but for aperiodic A（k）, which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of （A（k ＋ 1）,A（k））, k = 1, 2 iii） They are distinct from and less restrictive than recent results in the literature.展开更多
文摘Some sufficient conditions are obtained for the oscillation of first order neutral differential_difference equations with positive and negative periodic coefficients.
文摘The scattering theory in periodic structures has many applications in modern microoptics and industry. Periodic structures are often referred to as diffraction gratings. In this paper we consider a planar dielectric-layer modulated grating problem. The diffraction problem may be modeled by a Helmholtz equation with periodic coefficients. Results on existence and uniqueness of the solution for the diffraction problem are obtained by variational method and integral equation method, respectively. At the end of the paper, we also discuss the Born approximation to the solution of an equivalent integral equation.
基金Supported by CNPq-Conselho Nacional de Desenvolvimento Cient'fico e Tecnológico
文摘In this paper, we study the large time behavior of solutions of the parabolic semilinear equation δtu-div（a（x）△↓u） = -｜u｜^αu in （0,∞） × R^N, where α 〉 0 is constant and a∈ Cb^1（R^N） is a symmetric periodic matrix satisfying some ellipticity assumptions.Considering an integrable initial data u0 and α ∈ （2/N, 3/N）, we prove that the large time behavior of solutions is given by the solution U（t, x） of the homogenized linear problem δtU-div（a^h△↓U）=0,U（0） = C, where a^h is the homogenized matrix of a（x）, C is a positive constant and δ is the Dirac measure at 0.
基金the National Natural Science Foundation of China under grants 19901014 and 19932030 and the GAS K.C.Wong Poet-doctoral Resear
文摘Presents a study which proposed a dual approximate expression of the exact solution for mixed boundary value problems of second order elliptic partial differential equation with small periodic coefficients. Dual approximate error estimates; Main results.
文摘In this paper, the concept of generalized ω-periodic solution is given for Riccati's equationy'=a(t)y^2+b(t)y+c(t)with perriodic coefficients, the relation between generalized ω-periodicsolutions and the characteristic numbers of system x'_1=c(t)x_2, x'_2=-a(t)x_1-b(t)x_2 is indicated, andseveral necessary and sufficient conditions are given using the coefficients. Moreover, in the case of a(t)without zero, the relation between the number of continuous ω-periodic solutions of y'=a(t)y^2+b(t)y+c(t)+δand the parameter δ is given; thus the problem on the existence of continuous ω-periodic.solutions is basically solved.
文摘In this paper, we prove that, under appropriate hypotheses, the odd-order nonlinear neutral delay differential equation has the same oscillatory character as the associated linear equation with periodic coefficients Our method is based on the establishment of a new comparison theorem for the oscillation of Eq.(E). Hence we prove that the parameter set such that Eq.(E) has a nonoscillatory solution is closed in certain metric space, and avoid the difficulty to set the necessary and sufficient condition for the oscillation of Eq.(E).
基金The research was supported by the National Natural Science Foundation of China under grants 19901014 and 19932030 the Special Funds for Major State Basic Research Projects.
文摘This paper is concerned with the second order elliptic problems with small periodic coefficients on a bounded domain with a curved boundary. A two-scale curved element method which couples linear elements and isoparametric elements is proposed. The error estimate is obtained over the given smooth domain. Furthermore an additive Schwarz method is provided for the isoparametric element method.
基金Supported by the Brain Pool Program of Korea Federation of Science and Technology Societies(072-1-3-0164)the Natural Science Foundation of Guangdong Province in China(06025059)
文摘This article investigates the property of linearly dependence of solutions f（z） and f（z ＋ 2πi） for higher order linear differential equations with entire periodic coefficients.
基金This work is supported by the National Science Fund of Peop1e's Republic of China
文摘A thorough investigation of the systemd^2y(x):dx^2+p(x)y(x)=0with periodic impulse coefficientsp(x)={1,0≤x<x_0(2π>0> -η, x_0≤x<2π(η>p(x)=p(x+2π),-∞<x<∞is given, and the method can be applied to one with other periodic impulse coefficients.
文摘In this paper the homogenization method is improved to develop one kind of dual coupled approximate method, which reflects both the macro-scope properties of whole structure and its loadings, and micro-scope configuration properties of composite materials. The boundary value problem of woven membrane is considered, the dual asymptotic expression of the exact solution is given, and its approximation and error estimation are discussed. Finally the numerical example shows the effectiveness of this dual coupled method.
文摘We study the multiscale finite element method for solving multiscale elliptic problems with highly oscillating coefficients,which is designed to accurately capture the large scale behaviors of the solution without resolving the small scale characters.The key idea is to construct the multiscale base functions in the local partial differential equation with proper boundary conditions.The boundary conditions are chosen to extract more accurate boundary information in the local problem.We consider periodic and non-periodic coefficients with linear and oscillatory boundary conditions for the base functions.Numerical examples will be provided to demonstrate the effectiveness of the proposed multiscale finite element method.
文摘New conditions are derived for the l2-stability of time-varying linear and nonlinear discrete-time multiple-input multipleoutput （MIMO） systems, having a linear time time-invariant block with the transfer function F（z）, in negative feedback with a matrix of periodic/aperiodic gains A（k）, k = 0,1, 2,... and a vector of certain classes of non-monotone/monotone nonlinearities φp（-）, without restrictions on their slopes and also not requiring path-independence of their line integrals. The stability conditions, which are derived in the frequency domain, have the following features： i） They involve the positive definiteness of the real part （as evaluated on ｜z｜ = 1） of the product of Г （z） and a matrix multiplier function of z. ii） For periodic A（k）, one class of multiplier functions can be chosen so as to impose no constraint on the rate of variations A（k）, but for aperiodic A（k）, which allows a more general multiplier function, constraints are imposed on certain global averages of the generalized eigenvalues of （A（k ＋ 1）,A（k））, k = 1, 2 iii） They are distinct from and less restrictive than recent results in the literature.
