In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p...In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.展开更多
The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity proper...The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided.Then the existence and uniqueness of the DG solution are proven.Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established.Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants,the DG method achieves the uniform convergence in the L2 norm with respect to the singular perturbation parameter e when the space of polynomials with degree p is used.A numerical experiment validates the theoretical results.Furthermore,an ultra-convergence order 2p+1 at the nodes for the one-sided flux,uniform with respect to the singular perturbation parameter e,is observed numerically.展开更多
In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error esti...In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.展开更多
Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind sch...Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.展开更多
Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered...Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered at the origin 0, normalized by ?(z 0) = 0 and ?(z 0) = 1. Let us set $\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta $ , and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G,z 0) that minimizes the integral $\iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x }$ in the class $\mathop \prod \limits_n $ of all polynomials of degree ≤ n and satisfying the conditions P n (z 0) = 0 and P′ n (z 0) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to ? p (z) on $\bar G$ in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$ .展开更多
We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of ...We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.展开更多
In this paper, we introduced the notion of uniform convergence of the linear operators on the probabilistic nornied space, and the notion of probabilistic distance between the operators, which describes the above conv...In this paper, we introduced the notion of uniform convergence of the linear operators on the probabilistic nornied space, and the notion of probabilistic distance between the operators, which describes the above convergence completely. In terms of these notions, we obtained the essential features of the continuity of operators, and of the uniform convergence of operator sequences, and we also obtained the closure of continuity and complete continuity under the operation of the limit of uniform convergence.展开更多
Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f ： X →X of a sequence of continu...Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f ： X →X of a sequence of continuous topologically transitive （in strongly successive way） functions fn ： X →X, where X is a compact interval. Surprisingly, we find that the uniform limit function is chaotic in the sense of Devaney. Lastly, we give an example to show that the denseness property of Devaney＇s definition is lost on the limit function.展开更多
In this paper we study the dynamical behavior of a system ?approximated uniformly by a sequence ?of chaotic maps. We give examples to show that properties like sensitivity and denseness of periodic points need not be ...In this paper we study the dynamical behavior of a system ?approximated uniformly by a sequence ?of chaotic maps. We give examples to show that properties like sensitivity and denseness of periodic points need not be preserved under uniform convergence. We derive conditions under which some of the dynamical properties of the maps ?are preserved in .展开更多
In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under m...In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.展开更多
In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the m...In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.展开更多
文摘In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.
基金supported by the National Natural Science Foundation of China(12001189)supported by the National Natural Science Foundation of China(11171104,12171148)。
文摘The purpose of this work is to implement a discontinuous Galerkin(DG)method with a one-sided flux for a singularly perturbed Volterra integro-differential equation(VIDE)with a smooth kernel.First,the regularity property and a decomposition of the exact solution of the singularly perturbed VIDE with the initial condition are provided.Then the existence and uniqueness of the DG solution are proven.Then some appropriate projection-type interpolation operators and their corresponding approximation properties are established.Based on the decomposition of the exact solution and the approximation properties of the projection type interpolants,the DG method achieves the uniform convergence in the L2 norm with respect to the singular perturbation parameter e when the space of polynomials with degree p is used.A numerical experiment validates the theoretical results.Furthermore,an ultra-convergence order 2p+1 at the nodes for the one-sided flux,uniform with respect to the singular perturbation parameter e,is observed numerically.
文摘In this paper, using Lin＇s integral identity technique, we prove the optimal uniform convergence θ（Nx^-2ln^2Nx＋Ny^-2ln^2Ny） in the L^2-norm for singularly perturbed problems with parabolic layers. The error estimate is achieved by bilinear finite elements on a Shishkin type mesh. Here Nx and Ny are the number of elements in the x- and y-directions, respectively. Numerical results are provided supporting our theoretical analysis.
基金supported by the Department of Science & Technology, Government of India under research grant SR/S4/MS:318/06.
文摘Adaptive grid methods are established as valuable computational technique in approximating effectively the solutions of problems with boundary or interior layers. In this paper,we present the analysis of an upwind scheme for singularly perturbed differential-difference equation on a grid which is formed by equidistributing arc-length monitor function.It is shown that the discrete solution obtained converges uniformly with respect to the perturbation parameter.Numerical experiments illustrate in practice the result of convergence proved theoretically.
文摘Let G be a finite domain in the complex plane with K-quasicon formal boundary, z 0 be an arbitrary fixed point in G and p>0. Let π(z) be the conformal mapping from G onto the disk with radius r 0>0 and centered at the origin 0, normalized by ?(z 0) = 0 and ?(z 0) = 1. Let us set $\varphi _p \left( z \right): = \int_{x_0 }^x {\left[ {\phi \left( \zeta \right)} \right]^{2/8} } d\zeta $ , and let π n,p (z) be the generalized Bieberbach polynomial of degree n for the pair (G,z 0) that minimizes the integral $\iint\limits_c {\left| {\varphi _p \left( z \right) - P_x^1 (z)} \right|^p d0_x }$ in the class $\mathop \prod \limits_n $ of all polynomials of degree ≤ n and satisfying the conditions P n (z 0) = 0 and P′ n (z 0) = 1. In this work we prove the uniform convergence of the generalized Bieberbach polynomials π n,p (z) to ? p (z) on $\bar G$ in case of $p > 2 - \frac{{K^2 + 1}}{{2K^4 }}$ .
文摘We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.
文摘In this paper, we introduced the notion of uniform convergence of the linear operators on the probabilistic nornied space, and the notion of probabilistic distance between the operators, which describes the above convergence completely. In terms of these notions, we obtained the essential features of the continuity of operators, and of the uniform convergence of operator sequences, and we also obtained the closure of continuity and complete continuity under the operation of the limit of uniform convergence.
基金CSIR ( project no. F.NO. 8/3(45)/2005-EMR-I)for providing financial support to carry out the research work
文摘Recently, C. Tain and G. Chen introduced a new concept of sequence of time invariant function. In this paper we try to investigate the chaotic behavior of the uniform limit function f ： X →X of a sequence of continuous topologically transitive （in strongly successive way） functions fn ： X →X, where X is a compact interval. Surprisingly, we find that the uniform limit function is chaotic in the sense of Devaney. Lastly, we give an example to show that the denseness property of Devaney＇s definition is lost on the limit function.
文摘In this paper we study the dynamical behavior of a system ?approximated uniformly by a sequence ?of chaotic maps. We give examples to show that properties like sensitivity and denseness of periodic points need not be preserved under uniform convergence. We derive conditions under which some of the dynamical properties of the maps ?are preserved in .
基金The Major State Basic Research Program (19871051) of China the NNSF (19972039) of China and Yantai University Doctor Foundation (SX03B20).
文摘In this paper, we prove the existence, uniqueness and uniform convergence of the solution of finite volume element method based on the P1 conforming element for non-selfadjoint and indefinite elliptic problems under minimal elliptic regularity assumption.
文摘In this paper, we discuss the uniform convergence of the simple upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh for solving a singularly perturbed Robin boundary value problem, and investigate the midpoint upwind scheme on the Shishkin mesh and the Bakhvalov-Shishkin mesh to achieve better uniform convergence. The elaborate ε-uniform pointwise estimates are proved by using the comparison principle and barrier functions. The numerical experiments support the theoretical results for the schemes on the meshes.
