In this paper, we prove that the nonautonomous Schrodinger flow from a compact Riemannian manifold into a Kahler manifold admits a local solution. Under some certain conditions, the solution is unique and has higher r...In this paper, we prove that the nonautonomous Schrodinger flow from a compact Riemannian manifold into a Kahler manifold admits a local solution. Under some certain conditions, the solution is unique and has higher regularity.展开更多
We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potent...We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related matrix.Also some useful inequalities are established in this paper.Finally,extensive numerical results are reported to support our error estimates of the numerical methods.展开更多
This paper is concerned with the stability of a first order dynamic equation on time scales. In particular,using the properties of the generalized exponential function with time scales,different kinds of sufficient co...This paper is concerned with the stability of a first order dynamic equation on time scales. In particular,using the properties of the generalized exponential function with time scales,different kinds of sufficient conditions for the exponential stability are obtained.展开更多
This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is li...This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.展开更多
Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations.The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rat...Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations.The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems.We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall's inequality to improve the Fourier splitting method.This method was initiated by Maria Schonbek in the 1980's to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations.As applications,the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for n-dimensional incompressible Navier-Stokes equations,for the n-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.展开更多
基金Supported by the National Natural Science Foundation of China(Grant Nos.11731001 and 11471316)
文摘In this paper, we prove that the nonautonomous Schrodinger flow from a compact Riemannian manifold into a Kahler manifold admits a local solution. Under some certain conditions, the solution is unique and has higher regularity.
基金supported by Ministry of Education of Singapore grant R-146-000-120-112the National Natural Science Foundation of China(Grant No.11131005)the Doctoral Programme Foundation of Institution of Higher Education of China(Grant No.20110002110064).
文摘We study compact finite difference methods for the Schrodinger-Poisson equation in a bounded domain and establish their optimal error estimates under proper regularity assumptions on wave functionψand external potential V(x).The CrankNicolson compact finite difference method and the semi-implicit compact finite difference method are both of order O(h^(4)+τ^(2))in discrete l^(2),H^(1) and l^(∞) norms with mesh size h and time step τ.For the errors of compact finite difference approximation to the second derivative and Poisson potential are nonlocal,thus besides the standard energy method and mathematical induction method,the key technique in analysis is to estimate the nonlocal approximation errors in discrete l^(∞) and H^(1) norm by discrete maximum principle of elliptic equation and properties of some related matrix.Also some useful inequalities are established in this paper.Finally,extensive numerical results are reported to support our error estimates of the numerical methods.
基金Supported by the Natural Science Foundation of Guangdong Province (No.10151601501000003)Science Foundation of Huizhou University
文摘This paper is concerned with the stability of a first order dynamic equation on time scales. In particular,using the properties of the generalized exponential function with time scales,different kinds of sufficient conditions for the exponential stability are obtained.
基金This work is supported by NSFC(Grant Nos.11771035,11771162,11571128,61473126,91430216,91530204,11372354 and U1530401),a grant from the RGC of HK 11300517,China(Project No.CityU 11302915),China Postdoctoral Science Foundation under grant No.2016M602273,a grant DRA2015518 from 333 High-level Personal Training Project of Jiangsu Province,and the USA National Science Foundation grant DMS-1315259the USA Air Force Office of Scientific Research grant FA9550-15-1-0001.Jiwei Zhang also thanks the hospitality of Hong Kong City University during the period of his visiting.
文摘This paper is concerned with numerical solutions of time-fractional nonlinear parabolic problems by a class of L1-Galerkin finite element methods.The analysis of L1 methods for time-fractional nonlinear problems is limited mainly due to the lack of a fundamental Gronwall type inequality.In this paper,we establish such a fundamental inequality for the L1 approximation to the Caputo fractional derivative.In terms of the Gronwall type inequality,we provide optimal error estimates of several fully discrete linearized Galerkin finite element methods for nonlinear problems.The theoretical results are illustrated by applying our proposed methods to the time fractional nonlinear Huxley equation and time fractional Fisher equation.
文摘Consider the Cauchy problems for an n-dimensional nonlinear system of fluid dynamics equations.The main purpose of this paper is to improve the Fourier splitting method to accomplish the decay estimates with sharp rates of the global weak solutions of the Cauchy problems.We will couple together the elementary uniform energy estimates of the global weak solutions and a well known Gronwall's inequality to improve the Fourier splitting method.This method was initiated by Maria Schonbek in the 1980's to study the optimal long time asymptotic behaviours of the global weak solutions of the nonlinear system of fluid dynamics equations.As applications,the decay estimates with sharp rates of the global weak solutions of the Cauchy problems for n-dimensional incompressible Navier-Stokes equations,for the n-dimensional magnetohydrodynamics equations and for many other very interesting nonlinear evolution equations with dissipations can be established.
