In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme ...In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes.We relax the tangential continuity for velocity,which is the key ingredi-ent in achieving the uniform robustness.We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme,and the theories indicate that the error estimates for velocity are independent of pressure.Several numerical experiments are presented to confirm the theoretical findings.展开更多
In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy＇s law and Stokes equations in a single form of PDE but with strongly disconti...In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy＇s law and Stokes equations in a single form of PDE but with strongly discontinuous viscosity coefficient and zerothorder term coefficient. We present three different methods to construct uniformly stable finite element approximations. The first two methods are based on the original weak formulations of Darcy-Stokes-Brinkman equations. In the first method we consider the existing Stokes elements. We show that a stable Stokes element is also uniformly stable with respect to the coefficients and the jumps of Darcy-Stokes-Brinkman equations if and only if the discretely divergence-free velocity implies almost everywhere divergence-free one. In the second method we construct uniformly stable elements by modifying some well-known H（div）-conforming elements. We give some new 2D and 3D elements in a unified way. In the last method we modify the original weak formulation of Darcy-Stokes- Brinkman equations with a stabilization term. We show that all traditional stable Stokes elements are uniformly stable with respect to the coefficients and their jumps under this new formulation.展开更多
The development of mathematical modeling of infectious diseases is a key research area in various elds including ecology and epidemiology.One aim of these models is to understand the dynamics of behavior in infectious...The development of mathematical modeling of infectious diseases is a key research area in various elds including ecology and epidemiology.One aim of these models is to understand the dynamics of behavior in infectious diseases.For the new strain of coronavirus(COVID-19),there is no vaccine to protect people and to prevent its spread so far.Instead,control strategies associated with health care,such as social distancing,quarantine,travel restrictions,can be adopted to control the pandemic of COVID-19.This article sheds light on the dynamical behaviors of nonlinear COVID-19 models based on two methods:the homotopy perturbation method(HPM)and the modied reduced differential transform method(MRDTM).We invoke a novel signal ow graph that is used to describe the COVID-19 model.Through our mathematical studies,it is revealed that social distancing between potentially infected individuals who are carrying the virus and healthy individuals can decrease or interrupt the spread of the virus.The numerical simulation results are in reasonable agreement with the study predictions.The free equilibrium and stability point for the COVID-19 model are investigated.Also,the existence of a uniformly stable solution is proved.展开更多
The advancement in numerical models of serious resistant illnesses is a key research territory in different fields including the nature and the study of disease transmission.One of the aims of these models is to compr...The advancement in numerical models of serious resistant illnesses is a key research territory in different fields including the nature and the study of disease transmission.One of the aims of these models is to comprehend the elements of conduction of these infections.For the new strain of Covid-19(Coronavirus),there has been no immunization to protect individuals from the virus and to forestall its spread so far.All things being equal,control procedures related to medical services,for example,social distancing or separation,isolation,and travel limitations can be adjusted to control this pandemic.This article reveals some insights into the dynamic practices of nonlinear Coronavirus models dependent on the homotopy annoyance strategy(HPM).We summon a novel sign stream chart that is utilized to depict the Coronavirus model.Through the numerical investigations,it is uncovered that social separation of the possibly tainted people who might be conveying the infection and the healthy virus-free people can diminish or interrupt the spread of the infection.The mathematical simulation results are highly concurrent with the statistical forecasts.The free balance and dependability focus for the Coronavirus model is discussed and the presence of a consistently steady arrangement is demonstrated.展开更多
This paper is concerned with an almost periodic model of plankton allelopathy with impulsive effects. By using the comparison theorem and the Lyapunov method of the impulsive differential equations, sufficient conditi...This paper is concerned with an almost periodic model of plankton allelopathy with impulsive effects. By using the comparison theorem and the Lyapunov method of the impulsive differential equations, sufficient conditions which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the model are obtained. The main results in this paper improve the work in recent years. And the method used in this paper provides a new method to study the permanence, uniform asymptotical stability and almost periodic solution of the models with impulsive perturbations in biological populations. An example and numerical simulations are provided to illustrate the main results of this paper. Finally, a conclusion is also given to discuss how the impulsive effects influence the permanence, almost periodic solutions and uniform asymptotical stability of the model.展开更多
In this paper, we consider lower order rectangular finite element methods for the singularly perturbed Stokes problem. The model problem reduces to a linear Stokes problem when the perturbation parameter is large and ...In this paper, we consider lower order rectangular finite element methods for the singularly perturbed Stokes problem. The model problem reduces to a linear Stokes problem when the perturbation parameter is large and degenerates to a mixed formulation of Poisson＇s equation as the perturbation parameter tends to zero. We propose two 2D and two 3D nonconforming rectangular finite elements, and derive robust discretization error estimates. Numerical experiments are carried out to verify the theoretical results.展开更多
In this paper, we consider two-species n-patch almost periodic competitive systemswith diffusion, where one species can diffuse between any two of n patches, but the otheris confined to one of the patches and cannot d...In this paper, we consider two-species n-patch almost periodic competitive systemswith diffusion, where one species can diffuse between any two of n patches, but the otheris confined to one of the patches and cannot diffuse. We prove that the systems can have aunique positive almost periodic solution, which is globally uniformly asymptotically stableunder some appropriate conditions. In particular, if the system is a periodic system of period ω, it can have a positive globally uniformly asymptotically stable periodic solution ofperiod ω, which is a generalization of Theorem 4 in paper [6].展开更多
We consider Lienard system and obtain following conclusions: The zero solution of system x' + f(x)x' + g(x) = 0 is uniformly asymptotically stable if g(0) = 0, and (x) > 0. And system X' + f(x)x' ...We consider Lienard system and obtain following conclusions: The zero solution of system x' + f(x)x' + g(x) = 0 is uniformly asymptotically stable if g(0) = 0, and (x) > 0. And system X' + f(x)x' + g(x) = e(t) has uniformly asymptotically stable solutions if g(0) = 0, and Hence it has a unique almost odic solution when e(t) is almost periodic and it has a unique periodic solution when e(t) is periodic. In [1] Fink obtained above the second conclusion if sup In [2] we obtained same result if g(x) = cx.展开更多
By means of the properties of almost periodic system and Lyapunov function, we give some criteria which guarantee the existence and uniqueness and stability of almost periodic solutions of higher dimensional nonautono...By means of the properties of almost periodic system and Lyapunov function, we give some criteria which guarantee the existence and uniqueness and stability of almost periodic solutions of higher dimensional nonautonomous system. The result is more convenient and effective than the related result in[1].展开更多
In this paper, a new type of stability, namely φ0-strict stability is extended for the delay difference equations, and by using variational cone-valued Lyapunov-like functions some sufficient conditions for such stab...In this paper, a new type of stability, namely φ0-strict stability is extended for the delay difference equations, and by using variational cone-valued Lyapunov-like functions some sufficient conditions for such stability to hold are given.展开更多
In this paper,we consider almost periodic discrete two-species competitive sys-tems.By using Lyapunov functional,the existence conditions and uniqueness of almost periodic solutions for the this type of systems are ob...In this paper,we consider almost periodic discrete two-species competitive sys-tems.By using Lyapunov functional,the existence conditions and uniqueness of almost periodic solutions for the this type of systems are obtained.展开更多
This paper studies equation x' + cx' + g(x) = P(t,x). Under some suitable conditions the existence and uniqueness of almost periodic solution of this equation are given.
基金the Hong Kong RGC General Research Fund(Project numbers 14304719 and 14302018)CUHK Faculty of Science Direct Grant 2019-20。
文摘In this paper,we propose and analyze a uniformly robust staggered DG method for the unsteady Darcy-Forchheimer-Brinkman problem.Our formulation is based on velocity gradient-velocity-pressure and the resulting scheme can be flexibly applied to fairly general polygonal meshes.We relax the tangential continuity for velocity,which is the key ingredi-ent in achieving the uniform robustness.We present well-posedness and error analysis for both the semi-discrete scheme and the fully discrete scheme,and the theories indicate that the error estimates for velocity are independent of pressure.Several numerical experiments are presented to confirm the theoretical findings.
基金NSF DMS-0609727by the Center for Computational Mathematics and Applications of Penn State+3 种基金Jinchao Xu was also supported in part by NSFC-10501001Alexander H.Humboldt Foundation.Xiaoping Xie was supported by the National Natural Science Foundation of China (10771150)the National Basic Research Program of China (2005CB321701)the program for New Century Excellent Talents in University (NCET-07-0584)
文摘In this paper, we consider 2D and 3D Darcy-Stokes interface problems. These equations are related to Brinkman model that treats both Darcy＇s law and Stokes equations in a single form of PDE but with strongly discontinuous viscosity coefficient and zerothorder term coefficient. We present three different methods to construct uniformly stable finite element approximations. The first two methods are based on the original weak formulations of Darcy-Stokes-Brinkman equations. In the first method we consider the existing Stokes elements. We show that a stable Stokes element is also uniformly stable with respect to the coefficients and the jumps of Darcy-Stokes-Brinkman equations if and only if the discretely divergence-free velocity implies almost everywhere divergence-free one. In the second method we construct uniformly stable elements by modifying some well-known H（div）-conforming elements. We give some new 2D and 3D elements in a unified way. In the last method we modify the original weak formulation of Darcy-Stokes- Brinkman equations with a stabilization term. We show that all traditional stable Stokes elements are uniformly stable with respect to the coefficients and their jumps under this new formulation.
基金funded by“Taif University Researchers Supporting Project Number(TURSP-2020/16),Taif University,Taif,Saudi Arabia.”。
文摘The development of mathematical modeling of infectious diseases is a key research area in various elds including ecology and epidemiology.One aim of these models is to understand the dynamics of behavior in infectious diseases.For the new strain of coronavirus(COVID-19),there is no vaccine to protect people and to prevent its spread so far.Instead,control strategies associated with health care,such as social distancing,quarantine,travel restrictions,can be adopted to control the pandemic of COVID-19.This article sheds light on the dynamical behaviors of nonlinear COVID-19 models based on two methods:the homotopy perturbation method(HPM)and the modied reduced differential transform method(MRDTM).We invoke a novel signal ow graph that is used to describe the COVID-19 model.Through our mathematical studies,it is revealed that social distancing between potentially infected individuals who are carrying the virus and healthy individuals can decrease or interrupt the spread of the virus.The numerical simulation results are in reasonable agreement with the study predictions.The free equilibrium and stability point for the COVID-19 model are investigated.Also,the existence of a uniformly stable solution is proved.
基金support of“Taif University Deanship of Scientific Research Project number(1-441-23),Taif University,Taif,Saudi Arabia”.
文摘The advancement in numerical models of serious resistant illnesses is a key research territory in different fields including the nature and the study of disease transmission.One of the aims of these models is to comprehend the elements of conduction of these infections.For the new strain of Covid-19(Coronavirus),there has been no immunization to protect individuals from the virus and to forestall its spread so far.All things being equal,control procedures related to medical services,for example,social distancing or separation,isolation,and travel limitations can be adjusted to control this pandemic.This article reveals some insights into the dynamic practices of nonlinear Coronavirus models dependent on the homotopy annoyance strategy(HPM).We summon a novel sign stream chart that is utilized to depict the Coronavirus model.Through the numerical investigations,it is uncovered that social separation of the possibly tainted people who might be conveying the infection and the healthy virus-free people can diminish or interrupt the spread of the infection.The mathematical simulation results are highly concurrent with the statistical forecasts.The free balance and dependability focus for the Coronavirus model is discussed and the presence of a consistently steady arrangement is demonstrated.
基金Supported by the Scientific Research Fund of Yunnan Provincial Education Department(Grant No.2014Y388)
文摘This paper is concerned with an almost periodic model of plankton allelopathy with impulsive effects. By using the comparison theorem and the Lyapunov method of the impulsive differential equations, sufficient conditions which guarantee the permanence and existence of a unique uniformly asymptotically stable positive almost periodic solution of the model are obtained. The main results in this paper improve the work in recent years. And the method used in this paper provides a new method to study the permanence, uniform asymptotical stability and almost periodic solution of the models with impulsive perturbations in biological populations. An example and numerical simulations are provided to illustrate the main results of this paper. Finally, a conclusion is also given to discuss how the impulsive effects influence the permanence, almost periodic solutions and uniform asymptotical stability of the model.
基金supported by the Natural Science Foundation of China (10771150)the National Basic Research Program of China (2005CB321701)the Program for New Century Excellent Talents in University (NCET-07-0584)
文摘In this paper, we consider lower order rectangular finite element methods for the singularly perturbed Stokes problem. The model problem reduces to a linear Stokes problem when the perturbation parameter is large and degenerates to a mixed formulation of Poisson＇s equation as the perturbation parameter tends to zero. We propose two 2D and two 3D nonconforming rectangular finite elements, and derive robust discretization error estimates. Numerical experiments are carried out to verify the theoretical results.
文摘In this paper, we consider two-species n-patch almost periodic competitive systemswith diffusion, where one species can diffuse between any two of n patches, but the otheris confined to one of the patches and cannot diffuse. We prove that the systems can have aunique positive almost periodic solution, which is globally uniformly asymptotically stableunder some appropriate conditions. In particular, if the system is a periodic system of period ω, it can have a positive globally uniformly asymptotically stable periodic solution ofperiod ω, which is a generalization of Theorem 4 in paper [6].
文摘We consider Lienard system and obtain following conclusions: The zero solution of system x' + f(x)x' + g(x) = 0 is uniformly asymptotically stable if g(0) = 0, and (x) > 0. And system X' + f(x)x' + g(x) = e(t) has uniformly asymptotically stable solutions if g(0) = 0, and Hence it has a unique almost odic solution when e(t) is almost periodic and it has a unique periodic solution when e(t) is periodic. In [1] Fink obtained above the second conclusion if sup In [2] we obtained same result if g(x) = cx.
文摘By means of the properties of almost periodic system and Lyapunov function, we give some criteria which guarantee the existence and uniqueness and stability of almost periodic solutions of higher dimensional nonautonomous system. The result is more convenient and effective than the related result in[1].
基金Supported by National Natural Science Foundation of China(No.10371040).
文摘In this paper, a new type of stability, namely φ0-strict stability is extended for the delay difference equations, and by using variational cone-valued Lyapunov-like functions some sufficient conditions for such stability to hold are given.
基金the Natural Science Foundation of Fujian Province(Z0511014)the Foundation of Developing Science and Technology of Fuzhou University(2005-QX-18,2005-QX-21).
文摘In this paper,we consider almost periodic discrete two-species competitive sys-tems.By using Lyapunov functional,the existence conditions and uniqueness of almost periodic solutions for the this type of systems are obtained.
基金This work was supported by Fujian Education Department Science Foundation (K20009).
文摘This paper studies equation x' + cx' + g(x) = P(t,x). Under some suitable conditions the existence and uniqueness of almost periodic solution of this equation are given.
