This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapu...This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapunov functional method, we establish a sufficient condition on the stability and boundedness of the solutions of above equation. This will generalize the main results of reference [2].展开更多
By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>...By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>0(0≤t≤2π),a(0)=a(2π),F(x,y)=f(x)+α|y| β,α>0,β>0 are all constants,f∈C(R,R),e∈C[0,2π]. An example is given as an application.展开更多
In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditio...In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub（super） linearity are assumed. W＇e present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property （X^＋）, which is very important for the existence of periodic solutions and oscillation theory.展开更多
In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard-type equation with delays.
采用等温热压缩测试和TEM分析研究铝锂合金的流变行为与组织演化规律。结果表明:合金的热塑性变形过程受热激活控制,当变形温度低于410℃时,流变曲线具有明显的峰值应力,曲线由加工硬化、动态软化和稳定阶段3个阶段组成;当变形温度高于...采用等温热压缩测试和TEM分析研究铝锂合金的流变行为与组织演化规律。结果表明:合金的热塑性变形过程受热激活控制,当变形温度低于410℃时,流变曲线具有明显的峰值应力,曲线由加工硬化、动态软化和稳定阶段3个阶段组成;当变形温度高于410℃后,峰值应力不明显。随应变量的增加,合金组织演化规律为产生大量无规则缠结位错→"多边化"形成"位错墙"→分割原始晶粒成若干亚晶→亚晶合并长大并同时经受变形→重复上述过程。应变量的增加导致大量空位产生,刃型位错更易攀移、重组和对消,晶内形成亚晶组织。求解得到合金的材料常数如下:结构因子A为2.787×1016;变形激活能Q为217.397 k J/mol;应力指数n为6.11656;应力水平参数α为0.012568 mm2/N。应变速率和温度对合金流变应力的影响可以用包含Arrhenius等式的Z参数表示。展开更多
Based on the Taylor series method and Li’s spatial differential method, a high-order hybrid Taylor–Li scheme is proposed.The results of a linear advection equation indicate that, using the initial values of the squa...Based on the Taylor series method and Li’s spatial differential method, a high-order hybrid Taylor–Li scheme is proposed.The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with thirdorder accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme’s computation speed. A more rapid and highprecision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers’ equation.展开更多
In this paper, author obtain sufficient condition for the boundedness of solutions of the second order nonlinear differential systemdxdt=y-φ(x), dydt=-yf(x)-g(x).(E)The result can be applied to the Liénard-type ...In this paper, author obtain sufficient condition for the boundedness of solutions of the second order nonlinear differential systemdxdt=y-φ(x), dydt=-yf(x)-g(x).(E)The result can be applied to the Liénard-type system, which substantially extends and imporves some important results associated with the same problems in the literatures [1—10] et al..展开更多
The existence of monotone and non_monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector...The existence of monotone and non_monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Liénard system and the system given by symmetric transformation or quasi_symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi_stable, the existence of infinitely many monotone solusion is obtained.展开更多
In this article, we obtain Li-Yau-type gradient estimates with time dependent parameter for positive solutions of the heat equation that are different with the estimates by Li-Xu [21] and Qian [23]. As an application ...In this article, we obtain Li-Yau-type gradient estimates with time dependent parameter for positive solutions of the heat equation that are different with the estimates by Li-Xu [21] and Qian [23]. As an application of the estimate, we also obtained slight improvements of Davies' Li-Yau-type gradient estimate.展开更多
文摘This paper aims to investigate the retarded Liénard-type equation+f 1(x)+f 2(x)(t-τ)+f 3(x)2+φ(x)+g(x(t-τ))=0,where τ is a nonnegative constant, f 1,f 2,f 3,φ and g are continuous functions on R. Using Liapunov functional method, we establish a sufficient condition on the stability and boundedness of the solutions of above equation. This will generalize the main results of reference [2].
文摘By coincidence degree,the existence of solution to the boundary value problem of a generalized Liénard equationa(t)x'+F(x,x′)x′+g(x)=e(t), x(0)=x(2π),x′(0)=x′(2π)is proved,where a∈C 1[0,2π],a(t)>0(0≤t≤2π),a(0)=a(2π),F(x,y)=f(x)+α|y| β,α>0,β>0 are all constants,f∈C(R,R),e∈C[0,2π]. An example is given as an application.
文摘In this paper the generalized nonlinear Euler differential equation t^2k（tu＇）u＇＇＋ t（f（u） ＋ k（tu＇））u＇ ＋ g（u） = 0 is considered. Here the functions f（u）, g（u） and k（u） satisfy smoothness conditions which guarantee the uniqueness of solutions of initial value problems, however, no conditions of sub（super） linearity are assumed. W＇e present some necessary and sufficient conditions and some tests for the equivalent planar system to have or fail to have property （X^＋）, which is very important for the existence of periodic solutions and oscillation theory.
基金This work was supported by the NNSF of China (10071016) the Doctor Program Foundation of the Ministry of Education of China (20010532002) the Key Project of Chinese Ministry of Education (No.[2002]78).
文摘In this paper, we use the coincidence degree theory to establish new results on the existence of T-periodic solutions for the Liénard-type equation with delays.
文摘采用等温热压缩测试和TEM分析研究铝锂合金的流变行为与组织演化规律。结果表明:合金的热塑性变形过程受热激活控制,当变形温度低于410℃时,流变曲线具有明显的峰值应力,曲线由加工硬化、动态软化和稳定阶段3个阶段组成;当变形温度高于410℃后,峰值应力不明显。随应变量的增加,合金组织演化规律为产生大量无规则缠结位错→"多边化"形成"位错墙"→分割原始晶粒成若干亚晶→亚晶合并长大并同时经受变形→重复上述过程。应变量的增加导致大量空位产生,刃型位错更易攀移、重组和对消,晶内形成亚晶组织。求解得到合金的材料常数如下:结构因子A为2.787×1016;变形激活能Q为217.397 k J/mol;应力指数n为6.11656;应力水平参数α为0.012568 mm2/N。应变速率和温度对合金流变应力的影响可以用包含Arrhenius等式的Z参数表示。
基金supported by the National Natural Sciences Foundation of China(Grant Nos.41375112 and 41530426)the Chinese Academy of Sciences Key Technology Talent Program
文摘Based on the Taylor series method and Li’s spatial differential method, a high-order hybrid Taylor–Li scheme is proposed.The results of a linear advection equation indicate that, using the initial values of the square-wave type, a result with thirdorder accuracy occurs. However, using initial values associated with the Gaussian function type, a result with very high precision appears. The study demonstrates that, when the order of the time integral is more than three, the corresponding optimal spatial difference order could be higher than six. The results indicate that the reason for why there is no improvement related to an order of spatial difference above six is the use of a time integral scheme that is not high enough. The author also proposes a recursive differential method to improve the Taylor–Li scheme’s computation speed. A more rapid and highprecision program than direct computation of the high-order space differential item is employed, and the computation speed is dramatically boosted. Based on a multiple-precision library, the ultrahigh-order Taylor–Li scheme can be used to solve the advection equation and Burgers’ equation.
文摘In this paper, author obtain sufficient condition for the boundedness of solutions of the second order nonlinear differential systemdxdt=y-φ(x), dydt=-yf(x)-g(x).(E)The result can be applied to the Liénard-type system, which substantially extends and imporves some important results associated with the same problems in the literatures [1—10] et al..
文摘The existence of monotone and non_monotone solutions of boundary value problem on the real line for Liénard equation is studied. Applying the theory of planar dynamical systems and the comparison method of vector fields defined by Liénard system and the system given by symmetric transformation or quasi_symmetric transformation, the invariant regions of the system are constructed. The existence of connecting orbits can be proved. A lot of sufficient conditions to guarantee the existence of solutions of the boundary value problem are obtained. Especially, when the source function is bi_stable, the existence of infinitely many monotone solusion is obtained.
基金partially supported by the Yangfan project from Guangdong ProvinceNSFC(11571215)
文摘In this article, we obtain Li-Yau-type gradient estimates with time dependent parameter for positive solutions of the heat equation that are different with the estimates by Li-Xu [21] and Qian [23]. As an application of the estimate, we also obtained slight improvements of Davies' Li-Yau-type gradient estimate.
