The probabilistic solutions of the nonlinear stochastic dynamic(NSD)systems with polynomial type of nonlinearity are investigated with the subspace-EPC method.The space of the state variables of large-scale nonlinear ...The probabilistic solutions of the nonlinear stochastic dynamic(NSD)systems with polynomial type of nonlinearity are investigated with the subspace-EPC method.The space of the state variables of large-scale nonlinear stochastic dynamic system excited by white noises is separated into two subspaces.Both sides of the Fokker-Planck-Kolmogorov(FPK)equation corresponding to the NSD system is then integrated over one of the subspaces.The FPK equation for the joint probability density function of the state variables in another subspace is formulated.Therefore,the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of largescale NSD systems solvable with the exponential polynomial closure method.Examples about the NSD systems with polynomial type of nonlinearity are given to show the effectiveness of the subspace-EPC method in these cases.展开更多
： The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati tech...： The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many knownresults for second order dynamic equations. Some examples are given to illustrate the main results of this article.展开更多
The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-E...The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.展开更多
We establish a new Kamenev-type theorem for a class of second-order nonlinear damped delay dynamic equations on a time scale by using the generalized Riccati transformation technique. The criterion obtained improves r...We establish a new Kamenev-type theorem for a class of second-order nonlinear damped delay dynamic equations on a time scale by using the generalized Riccati transformation technique. The criterion obtained improves related contributions to the subject. An example is provided to illustrate assumptions in our theorem are less restrictive.展开更多
In this paper,we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical syste...In this paper,we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems.Two critical values which can characterize the scale of dissipation effect are obtained.If dissipation effect is not less than a certain critical value,the traveling wave solutions appear as kink profile;while if it is less than this critical value,they appear as damped oscillatory.All expressions of bounded traveling wave solutions are presented,including exact expressions of bell and kink profile solitary wave solutions,as well as approximate expressions of damped oscillatory solutions.For approximate damped oscillatory solution,using homogenization principle,we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions.It can be seen that the error is an infinitesimal decreasing in the exponential form.展开更多
非线性动力方程直接积分法的基础是构造t时刻与t+△t时刻状态量间的关系,由此形成基本量的非线性方程组,再在每个时间步内采用Newton-Raphson或BFGS等迭代方法求解.该文基于Bathe复合积分法(composite implicit time integration),提出...非线性动力方程直接积分法的基础是构造t时刻与t+△t时刻状态量间的关系,由此形成基本量的非线性方程组,再在每个时间步内采用Newton-Raphson或BFGS等迭代方法求解.该文基于Bathe复合积分法(composite implicit time integration),提出了非线性阻尼系统基于速度变量的复合时间积分迭代格式.以非线性黏滞阻尼Sdof系统为例,按上述方法以及基于BFGS迭代的Newmark-β法编制Fortran程序,结果与Adina软件对比,验证了该文方法的有效性.展开更多
基金supported by the Research Committee of the University of Macao(Grant No.MYRG138-FST11-EGK).
文摘The probabilistic solutions of the nonlinear stochastic dynamic(NSD)systems with polynomial type of nonlinearity are investigated with the subspace-EPC method.The space of the state variables of large-scale nonlinear stochastic dynamic system excited by white noises is separated into two subspaces.Both sides of the Fokker-Planck-Kolmogorov(FPK)equation corresponding to the NSD system is then integrated over one of the subspaces.The FPK equation for the joint probability density function of the state variables in another subspace is formulated.Therefore,the FPK equation in low dimensions is obtained from the original FPK equation in high dimensions and it makes the problem of obtaining the probabilistic solutions of largescale NSD systems solvable with the exponential polynomial closure method.Examples about the NSD systems with polynomial type of nonlinearity are given to show the effectiveness of the subspace-EPC method in these cases.
基金Supported by the Scientific Research Fund of Hunan Provincial Education Department(09A082)
文摘： The oscillation for a class of second order nonlinear variable delay dynamic equation on time scales with nonlinear neutral term and damping term was discussed in this article. By using the generalized Riccati technique, integral averaging technique and the time scales theory, some new sufficient conditions for oscillation of the equation are proposed. These results generalize and extend many knownresults for second order dynamic equations. Some examples are given to illustrate the main results of this article.
文摘The probabilistic solutions to some nonlinear stochastic dynamic (NSD) systems with various polynomial types of nonlinearities in displacements are analyzed with the subspace-exponential polynomial closure (subspace-EPC) method. The space of the state variables of the large-scale nonlinear stochastic dynamic system excited by Gaussian white noises is separated into two subspaces. Both sides of the Fokker-Planck-Kolmogorov (FPK) equation corresponding to the NSD system are then integrated over one of the subspaces. The FPK equation for the joint probability density function of the state variables in the other subspace is formulated. Therefore, the FPK equations in low dimensions are obtained from the original FPK equation in high dimensions and the FPK equations in low dimensions are solvable with the exponential polynomial closure method. Examples about multi-degree-offreedom NSD systems with various polynomial types of nonlinearities in displacements are given to show the effectiveness of the subspace-EPC method in these cases.
基金supported by the Applied Mathematics Enhancement Program of Linyi University
文摘We establish a new Kamenev-type theorem for a class of second-order nonlinear damped delay dynamic equations on a time scale by using the generalized Riccati transformation technique. The criterion obtained improves related contributions to the subject. An example is provided to illustrate assumptions in our theorem are less restrictive.
基金This research is supported by the National Natural Science Foundation of China(No.11071164)Shanghai Natural Science Foundation Project(No.10ZR1420800)Leading Academic Discipline Project of Shanghai Municipal Government(No.S30501).
文摘In this paper,we analyze the relation between the shape of the bounded traveling wave solutions and dissipation coefficient of nonlinear wave equation with cubic term by the theory and method of planar dynamical systems.Two critical values which can characterize the scale of dissipation effect are obtained.If dissipation effect is not less than a certain critical value,the traveling wave solutions appear as kink profile;while if it is less than this critical value,they appear as damped oscillatory.All expressions of bounded traveling wave solutions are presented,including exact expressions of bell and kink profile solitary wave solutions,as well as approximate expressions of damped oscillatory solutions.For approximate damped oscillatory solution,using homogenization principle,we give its error estimate by establishing the integral equation which reflects the relations between the exact and approximate solutions.It can be seen that the error is an infinitesimal decreasing in the exponential form.
基金the Natural Science Foundation of Hunan Province(12JJ6006)the Hunan Province Science and Technology Project(2012FJ3107)+1 种基金the National Natural Science Foundation ofChina(11071222)the Scientific Research Fund of Education Department of Guangxi Zhuang Autonomous Region(2013YB223)
文摘非线性动力方程直接积分法的基础是构造t时刻与t+△t时刻状态量间的关系,由此形成基本量的非线性方程组,再在每个时间步内采用Newton-Raphson或BFGS等迭代方法求解.该文基于Bathe复合积分法(composite implicit time integration),提出了非线性阻尼系统基于速度变量的复合时间积分迭代格式.以非线性黏滞阻尼Sdof系统为例,按上述方法以及基于BFGS迭代的Newmark-β法编制Fortran程序,结果与Adina软件对比,验证了该文方法的有效性.
