The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solu...The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solution and the optimal error estimate of its derivative with respect to time are derived by using some novel techniques. Moreover, employing a postprocessing technique, the global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is studied.展开更多
This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the F...This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.展开更多
This paper deals with the stability of systems with discontinuous righthand side （with solutions in Filippov＇s sense） via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of “set-valu...This paper deals with the stability of systems with discontinuous righthand side （with solutions in Filippov＇s sense） via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of “set-valued derivative” of vector Lyapunov functions is introduced, some generalized comparison principles on discontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.展开更多
The Wilson finite element method is considered to solve a class of two- dimensional second order elliptic boundary value problems. By using of the particular structure of the element and some new techniques, we obtain...The Wilson finite element method is considered to solve a class of two- dimensional second order elliptic boundary value problems. By using of the particular structure of the element and some new techniques, we obtain the superclose and global superconvergence on anisotropic meshes. Numerical example is also given to confirm our theoretical analysis.展开更多
Based on the modified Sawad^Kotera equation, we introduce a 3 ~ 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a genera...Based on the modified Sawad^Kotera equation, we introduce a 3 ~ 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawad-Kotera equation, by which a Lax pair of the modified Sawada-Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawad-Kotera equation, the Kaup-Kupershmidt equation, and the modified Sawad-Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada-Kotera equation are constructed with the aid of their Lax pairs.展开更多
In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the unique...In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.展开更多
Daxboux transformation with multi-parameters for the Boussinesq-Burgers （B-B） equation is derived. For an application, some important explicit solutions of the B-B equation are obtained, including 2N-soliton solutio...Daxboux transformation with multi-parameters for the Boussinesq-Burgers （B-B） equation is derived. For an application, some important explicit solutions of the B-B equation are obtained, including 2N-soliton solution and periodic solution. Finally, some elegant and interesting figures are plotted.展开更多
The convergence analysis of the lower order nonconforming element pro- posed by Park and Sheen is applied to the second-order elliptic problem under anisotropic meshes. The corresponding error estimation is obtained. ...The convergence analysis of the lower order nonconforming element pro- posed by Park and Sheen is applied to the second-order elliptic problem under anisotropic meshes. The corresponding error estimation is obtained. Moreover, by using the interpo- lation postprocessing technique, a global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is derived. Numerical results are also given to verify the theoretical analysis.展开更多
This paper is devoted to the study of the underlying linearities of the coupled Harry-Dym （cHD） soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of f...This paper is devoted to the study of the underlying linearities of the coupled Harry-Dym （cHD） soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of finite-dimensional Hamiltonian systems associated with soliton equations are presented, constituting the decomposition of the cHD soliton hierarchy. After suitably introducing the Abel-Jacobi coordinates on a Riemann surface, the cHD soliton hierarchy can be ultimately reduced to linear superpositions, expressed by the Abel-Jacobi variables.展开更多
In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double d...In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt ＋ bux4 - duxxt = f（u）xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.展开更多
In this paper, global input-to-state stability （ISS） for discrete-time piecewise affine systems with time-delay are considered Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhiu and Lya...In this paper, global input-to-state stability （ISS） for discrete-time piecewise affine systems with time-delay are considered Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhiu and Lyapunov-Krasovskii methods are used The theorems of Lyapunov-Razumikhin type and Lyapunov-Krasovskii type for piecewise affine systems with time-delay are shown respectively.展开更多
In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractio...In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.展开更多
Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very import...Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.展开更多
Let X = （X1, ···, Xm） be an infinitely degenerate system of vector fields. The aim of this paper is to study the existence of infinitely many solutions for the sum of operators X =sum （ ） form j=1 t...Let X = （X1, ···, Xm） be an infinitely degenerate system of vector fields. The aim of this paper is to study the existence of infinitely many solutions for the sum of operators X =sum （ ） form j=1 to m Xj Xj.展开更多
Sufficient conditions for the existence of at least two positive solutions of a nonlinear m -points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An exa...Sufficient conditions for the existence of at least two positive solutions of a nonlinear m -points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An example is provided to illustrate the theory.展开更多
With the help of the zero-curvature equation and the super trace identity, we derive a super extensionof the Kaup-Newell hierarchy associated with a 3×3 matrix spectral problem and establish its super bi-Hamilton...With the help of the zero-curvature equation and the super trace identity, we derive a super extensionof the Kaup-Newell hierarchy associated with a 3×3 matrix spectral problem and establish its super bi-Hamiltonianstructures.Furthermore, infinite conservation laws of the super Kaup-Newell equation are obtained by using spectralparameter expansions.展开更多
The problem of adaptive regulation of a class of high-order parametric nonholonomic systems in chained-form was discussed. Using adding a power integrator technique and state scaling with discontinuous projection tech...The problem of adaptive regulation of a class of high-order parametric nonholonomic systems in chained-form was discussed. Using adding a power integrator technique and state scaling with discontinuous projection technique, a discontinuous adaptive dynamic controller was constructed. The controller guarantees the estimated value of unknown parameter is in the prescribed extent.展开更多
基金This research is supported by the NSF of China (10371113 10471133),SF of Henan ProvinceSF of Education Committee of Henan Province (2006110011)
文摘The main aim of this paper is to study the approximation to viscoelasticity type equations with a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes. The superclose property of the exact solution and the optimal error estimate of its derivative with respect to time are derived by using some novel techniques. Moreover, employing a postprocessing technique, the global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is studied.
基金supported by the Mathematical Tianyuan Foundation (No. 10826078)the National Natural Science Foundation of China (No. 60874006)
文摘This paper discusses the problem of finite-time stability with respect to a closed, but not necessarily compact, invariant set for a class of nonlinear systems with discontinuous right-hand sides in the sense of the Filippov solutions. When the Lyapunov function is Lipschitz continuous and regular, the Lyapunov theorem on finite-time stability with respect to a closed invariant set is presented.
文摘This paper deals with the stability of systems with discontinuous righthand side （with solutions in Filippov＇s sense） via locally Lipschitz continuous and regular vector Lyapunov functions. A new type of “set-valued derivative” of vector Lyapunov functions is introduced, some generalized comparison principles on discontinuous systems are shown. Furthermore, Lyapunov stability theory is developed for a class of discontinuous systems based on locally Lipschitz continuous and regular vector Lyapunov functions.
基金Project supported by the National Natural Science Foundation of China (No. 10371113)
文摘The Wilson finite element method is considered to solve a class of two- dimensional second order elliptic boundary value problems. By using of the particular structure of the element and some new techniques, we obtain the superclose and global superconvergence on anisotropic meshes. Numerical example is also given to confirm our theoretical analysis.
基金Project supported by the National Natural Science Foundation of China (Grant No. 11171312)the Specialized Research Fund for the Doctoral Program of Higher Education, China (Grant No. 200804590008)
文摘Based on the modified Sawad^Kotera equation, we introduce a 3 ~ 3 matrix spectral problem with two potentials and derive a hierarchy of new nonlinear evolution equations. The second member in the hierarchy is a generalization of the modified Sawad-Kotera equation, by which a Lax pair of the modified Sawada-Kotera equation is obtained. With the help of the Miura transformation, explicit solutions of the Sawad-Kotera equation, the Kaup-Kupershmidt equation, and the modified Sawad-Kotera equation are given. Moreover, infinite sequences of conserved quantities of the first two nonlinear evolution equations in the hierarchy and the modified Sawada-Kotera equation are constructed with the aid of their Lax pairs.
基金supported by the National Natural Science Foundation of China(11226175,11271336 and 11171311)Specialized Reseach Fund for the Docotoral Program of Higher Education(20124301120002)Foundation of He’nan Educational Committee(2009C110006)
文摘In this paper, the existence and the uniqueness of the local generalized solution and the local classical solution of the Cauchy problem for the generalized BBM-Burgers equationare proved. The existence and the uniqueness of the global generalized solution and the global classical solution for the Cauchy problem of equation （1） are proved when n = 3, 2, 1. Moreover, the decay property of the solution is discussed.
文摘Daxboux transformation with multi-parameters for the Boussinesq-Burgers （B-B） equation is derived. For an application, some important explicit solutions of the B-B equation are obtained, including 2N-soliton solution and periodic solution. Finally, some elegant and interesting figures are plotted.
基金Project supported by the National Natural Science Foundation of China(Nos.10371113,10471133 and 10590353)
文摘The convergence analysis of the lower order nonconforming element pro- posed by Park and Sheen is applied to the second-order elliptic problem under anisotropic meshes. The corresponding error estimation is obtained. Moreover, by using the interpo- lation postprocessing technique, a global superconvergence property for the discretization error of the postprocessed discrete solution to the solution itself is derived. Numerical results are also given to verify the theoretical analysis.
基金Project supported by the National Natural Science Foundation of China （Grant No 10471132）, and the National Key Basic Research Special Foundation of China （Grant No 113000531034）.Acknowledgments The authors are obliged to the anonymous referee for his valuable remarks and suggestions.
文摘This paper is devoted to the study of the underlying linearities of the coupled Harry-Dym （cHD） soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of finite-dimensional Hamiltonian systems associated with soliton equations are presented, constituting the decomposition of the cHD soliton hierarchy. After suitably introducing the Abel-Jacobi coordinates on a Riemann surface, the cHD soliton hierarchy can be ultimately reduced to linear superpositions, expressed by the Abel-Jacobi variables.
文摘In this article, the existence, uniqueness and regularities of the global generalized solution and global classical solution for the periodic boundary value problem and the Cauchy problem of the general cubic double dispersion equationutt - uxx - auxxtt ＋ bux4 - duxxt = f（u）xxare proved, and the sufficient conditions of blow-up of the solutions for the Cauchy problems in finite time are given.
基金supported by National Natural Science Foundation of China (No. 60874006)Natural Science Foundation of Hei-longjiang Province for Youth (No. QC2009C99)
文摘In this paper, global input-to-state stability （ISS） for discrete-time piecewise affine systems with time-delay are considered Piecewise quadratic ISS-Lyapunov functions are adopted. Both Lyapunov-Razumikhiu and Lyapunov-Krasovskii methods are used The theorems of Lyapunov-Razumikhin type and Lyapunov-Krasovskii type for piecewise affine systems with time-delay are shown respectively.
基金Foundation item: Supported by the NSF of China(10371113)Supported by the Foundation of Overseas Scholar of China(2001(119))Supported by the project of Creative Engineering of Province of China(2002(219))
基金supported by the National Natural Science Foundation of China (No. 10671182)
文摘In this paper, existence and uniqueness of the generalized global solution and the classical global solution to the initial value problem for a class of fourth-order nonlinear wave equations are studied in the fractional order Sobolev space using the contraction mapping principle and the extension theorem. The sufficient conditions for the blow up of the solution to the initial value problem are given.
基金supported by the National Natural Science Foundation of China (10702065 and 10532050)China National Funds for Distinguished Young Scientists (10625211)the Program of Shanghai Subject Chief Scientist (08XD14044)
文摘Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluc-tuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratiodependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.
基金supported by Natural Science Foundation of China (10971199)Natural Science Foundations of Henan Province (092300410067)
文摘Let X = （X1, ···, Xm） be an infinitely degenerate system of vector fields. The aim of this paper is to study the existence of infinitely many solutions for the sum of operators X =sum （ ） form j=1 to m Xj Xj.
文摘Sufficient conditions for the existence of at least two positive solutions of a nonlinear m -points boundary value problems are established. The results are obtained by using a new fixed point theorem in cones. An example is provided to illustrate the theory.
基金Supported by National Natural Science Foundation of China under Grant No.10871182 Innovation Scientists and Technicians Troop Construction Projects of Henan Province (084200410019)SRFDP (200804590008)
文摘With the help of the zero-curvature equation and the super trace identity, we derive a super extensionof the Kaup-Newell hierarchy associated with a 3×3 matrix spectral problem and establish its super bi-Hamiltonianstructures.Furthermore, infinite conservation laws of the super Kaup-Newell equation are obtained by using spectralparameter expansions.
基金Supported by the National Natural Science Foundation of China under Grant No 10471132, and Innovation Scientists and Technicians Troop Construction Projects of Henan Province.
基金Project supported by the Scientific Research Foundation of Education Bureau of Henan Province (No.2003110002).
文摘The problem of adaptive regulation of a class of high-order parametric nonholonomic systems in chained-form was discussed. Using adding a power integrator technique and state scaling with discontinuous projection technique, a discontinuous adaptive dynamic controller was constructed. The controller guarantees the estimated value of unknown parameter is in the prescribed extent.
