Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this a...Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.展开更多
The discrete complex cubic Ginzburg-Landau equation is an important model to describe a number of physical systems such as Taylor and frustrated vortices in hydrodynamics and semiconductor laser arrays in optics. In t...The discrete complex cubic Ginzburg-Landau equation is an important model to describe a number of physical systems such as Taylor and frustrated vortices in hydrodynamics and semiconductor laser arrays in optics. In this paper, the exact solutions of the discrete complex cubic Ginzburg-Landau equation are derived using homogeneous balance principle and the GI/G-expansion method, and the linear stability of exact solutions is discussed.展开更多
In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A suffic...In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.展开更多
In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arb...In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.展开更多
In this paper, we consider an almost periodic system which includes a system of the type , where k is a positive integer, aij are almost periodic in n and satisfy aij(n)≥0 for i≠j,? for 1≤j≤m. In the special case ...In this paper, we consider an almost periodic system which includes a system of the type , where k is a positive integer, aij are almost periodic in n and satisfy aij(n)≥0 for i≠j,? for 1≤j≤m. In the special case where aij(n) are constant functions, above system is a mathematical model of gas dynamics and was treated by T. Carleman and R. D. Jenks for differential systems. In the main theorem, we show that if the m X m matrix (aij(n)) is irreducible, then there exists a positive almost periodic solution which is unique and has some stability. Moreover, we can see that this result gives R. D. Jenks’ result for differential model in the case where aij(n) are constant functions. In Section 3, we consider the linear system with variable cofficients . Even in nonlinear problems, this linear system plays an important role, as their variational equations, and it is requested to determine the uniform asymptotically stability of the zero solution from the information about A(n). In order to obtain the existence of almost periodic solutions of both linear and nonlinear almost periodic discrete systems: above linear system and? for 1≤i≤m, respectively, we shall consider between certain stability properties, which are referred to as uniformly asymptotically stable, and the diagonal dominance matrix condition.展开更多
In this paper,we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients.When there are more than one coefficient having the sa...In this paper,we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients.When there are more than one coefficient having the same maximal order and the same maximal type,the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained.Meanwhile,the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.展开更多
This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state...This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state at some particular concentration. The movement states’ description is found for the primary and secondary level. The primary system is found to be predominantly an unstable system while the secondary system is stable. This is derived from the state of dynamic eigenvalues associated with the system. The saturated solution-particle is projected to be stable both for the primary potential and secondary state. A volume conserving linear system has been suggested to describe the dynamical state of movement of a solution particle.展开更多
An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations i...An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.展开更多
In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special c...In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special cases. Compared with the expansion method and He＇s homotopy perturbation method, respectively numerical examples are given to certify the effectiveness of the method. The results show that the block-by-block method is very effective, simple, and of high accuracy in solving the system of linear Volterra integral equations of the second kind.展开更多
With the help of the symbolic computation system Maple, the Riccati equation mapping approach and a linear variable separation approach, a new family of complex solutions for the （2＋ 1）-dimensional Boiti-Leon-Pempi...With the help of the symbolic computation system Maple, the Riccati equation mapping approach and a linear variable separation approach, a new family of complex solutions for the （2＋ 1）-dimensional Boiti-Leon-Pempinelli system （BLP） is derived. Based on the derived solitary wave solution, some novel complex wave localized excitations are obtained.展开更多
There are given sufficient conditions for the ultimate boundedness of solutions and for the existence of periodic solutions of a certain vector differential equation of third-order.
With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ ...With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ my ＋ nt ＋ Г（x,y, t） for the （2＋1）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system （GCBS） are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.展开更多
This paper considers the Cauchy problem with a kind of non-smooth initial data for general inhomogeneous quasilinear hyperbolic systems with characteristics with constant multiplicity. Under the matching condition, ba...This paper considers the Cauchy problem with a kind of non-smooth initial data for general inhomogeneous quasilinear hyperbolic systems with characteristics with constant multiplicity. Under the matching condition, based on the refined fomulas on the decomposition of waves, we obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution to the Cauchy problem.展开更多
In this paper,we consider nonnegative classical solutions of a Quasi-linear reaction-diffusion system with nonlinear boundary conditions.We prove the uniqueness of a nonnegative classical solution to this problem.
In this paper, astochastic predator-prey systems with nonlinear harvesting and impulsive effect are investigated. Firstly, we show the existence and uniqueness of the global positive solution of the system. Secondly, ...In this paper, astochastic predator-prey systems with nonlinear harvesting and impulsive effect are investigated. Firstly, we show the existence and uniqueness of the global positive solution of the system. Secondly, by constructing appropriate Lyapunov function and using comparison theorem with an impulsive differential equation, we study that a positive periodic solution exists. Thirdly, we prove that system is globally attractive. Finally, numerical simulations are presented to show the feasibility of the obtained results.展开更多
This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester ma...This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches. Keywords Second-order linear systems - Eigenstructure assignment - Proportional plus derivative feedback - Parametric solution - Singular value decompoition - Right factorization This work was supported in part by the Chinese Outstanding Youth Foundation (No.69504002).展开更多
In this article, the notion of pinning control for directed networks of dynamical systems is introduced, where the nodes could be either single-input single-output （SISO） or multi-input multi-output （MIMO） dynamic...In this article, the notion of pinning control for directed networks of dynamical systems is introduced, where the nodes could be either single-input single-output （SISO） or multi-input multi-output （MIMO） dynamical systems, and could be non-identical and nonlinear in general but will be specified to be identical linear time-invariant （LTI） systems here in the study of network controllability. Both state and structural controllability problems will be discussed, illustrating how the network topology, node-system dynamics, external control inputs and inner dynamical interactions altogether affect the controllability of a general complex network of LTI systems, with necessary and sufficient conditions presented for both SISO and MIMO settings. To that end, the controllability of a special temporally switching directed network of linear time-varying （LTV） node systems will be addressed, leaving some more general networks and challenging issues to the end for research outlook.展开更多
A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provi...A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX-XF = BY .展开更多
This paper presents the theory and applications of a new computational technique referred to as Differential Transform Method (DTM) for solving second order linear ordinary differential equations, for both homogeneous...This paper presents the theory and applications of a new computational technique referred to as Differential Transform Method (DTM) for solving second order linear ordinary differential equations, for both homogeneous and nonhomogeneous cases. For the robustness and efficiency of the method, four examples are considered. The results indicate that the DTM is reliable and accurate when compared to the exact solutions of the solved problems.展开更多
Modern science has solved the systematic dynamic problems involving complexity such as complex systems by logical solution algebraically such as the old chaos theory derived from algebraic statistics based on determin...Modern science has solved the systematic dynamic problems involving complexity such as complex systems by logical solution algebraically such as the old chaos theory derived from algebraic statistics based on determinism. It is very vague and difficult for us. Moreover, modern science treats systems dynamic problems as logical static problems according to determinism. However, this study proposes a novel solution for systematic problems, it is accomplished by systems analysis theory based on a third science (indeterminism) as it is more precise and easier to understand compared to chaos theory. Furthermore, it is suitable for non-physicists, because it is supported by computer simulations without any mathematical processing. Thus, it can be solved especially regressive systems solution. Therefore, this provided an innovative scientific result to non-physicists for their unsolved systematic problems in modern science.展开更多
文摘Several approximate methods have been used to find approximate solutions of elliptic systems of first order equations. One common method is the Newton imbedding approach, i.e. the parameter extension method. In this article, we discuss approximate solutions to discontinuous Riemann-Hilbert boundary value problems, which have various applications in mechanics and physics. We first formulate the discontinuous Riemann-Hilbert problem for elliptic systems of first order complex equations in multiply connected domains and its modified well-posedness, then use the parameter extensional method to find approximate solutions to the modified boundary value problem for elliptic complex systems of first order equations, and then provide the error estimate of approximate solutions for the discontinuous boundary value problem.
基金Supported in part by the Basic Science and the Front Technology Research Foundation of Henan Province of China under Grant No.092300410179the Doctoral Scientific Research Foundation of Henan University of Science and Technology under Grant No.09001204
文摘The discrete complex cubic Ginzburg-Landau equation is an important model to describe a number of physical systems such as Taylor and frustrated vortices in hydrodynamics and semiconductor laser arrays in optics. In this paper, the exact solutions of the discrete complex cubic Ginzburg-Landau equation are derived using homogeneous balance principle and the GI/G-expansion method, and the linear stability of exact solutions is discussed.
文摘In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x） ｜ t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.
文摘In this paper, we propose DQMR algorithm for the Drazin-inverse solution of consistent or inconsistent linear systems of the form Ax=b where is a singular and in general non-hermitian matrix that has an arbitrary index. DQMR algorithm for singular systems is analogous to QMR algorithm for non-singular systems. We compare this algorithm with DGMRES by numerical experiments.
文摘In this paper, we consider an almost periodic system which includes a system of the type , where k is a positive integer, aij are almost periodic in n and satisfy aij(n)≥0 for i≠j,? for 1≤j≤m. In the special case where aij(n) are constant functions, above system is a mathematical model of gas dynamics and was treated by T. Carleman and R. D. Jenks for differential systems. In the main theorem, we show that if the m X m matrix (aij(n)) is irreducible, then there exists a positive almost periodic solution which is unique and has some stability. Moreover, we can see that this result gives R. D. Jenks’ result for differential model in the case where aij(n) are constant functions. In Section 3, we consider the linear system with variable cofficients . Even in nonlinear problems, this linear system plays an important role, as their variational equations, and it is requested to determine the uniform asymptotically stability of the zero solution from the information about A(n). In order to obtain the existence of almost periodic solutions of both linear and nonlinear almost periodic discrete systems: above linear system and? for 1≤i≤m, respectively, we shall consider between certain stability properties, which are referred to as uniformly asymptotically stable, and the diagonal dominance matrix condition.
基金Supported by the National Natural Science Foundation of China(No.11761035)the Natural Science Foundation of Jiangxi Province in China(No.20171BAB201002)
文摘In this paper,we investigate the growth of meromorphic solutions of some kind of non-homogeneous linear difference equations with special meromorphic coefficients.When there are more than one coefficient having the same maximal order and the same maximal type,the estimates on the lower bound of the order of meromorphic solutions of the involved equations are obtained.Meanwhile,the above estimates are sharpened by combining the relative results of the corresponding homogeneous linear difference equations.
文摘This work seeks to describe intra-solution particle movement system. It makes use of data obtained from simulations of patients on efavirenz. A system of ordinary differential equations is used to model movement state at some particular concentration. The movement states’ description is found for the primary and secondary level. The primary system is found to be predominantly an unstable system while the secondary system is stable. This is derived from the state of dynamic eigenvalues associated with the system. The saturated solution-particle is projected to be stable both for the primary potential and secondary state. A volume conserving linear system has been suggested to describe the dynamical state of movement of a solution particle.
文摘An exact solution of a linear difference equation in a finite number of steps has been obtained. This refutes the conventional wisdom that a simple iterative method for solving a system of linear algebraic equations is approximate. The nilpotency of the iteration matrix is the necessary and sufficient condition for getting an exact solution. The examples of iterative equations providing an exact solution to the simplest algebraic system are presented.
基金Supported by the National Natural Science Foundation of China(10962008)
文摘In this paper, we provide a generalized block-by-block method for constructing block-by-block systems to solve the system of linear Volterra integral equations of the second kind, and then deduce some of the special cases. Compared with the expansion method and He＇s homotopy perturbation method, respectively numerical examples are given to certify the effectiveness of the method. The results show that the block-by-block method is very effective, simple, and of high accuracy in solving the system of linear Volterra integral equations of the second kind.
基金Project supported by the National Natural Science Foundation of China(Grant No.11375079)the Natural Science Foundation of Zhejiang Province,China(Grant Nos.Y6100257 and Y6110140)
文摘With the help of the symbolic computation system Maple, the Riccati equation mapping approach and a linear variable separation approach, a new family of complex solutions for the （2＋ 1）-dimensional Boiti-Leon-Pempinelli system （BLP） is derived. Based on the derived solitary wave solution, some novel complex wave localized excitations are obtained.
文摘There are given sufficient conditions for the ultimate boundedness of solutions and for the existence of periodic solutions of a certain vector differential equation of third-order.
基金supported by the National Natural Science Foundation of China(Grant No.11375079)the Scientific Research Fund of Zhejiang Provincial Education Department of China(Grant No.Y 201120994)the Natural Science Foundation of Zhejiang Province,China(Grant Nos.Y6100257,LY14A010005,and Y6110140)
文摘With the help of the symbolic computation system, Maple and Riccati equation （ξ＇ = ao ＋ a1ξ＋ a2ξ2）, expansion method, and a linear variable separation approach, a new family of exact solutions with q = lx ＋ my ＋ nt ＋ Г（x,y, t） for the （2＋1）-dimensional generalized Calogero-Bogoyavlenskii-Schiff system （GCBS） are derived. Based on the derived solitary wave solution, some novel localized excitations such as fusion, fission, and annihilation of complex waves are investigated.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 11071141 11271192)+4 种基金China Postdoctoral Science Foundation (Grant No. 20100481161)the Postdoctoral Foundation of Jiangsu Province (GrantNo. 1001042C)Qing Lan Project of Jiangsu Provincethe Natural Science Foundation of the Jiangsu Higher Education Committee of China (Grant No. 11KJA110001)the Natural Science Foundation of Jiangsu Provience (Grant No. BK2011777)
文摘This paper considers the Cauchy problem with a kind of non-smooth initial data for general inhomogeneous quasilinear hyperbolic systems with characteristics with constant multiplicity. Under the matching condition, based on the refined fomulas on the decomposition of waves, we obtain a necessary and sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution to the Cauchy problem.
基金Supported by the National Natural Science Foundation of China(90410011)
文摘In this paper,we consider nonnegative classical solutions of a Quasi-linear reaction-diffusion system with nonlinear boundary conditions.We prove the uniqueness of a nonnegative classical solution to this problem.
文摘In this paper, astochastic predator-prey systems with nonlinear harvesting and impulsive effect are investigated. Firstly, we show the existence and uniqueness of the global positive solution of the system. Secondly, by constructing appropriate Lyapunov function and using comparison theorem with an impulsive differential equation, we study that a positive periodic solution exists. Thirdly, we prove that system is globally attractive. Finally, numerical simulations are presented to show the feasibility of the obtained results.
文摘This paper considers eigenstructure assignment in second-order linear systems via proportional plus derivative feedback. It is shown that the problem is closely related to a type of so-called second-order Sylvester matrix equations. Through establishing two general parametric solutions to this type of matrix equations, two complete parametric methods for the proposed eigenstructure assignment problem are presented. Both methods give simple complete parametric expressions for the feedback gains and the closed-loop eigenvector matrices. The first one mainly depends on a series of singular value decompositions, and is thus numerically simple and reliable; the second one utilizes the right factorization of the system, and allows the closed-loop eigenvalues to be set undetermined and sought via certain optimization procedures. An example shows the effectiveness of the proposed approaches. Keywords Second-order linear systems - Eigenstructure assignment - Proportional plus derivative feedback - Parametric solution - Singular value decompoition - Right factorization This work was supported in part by the Chinese Outstanding Youth Foundation (No.69504002).
文摘In this article, the notion of pinning control for directed networks of dynamical systems is introduced, where the nodes could be either single-input single-output （SISO） or multi-input multi-output （MIMO） dynamical systems, and could be non-identical and nonlinear in general but will be specified to be identical linear time-invariant （LTI） systems here in the study of network controllability. Both state and structural controllability problems will be discussed, illustrating how the network topology, node-system dynamics, external control inputs and inner dynamical interactions altogether affect the controllability of a general complex network of LTI systems, with necessary and sufficient conditions presented for both SISO and MIMO settings. To that end, the controllability of a special temporally switching directed network of linear time-varying （LTV） node systems will be addressed, leaving some more general networks and challenging issues to the end for research outlook.
基金supported by the Major Program of National Nat-ural Science Foundation of China (No. 60710002) Program for Changjiang Scholars and Innovative Research Team in University
文摘A closed-form solution to the linear matrix equation AX-EXF = BY with X and Y unknown and matrix F being in a companion form is proposed, and two equivalent forms of this solution are also presented. The results provide great convenience to the computation and analysis of the solutions to this class of equations, and can perform important functions in many analysis and design problems in descriptor system theory. The results proposed here are parallel to and more general than our early work about the linear matrix equation AX-XF = BY .
文摘This paper presents the theory and applications of a new computational technique referred to as Differential Transform Method (DTM) for solving second order linear ordinary differential equations, for both homogeneous and nonhomogeneous cases. For the robustness and efficiency of the method, four examples are considered. The results indicate that the DTM is reliable and accurate when compared to the exact solutions of the solved problems.
文摘Modern science has solved the systematic dynamic problems involving complexity such as complex systems by logical solution algebraically such as the old chaos theory derived from algebraic statistics based on determinism. It is very vague and difficult for us. Moreover, modern science treats systems dynamic problems as logical static problems according to determinism. However, this study proposes a novel solution for systematic problems, it is accomplished by systems analysis theory based on a third science (indeterminism) as it is more precise and easier to understand compared to chaos theory. Furthermore, it is suitable for non-physicists, because it is supported by computer simulations without any mathematical processing. Thus, it can be solved especially regressive systems solution. Therefore, this provided an innovative scientific result to non-physicists for their unsolved systematic problems in modern science.
