In this paper,we mainly investigate the dynamical properties of entire solutions of complex differential equations.With some conditions on coefficients,we prove that the set of common limiting directions of Julia sets...In this paper,we mainly investigate the dynamical properties of entire solutions of complex differential equations.With some conditions on coefficients,we prove that the set of common limiting directions of Julia sets of solutions,their derivatives and their primitives must have a definite range of measure.展开更多
In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and pr...In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.展开更多
In order to intercept the future targets that are characterized by high maneuverability, multiple interceptors may be launched and aimed at single target. The scenario of two missiles P and Q intercepting a single tar...In order to intercept the future targets that are characterized by high maneuverability, multiple interceptors may be launched and aimed at single target. The scenario of two missiles P and Q intercepting a single target is modeled as a two-pursuit single-evader non-zero-sum linear quadratic differential game. The intercept space is decomposed into three subspaces which are mutually disjoint and their union covers the entire intercept space. The effect of adding the second interceptor arises in the intercept space of both P and Q （PQ-intercept space）. A guidance law is derived from the Nash equilibrium strategy set （NESS） of the game. Simulation studies are focused on the PQ-intercept space. It is indicated that 1） increasing the target＇s maneuverability will enlarge PQ-intercept space; 2） the handover conditions will be released if the initial zero-effort-miss （ZEM） of both interceptors has opposite sign; 3） overvaluation of the target＇s maneuverability by choosing a small weight coefficient will generate robust performance with respect to the target maneuvering command switch time and decrease the fuel requirement; and 4） cooperation between interceptors increases the interception probability.展开更多
In this article, the authors study the growth of certain second order linear differential equation f″＋A（z）f′＋B（z）f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar ...In this article, the authors study the growth of certain second order linear differential equation f″＋A（z）f′＋B（z）f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.展开更多
In this paper, we investigate the growth of solutions of higher order linear differential equations with meromorphic coefficients. Under certain conditions, we obtain precise estimation of growth order and hyper-order...In this paper, we investigate the growth of solutions of higher order linear differential equations with meromorphic coefficients. Under certain conditions, we obtain precise estimation of growth order and hyper-order of solutions of the equation.展开更多
This paper studies a bounded discriminating domain for hybrid linear differential game with two players and two targets using viability theory. First of all, we prove that the convex hull of a closed set is also a dis...This paper studies a bounded discriminating domain for hybrid linear differential game with two players and two targets using viability theory. First of all, we prove that the convex hull of a closed set is also a discriminating domain if the set is a discriminating domain. Secondly, in order to determine that a bounded polyhedron is a discriminating domain, we give a result that it only needs to verify that the extreme points of the polyhedron meet the viability conditions. The difference between our result and the existing ones is that our result just needs to verify the finite points （extreme points） and the existing ones need to verify all points in the bounded polyhedron.展开更多
The measurements and analysis of deformation of engineering structures such as dams, bridges and high-rise buildings are important tasks for civil engineers. It is evident that, all civil engineering structures are su...The measurements and analysis of deformation of engineering structures such as dams, bridges and high-rise buildings are important tasks for civil engineers. It is evident that, all civil engineering structures are susceptible for deterioration over a period of time. Bridges in particular, deteriorate due to loading conditions, environmental changes, earth movement, material used during construction, age and corrosion of steel. Continuous monitoring of such structure is the most important aspect as it provides quantitative information, assesses the state of the structure, detects unsafe positions and proposes early safety measures to be taken before it can threaten the safety of vehicles, goods and human life. Despite government’s efforts to construct roads and highways, bridge deformation monitoring has not been given priority in most of African countries and ultimately causes some bridges to collapse unexpectedly. The purpose of this research is to integrate Global Positioning System (GPS) and Linear Variable Differential Transducers (LVDT) to monitor deformation of a bridge. The horizontal positions of reference and monitoring points were determined using Global Positioning System (GPS) while the vertical deflections, accelerations and strain were determined using Linear Variable Differential Transducers (LVDT). The maximum displacements obtained between zero and first epochs in x, y and z components were 0.798 m, at point LT08, 0.865 m at point BR13, and 0.56 m at point LT02 respectively. The maximum deflections for LVDT 1, 2 and 3 are 28.563 mm, 31.883 mm and 40.926 mm respectively. Finally, the correlation coefficient for the observations was 0.679 with standard deviations of 0.0168 and 0.0254 in x and y respectively. Our results identified some slight displacements in horizontal components at the bridge.展开更多
The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory ...The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.展开更多
In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fraction...In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the（3＋1）-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.展开更多
In this paper, some properties of solutions of linear differential equations f^（k）＋A（z）f = 0 and f（k）＋ A（z）f = F（z） are discussed. Our results are a generalization of the original results.
A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forc...A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.展开更多
This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect...This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.展开更多
In this paper, the principle techinique of the differentiator method, and some examples using the method to obtain the general solution and special solution of the differential equation are introduced. The essential d...In this paper, the principle techinique of the differentiator method, and some examples using the method to obtain the general solution and special solution of the differential equation are introduced. The essential difference between this method and the others is that by this method special and general solutions can be obtained directly with the operations of the differentor in the differential equation and without the enlightenment of other scientific knowledge.展开更多
In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part,...In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.展开更多
The aim of this paper is to consider the following two problems:(1) Establish interrelationships between the growth of coefficients and the geometric distribution of zeros of solutions of non-homogeneous linear differ...The aim of this paper is to consider the following two problems:(1) Establish interrelationships between the growth of coefficients and the geometric distribution of zeros of solutions of non-homogeneous linear differential equation f′"+A_2(z)f"+A_1(z)f′+A_0(z)f=A_3(z)where A(z),..., A(z) are analytic functions in the unit disc D;(2) Find some sufficient conditions on the analytic coefficients of the differential equation for all solutions to belong to the Zygmund type space.The f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+A_0(z)f=0 results we obtain are a generalization of some earlier results by Heittokangas, Gröhn,Korhoneon and Rättyä.展开更多
Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian ...Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.展开更多
This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the ps...This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given.展开更多
In this paper, the authors investigate the zeros and growth of solutions of second order linear differential equations with entire coefficients of [p, q]-order and obtain some results which improve and generalize some...In this paper, the authors investigate the zeros and growth of solutions of second order linear differential equations with entire coefficients of [p, q]-order and obtain some results which improve and generalize some previous results.展开更多
In this paper a generalized version of the classical Hardy-Littlewood-Polya inequality is given.Furthermore,the Stechkin's problem for a linear differential operator is solved in L_2(R), and the optimal recovery p...In this paper a generalized version of the classical Hardy-Littlewood-Polya inequality is given.Furthermore,the Stechkin's problem for a linear differential operator is solved in L_2(R), and the optimal recovery problem for such differential operator is considered.展开更多
In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the origi...In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.展开更多
基金supported by Shanghai Center for Mathematical Sci-ences,China Scholarship Council(201206105015)National Science Foundation of China(11171119,11001057,11571049)Natural Science Foundation of Guangdong Province in China(2014A030313422)
文摘In this paper,we mainly investigate the dynamical properties of entire solutions of complex differential equations.With some conditions on coefficients,we prove that the set of common limiting directions of Julia sets of solutions,their derivatives and their primitives must have a definite range of measure.
基金supported by Shanghai Center for Mathematical Science China Scholarship Council(201206105015)the National Science Foundation of China(11171119,11001057,11571049)the Natural Science Foundation of Guangdong Province in China(2014A030313422)
文摘In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.
文摘In order to intercept the future targets that are characterized by high maneuverability, multiple interceptors may be launched and aimed at single target. The scenario of two missiles P and Q intercepting a single target is modeled as a two-pursuit single-evader non-zero-sum linear quadratic differential game. The intercept space is decomposed into three subspaces which are mutually disjoint and their union covers the entire intercept space. The effect of adding the second interceptor arises in the intercept space of both P and Q （PQ-intercept space）. A guidance law is derived from the Nash equilibrium strategy set （NESS） of the game. Simulation studies are focused on the PQ-intercept space. It is indicated that 1） increasing the target＇s maneuverability will enlarge PQ-intercept space; 2） the handover conditions will be released if the initial zero-effort-miss （ZEM） of both interceptors has opposite sign; 3） overvaluation of the target＇s maneuverability by choosing a small weight coefficient will generate robust performance with respect to the target maneuvering command switch time and decrease the fuel requirement; and 4） cooperation between interceptors increases the interception probability.
基金the National Natural Science Foundation of China(10161006,10571044)the Natural Science Foundation of Guangdong Prov(06025059)
文摘In this article, the authors study the growth of certain second order linear differential equation f″＋A（z）f′＋B（z）f=0 and give precise estimates for the hyperorder of solutions of infinite order. Under similar conditions, higher order differential equations will be considered.
文摘In this paper, we investigate the growth of solutions of higher order linear differential equations with meromorphic coefficients. Under certain conditions, we obtain precise estimation of growth order and hyper-order of solutions of the equation.
基金supported by National Science Foundation of China(11171221)Doctoral Program Foundation of Institutions of Higher Education of China(20123120110004)+2 种基金Natural Science Foundation of Shanghai(14ZR1429200)Innovation Program of Shanghai Municipal Education Commission(15ZZ073)Key Research Project Plan of Institutions of Higher of Henan Province(17A120010)
文摘This paper studies a bounded discriminating domain for hybrid linear differential game with two players and two targets using viability theory. First of all, we prove that the convex hull of a closed set is also a discriminating domain if the set is a discriminating domain. Secondly, in order to determine that a bounded polyhedron is a discriminating domain, we give a result that it only needs to verify that the extreme points of the polyhedron meet the viability conditions. The difference between our result and the existing ones is that our result just needs to verify the finite points （extreme points） and the existing ones need to verify all points in the bounded polyhedron.
文摘The measurements and analysis of deformation of engineering structures such as dams, bridges and high-rise buildings are important tasks for civil engineers. It is evident that, all civil engineering structures are susceptible for deterioration over a period of time. Bridges in particular, deteriorate due to loading conditions, environmental changes, earth movement, material used during construction, age and corrosion of steel. Continuous monitoring of such structure is the most important aspect as it provides quantitative information, assesses the state of the structure, detects unsafe positions and proposes early safety measures to be taken before it can threaten the safety of vehicles, goods and human life. Despite government’s efforts to construct roads and highways, bridge deformation monitoring has not been given priority in most of African countries and ultimately causes some bridges to collapse unexpectedly. The purpose of this research is to integrate Global Positioning System (GPS) and Linear Variable Differential Transducers (LVDT) to monitor deformation of a bridge. The horizontal positions of reference and monitoring points were determined using Global Positioning System (GPS) while the vertical deflections, accelerations and strain were determined using Linear Variable Differential Transducers (LVDT). The maximum displacements obtained between zero and first epochs in x, y and z components were 0.798 m, at point LT08, 0.865 m at point BR13, and 0.56 m at point LT02 respectively. The maximum deflections for LVDT 1, 2 and 3 are 28.563 mm, 31.883 mm and 40.926 mm respectively. Finally, the correlation coefficient for the observations was 0.679 with standard deviations of 0.0168 and 0.0254 in x and y respectively. Our results identified some slight displacements in horizontal components at the bridge.
基金Supported by the National Natural Science Foundation of China(11101096 )Guangdong Natural Science Foundation (S2012010010376, S201204006711)
文摘The main purpose of this article is to study the existence theories of global meromorphic solutions for some second-order linear differential equations with meromorphic coefficients, which perfect the solution theory of such equations.
文摘In this paper, the fractional auxiliary sub-equation expansion method is proposed to solve nonlinear fractional differential equations. To illustrate the effectiveness of the method, we discuss the space-time fractional Kd V equation, the space-time fractional RLW equation, the space-time fractional Boussinesq equation, and the（3＋1）-spacetime fractional ZK equation. The solutions are expressed in terms of fractional hyperbolic and fractional trigonometric functions. These solutions are useful to understand the mechanisms of the complicated nonlinear physical phenomena and fractional differential equations. Among these solutions, some are found for the first time. The analytical solution of homogenous linear FDEs with constant coefficients are obtained by using the series and the Mittag–Leffler function methods. The obtained results recover the well-know solutions when α = 1.
基金Supported by the National Natural Science Foundation of China(Grant Nos.1130123211171119)+1 种基金the Youth Science Foundation of Education Bureau of Jiangxi Province(Grant No.GJJ12207)the Natural Science Foundation of Jiangxi Province(Grant No.20132BAB211009)
文摘In this paper, some properties of solutions of linear differential equations f^（k）＋A（z）f = 0 and f（k）＋ A（z）f = F（z） are discussed. Our results are a generalization of the original results.
基金supported by the Natural Science Basic Research Plan in Shaanxi Province of China(2013JM1022)the Fundamental Research Funds for the Central Universities(K50511700004)
文摘A discrete differential evolution algorithm combined with the branch and bound method is developed to solve the integer linear bilevel programming problems, in which both upper level and lower level variables are forced to be integer. An integer coding for upper level variables is adopted, and then a discrete differential evolution algorithm with an improved feasibility-based comparison is developed to directly explore the integer solution at the upper level. For a given upper level integer variable, the lower level integer programming problem is solved by the existing branch and bound algorithm to obtain the optimal integer solution at the lower level. In the same framework of the algorithm, two other constraint handling methods, i.e. the penalty function method and the feasibility-based comparison method are also tested. The experimental results demonstrate that the discrete differential evolution algorithm with different constraint handling methods is effective in finding the global optimal integer solutions, but the improved constraint handling method performs better than two compared constraint handling methods.
基金supported by the National Natural Science Foundation of China (11101096)
文摘This article discusses the problems on the existence of meromorphic solutions of some higher order linear differential equations with meromorphic coefficients. Some nice results are obtained. And these results perfect the complex oscillation theory of meromorphic solutions of linear differential equations.
文摘In this paper, the principle techinique of the differentiator method, and some examples using the method to obtain the general solution and special solution of the differential equation are introduced. The essential difference between this method and the others is that by this method special and general solutions can be obtained directly with the operations of the differentor in the differential equation and without the enlightenment of other scientific knowledge.
基金Provincial Science and Technology Foundation of Guizhou
文摘In this paper, we give some sufficient conditions of the instability for the fourth order linear differential equation with varied coefficient, at least one of the characteristic roots of which has positive real part, by means of Liapunov's second method.
基金National Natural Science Foundation of China(Grant No.11661043)。
文摘The aim of this paper is to consider the following two problems:(1) Establish interrelationships between the growth of coefficients and the geometric distribution of zeros of solutions of non-homogeneous linear differential equation f′"+A_2(z)f"+A_1(z)f′+A_0(z)f=A_3(z)where A(z),..., A(z) are analytic functions in the unit disc D;(2) Find some sufficient conditions on the analytic coefficients of the differential equation for all solutions to belong to the Zygmund type space.The f^{(k)}+A_{k-1}(z)f^{(k-1)}+\cdots+A_1(z)f'+A_0(z)f=0 results we obtain are a generalization of some earlier results by Heittokangas, Gröhn,Korhoneon and Rättyä.
基金The supports of the National Natural Science Foundation of China(Grant Nos.51725804 and U1711264)the Research Fund for State Key Laboratories of Ministry of Science and Technology of China(SLDRCE19-B-23)the Shanghai Post-Doctoral Excellence Program(2022558)。
文摘Stochastic fractional differential systems are important and useful in the mathematics,physics,and engineering fields.However,the determination of their probabilistic responses is difficult due to their non-Markovian property.The recently developed globally-evolving-based generalized density evolution equation(GE-GDEE),which is a unified partial differential equation(PDE)governing the transient probability density function(PDF)of a generic path-continuous process,including non-Markovian ones,provides a feasible tool to solve this problem.In the paper,the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established.In particular,it is proved that in the GE-GDEE corresponding to the state-quantities of interest,the intrinsic drift coefficient is a time-varying linear function,and can be analytically determined.In this sense,an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original highdimensional linear fractional differential system can be constructed such that their transient PDFs are identical.Specifically,for a multi-dimensional linear fractional differential system,if only one or two quantities are of interest,GE-GDEE is only in one or two dimensions,and the surrogate system would be a one-or two-dimensional linear integer-order system.Several examples are studied to assess the merit of the proposed method.Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems,the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian,and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.
文摘This note contains three main results.Firstly,a complete solution of the Linear Non-Homogeneous Matrix Differential Equations(LNHMDEs)is presented that takes into account both the non-zero initial conditions of the pseudo state and the nonzero initial conditions of the input.Secondly,in order to characterise the dynamics of the LNHMDEs correctly,some important concepts such as the state,slow state(smooth state)and fast state(impulsive state)are generalized to the LNHMDE case and the solution of the LNHMDEs is separated into the smooth(slow)response and the fast(implusive)response.As a third result,a new characterization of the impulsive free initial conditions of the LNHMDEs is given.
基金Supported by the Natural Science Foundation of Jiangxi Province(Grant Nos.20132BAB21100220122BAB211005)+1 种基金the Youth Foundation of Education Bureau of Jiangxi Province(Grant Nos.GJJ14271GJJ14272)
文摘In this paper, the authors investigate the zeros and growth of solutions of second order linear differential equations with entire coefficients of [p, q]-order and obtain some results which improve and generalize some previous results.
基金Supported by the National Fund of Natural Sciences.
文摘In this paper a generalized version of the classical Hardy-Littlewood-Polya inequality is given.Furthermore,the Stechkin's problem for a linear differential operator is solved in L_2(R), and the optimal recovery problem for such differential operator is considered.
文摘In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.
