Non-periodic averaging principles for measure functional differential equations and functional dynamic equations on time scales involving impulses

We present a non-periodic averaging principle for measure functional differential equations and, using the correspondence between solutions of measure functional differential equations and solutions of functional dynamic equations on time scales (see Federson et al., 2012 [8]), we obtain a non-periodic averaging result for functional dynamic equations on time scales. Moreover, using the relation between measure functional differential equations and impulsive measure functional differential equations, we get a non-periodic averaging theorem for these equations. Also, it is a known fact that we can relate impulsive measure functional differential equations and impulsive functional dynamic equations on time scales (see Federson et al., 2013 [9]). Therefore, applying this correspondence to our averaging principle, we obtain a non-periodic averaging theorem for impulsive functional dynamic equations on time scales.     

Propagation of socillations in the Solutions of 1-D Compressible Fluid Equations

We study the propagation of initial osillations in the solutions of one-dimensional inviscid gas dynamic equations and the compressible Navier-Stokes equations. Using Multiple scale analysis, we derivbe the homogenized equations which take the form of n averaged system coupled with a dynamic cell-problem. We prove rigorous error estimates to justify the validity of these equations. We also show that the weak limits of the osicllatory solytions satisfy gas dynamic equations with an equation of state depeding on the microstructurer of the inital data     

Dynamic equations on time scales and generalized ordinary differential equations

The aim of this paper is to show that dynamic equations on time scales can be treated in the framework of generalized ordinary differential equations as introduced by J. Kurzweil. We also use some known results for generalized ordinary differential equations to obtain new theorems related to stability and continuous dependence on parameters for dynamic equations on time scales.     

Oscillation of even order nonlinear delay dynamic equations on time scales

One of the important methods for studying the oscillation of higher order differential equations is to make a comparison with second order differential equations. The method involves using Taylor’s Formula. In this paper we show how such a method can be used for a class of even order delay dynamic equations on time scales via comparison with second order dynamic inequalities. In particular, it is shown that nonexistence of an eventually positive solution of a certain second order delay dynamic inequality is sufficient for oscillation of even order dynamic equations on time scales. The arguments are based on Taylor monomials on time scales.     

Classroom note: Application of the Sumudu transform to discrete dynamic systems

The Sumudu transform is an integral transform introduced to solve differential equations and control engineering problems. The transform possesses many interesting properties that make visualization easier and application has been demonstrated in the solution of partial differential equations, integral equations, integro-differential equations and dynamic/dynamic-differential systems. In this note, application of the method of Sumudu transform to the solution of discrete dynamic systems is described. Derivations find ready application in recurrence relations. Solutions are obtained to certain first degree, second degree and third degree dynamic systems.     

Asymptotic Behavior of Solutions of Dynamic Equations

We consider linear dynamic systems on time scales, which contain as special cases linear differential systems, difference systems, or other dynamic systems. We give an asymptotic representation for a fundamental solution matrix that reduces the study of systems in the sense of asymptotic behavior to the study of scalar dynamic equations. In order to understand the asymptotic behavior of solutions of scalar linear dynamic equations on time scales, we also investigate the behavior of solutions of the simplest types of such scalar equations, which are natural generalizations of the usual exponential function.     

ON THE BOUNDEDNESS OF A CLASS OF NONLINEAR DYNAMIC EQUATIONS OF THE THIRD ORDER

In this paper, a modified nonlinear dynamic inequality on time scales is used to study the boundedness of a class of nonlinear third-order dynamic equations on time scales. These theorems contain as special cases results for dynamic differential equations, difference equations and $q$-difference equations.     

复合材料选层板的非线性动力稳定性理论

本文用Hamilton原理建立了复合材料迭层板动力稳定性的一般方程.用此方程可以考虑横向剪切变形、板的初始几何不完善性、纵向惯性力和转动惯性力、以及纤维的铺设角度等各种因素对于迭层板的动力稳定性的影响.所求得的基本方程的解表明迭层板的一些重要的动力失稳特征只有通过对这些因素的考虑和分析才能够得出.     

Delay dynamic equations on time scales

This article is concerned with delay dynamic equations on time scales. Linear and nonlinear delay dynamic equations are discussed. By using a different theorem to that used S. Hilger, Analysis on measure chains. A unified approach to continous and discrete calculus. Results Math. 18 (1990), pp. 18–56], some criteria for the existence, uniqueness and continuous dependence of the solution for the nonlinear delay dynamic equations are established.     

Periodic solutions of functional dynamic equations with infinite delay

In this paper, sufficient criteria are established for the existence of periodic solutions of some functional dynamic equations with infinite delays on time scales, which generalize and incorporate as special cases many known results for differential equations and for difference equations when the time scale is the set of the real numbers or the integers, respectively. The approach is mainly based on the Krasnosel’ski? fixed point theorem, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in studying dynamic equations on time scales. This study shows that one can unify such existence studies in the sense of dynamic equations on general time scales.     

Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments

We study oscillatory behavior of solutions to a class of second-order half-linear dynamic equations with deviating arguments under the assumptions that allow applications to dynamic equations with delayed and advanced arguments. Several improved Fite–Hille–Wintner-type criteria are obtained that do not need some restrictive assumptions required in related results. Illustrative examples and conclusions are presented to show that these criteria are sharp for differential equations and provide sharper estimates for oscillation of corresponding $q$-difference equations.     

ASYMPTOTIC ANALYSIS OF DYNAMIC PROBLEMS FOR LINEARLY ELASTIC SHELLS—— JUSTIFICATION OF EQUATIONS FOR DYNAMIC KOITER SHELLS

91. IntroductionIn this paper, or,g,a,r,' take their values in the set {1, 2}, i, i, k, l,' take their valuesin the set {1, 2, 3}.In [1] and 12] under certain conditions, starting from the three-dimensional dynamicequations Of elastic shells we have given the justifications of dynamic equations of membraneshells and fie-cural shells respectively. In this paper, we shall show that, starting from thedynamic equations of KOiter shells, we can alsO get the dylltalc equstions of membraneshell…     

Existence of solutions to second-order BVPs on time scales

We consider a boundary value problem (BVP) for systems of second-order dynamic equations on time scales. Using methods involving dynamic inequalities, we formulate conditions under which all solutions to a certain family of systems of dynamic equations satisfy certain a priori bounds. These results are then applied to guarantee the existence of solutions to BVPs for systems of dynamic equations on time scales.     

Dynamic Systems Admitting the Normal Shift and Wave Equations

We consider wave equations on Riemannian manifolds and investigate wave front dynamics in the semiclassical approximation. The problem of finding wave equations whose wave front dynamics is described by Newtonian dynamic systems admitting the normal shift is solved. A subclass of these dynamic systems that can be defined by modified Lagrange and Hamilton equations is described explicitly.     

Phase spaces and periodic solutions of set functional dynamic equations with infinite delay

Very recently, a new theory known as set dynamic equations on time scales has been built. In this paper, a phase space is built for set functional dynamic equations with infinite delay on time scales and sufficient criteria are established for the existence of periodic solutions of such equations, which generalize and incorporate as special cases some known results for set differential equations and for set difference equations when the time scale is the real number set or the integer set, respectively, moreover, for differential inclusions and difference inclusions if the variable under consideration is a single valued mapping. Our results show that one can unify the study of some continuous or discrete problems in the sense of (set) dynamic equations on general time scales.     

The symmetry approach on economic systems

This paper studies the symmetry on the dynamic equations of economic systems. Symmetry group transforms a solution of the equations to other solutions. Thus we can understand the transformational relations among these solutions. Symmetry describes the invariance of the dynamic equations. Then the symmetry provides more information for the inner structure of the economic systems. The complexity of the socioeconomic phenomena is discussed by using dynamic systems with symmetry. For the calculation of the symmetry, the prolongation technique is adopted. This method reformulates the basic problem of finding solutions of differential equations in a more geometrical form which is suited to the investigation of symmetry groups. Some examples of dynamic equations by using symmetry analysis are also illustrated.     

Measure functional differential equations and functional dynamic equations on time scales

We study measure functional differential equations and clarify their relation to generalized ordinary differential equations. We show that functional dynamic equations on time scales represent a special case of measure functional differential equations. For both types of equations, we obtain results on the existence and uniqueness of solutions, continuous dependence, and periodic averaging.     

Modelling of closed-chain manipulators on an excavator vehicle

The main focus of this paper is to develop a physics-based model for a closed-chain manipulator in an excavator vehicle. The derivation of closed-chain manipulator dynamic equations with a structure similar to open-chain manipulator equations is an important research problem, particularly with reference to controller design. In this paper, an approach for deriving closed-chain manipulator equations with an open-chain structure, based on trigonometric t-formulae, is presented. Holonomic loop closure constraints are employed in order to derive the closed-chain mechanism dynamics from the reduced system dynamics. The closed-chain equations, with a structure similar to serial link equations, are presented. The model incorporates the dynamic properties of the manipulator and bucket. The dynamic model for the excavation system is validated against measured data obtained from a full-scale closed-chain excavator vehicle. A dynamic model is important for the design of control strategies for trajectory tracking, a key requirement for automating the excavation task. It is noted that even though the results presented in this paper are focused on a particular excavator vehicle, the research is generic and can be adapted to any closed-chain manipulator.     

时标上一类三阶中立型动力方程的振动准则

利用广义Riccati代换给出了时标上一类三阶中立型动力方程的振动准则,其结果不仅得到了几个振动解存在的条件,也拓展了一些已有的结论,在最后还给出了几个例子来验证结论.