Higher-order Dynamic Equations Research Articles

Oscillation and Nonoscillatory Criteria of Higher Order Dynamic Equations on Time Scales

In this paper, we consider two universal higher order dynamic equations with several delay functions. We will establish two oscillatory criteria of the first equation and a sufficient and necessary condition for the second equation with a nonoscillatory solution by employing fixed point theorem.

Study of nonlinear Pachpatte\u2019s inequalities on time scales

In this paper, we develop a fundamental dynamic inequality, a generalization of comparison theorem and reproduce the proofs of some nonlinear integral Pachpatte’s inequalities by using their continuous analogue. We also unify and extend these improved integral Pachpatte’s inequalities and their corresponding discrete analogues on arbitrary time scales. The results are used to make qualitative analysis of higher order dynamic equations.

Synthesis on the Acceleration Energies in the Advanced Mechanics of the Multibody Systems

The present paper’s objective is to highlight some new developments of the main author in the field of advanced dynamics of systems and higher order dynamic equations. These equations have been developed on the basis of the matrix exponentials which prove to have undeniable advantages in the matrix study of any complex mechanical system. The present paper proposes some new approaches, based on differential principles from analytical mechanics, by using some important dynamics notions, regarding the acceleration energies of the first, second and third order. This study extended the equations of the higher order, which provide the possibility of applying the initial motion conditions in the positions, velocities and accelerations of the first and second order. In order to determine the time variation laws for the generalized variables, the driving forces and acceleration energies of the higher order are applied by the time polynomial functions of the fifth order. According to inverse kinematics also named control kinematics of the robots, the applications of polynomial functions lead to the kinematic control functions of mechanical motions, especially the transitory motions. They influence the dynamic behavior of multibody systems, in which robot structures are included.

On asymptotic relationships between two higher order dynamic equations on time scales

We consider the n th order dynamic equations x Δ n + p 1 ( t ) x Δ n − 1 + ⋯ + p n ( t ) x = 0 and y Δ n + p 1 ( t ) y Δ n − 1 + ⋯ + p n ( t ) y = f ( t , y ( τ ( t ) ) ) on a time scale T , where τ is a composition of the forward jump operators, p i are real rd-continuous functions and f is a continuous function; T is assumed to be unbounded above. We establish conditions that guarantee asymptotic equivalence between some solutions of these equations. No restriction is placed on whether the solutions are oscillatory or nonoscillatory. Applications to second order Emden–Fowler type dynamic equations and Euler type dynamic equations are shown.

Lyapunov-type inequality for a higher order dynamic equation on time scales.

The purpose of this work is to establish a Lyapunov-type inequality for the following dynamic equation \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S_n^\ riangle (t,x(t))+u(t)x^{p}(t)=0$$\\end{document}Sn▵(t,x(t))+u(t)xp(t)=0on some time scale T under the anti-periodic boundary conditions S_k(a,x(a))+S_k(b,x(b))=0, (0le kle n-1), where S_0(t,x(t))=x(t), S_k(t,x(t))=a_k(t)S^triangle _{k-1}(t,x(t)) for 1le kle n-1 and S_n(t,x(t))=a_n(t)[S_{n-1}^Delta (t,x(t))]^p, a_kin C_{rd}({mathbf{T}},(-infty ,0)cup (0,infty )),(1le kle n) with a_n(a)=a_n(b) and uin C_{rd}({mathbf{T}}, {mathbf{R}}), p is the quotient of two odd positive integers and a,bin {mathbf{T}} with a<b.

Lyapunov Inequality For Dynamic Equation With Order n +1 On Time Scales

In this paper, we establish a Lyapunov-type inequality for the higher order dynamic equation on an arbitrary time scale 𝕋 where n is a positive integer, α ≥ 1 is the quotient of two odd positive integers and with ak ε Crd(𝕋, (0,∞)) (1 ≤ k≤ n) and p ε Crd(𝕋,ℝ).

Comparison tests for the asymptotic behaviour of higher-order dynamic equations of neutral type

Abstract In this paper, we study oscillation and asymptotic behaviour of higher-order neutral delay dynamic equations, and establish comparison with first-order delay dynamic equations. We also present some examples to show applicability and significance of the new results.

Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales

Abstract In this paper, we investigate the oscillation of the following higher-order dynamic equation: { r n ( t ) [ ( r n − 1 ( t ) ( ⋯ ( r 1 ( t ) x △ ( t ) ) △ ⋯ ) △ ) △ ] γ } △ + F ( t , x ( τ ( t ) ) ) = 0 on an arbitrary time scale T, where n ≥ 2 , 1 r k ( t ) ( 1 ≤ k ≤ n ) are positive rd-continuous functions on T, and γ is the quotient of two odd positive integers, τ : T → T with τ ( t ) &gt; t and F ∈ C ( T × R , R ) . We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero. MSC:34K11, 39A10, 39A99.

Oscillatory behaviour of a higher-order dynamic equation

In this paper we are concerned with the oscillation of solutions of a certain more general higher-order nonlinear neutral-type functional dynamic equation with oscillating coefficients. We obtain some sufficient criteria for oscillatory behaviour of its solutions.

Oscillation for Higher Order Dynamic Equations on Time Scales

We investigate the oscillation of the following higher order dynamic equation:{an(t)[(an-1(t)(⋯(a1(t)xΔ(t))Δ⋯)Δ)Δ]α}Δ+p(t)xβ(t)=0, on some time scaleT, wheren≥2,ak(t) (1≤k≤n)andp(t)are positive rd-continuous functions onTandα,βare the quotient of odd positive integers. We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero.

Asymptotic Behavior of Solutions of Higher-Order Dynamic Equations on Time Scales

We investigate the asymptotic behavior of solutions of the following higher-order dynamic equation , on an arbitrary time scale , where the function is defined on . We give sufficient conditions under which every solution of this equation satisfies one of the following conditions: (1) ; (2) there exist constants with , such that , where are as in Main Results.

3D time-domain regular grid infinite element in elastic foundation

The normal dynamic infinite element in elastic foundation cannot be used in time-domain due to it’s inclusive frequency term. A novel 3D regular grid infinite element is constructed to deal with the time-domain problem. This new infinite element method can easily transform the frequency terms of mass matrix and stiffness matrix to the terms in higher-order dynamic equation, thus a higher-order dynamic equilibrium equation is formed. Based on the second-order Wilson-θ dynamic equation, a new time-domain numeric formula of higher-order dynamic equation is deduced, and the time-domain calculation coupling with finite element and infinite element can be realized. The classic 3D fluctuation problem in elastic foundation is employed as an illustrative example to investigate the accuracy and validity of this new infinite element. The result indicates that the new dynamic infinite element has a high accuracy.

Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients

In this paper, we present a criterion on the oscillation of unbounded solutions for higher-order dynamic equations of the following form: x(t) + A(t)x( (t)) n large t. We give change of order formula for double(iterated) integrals to prove our main result. Some simple examples are given to illustrate the applicability of our results too. In the literature, almost all of the results for (?) with T = R and T = Z hold for bounded solutions. Our results are new and not stated in the literature even for the particular cases T = R and/or T = Z.

Positive Solutions to Singular and Delay Higher-Order Differential Equations on Time Scales

We are concerned with singular three-point boundary value problems for delay higher-order dynamic equations on time scales. Theorems on the existence of positive solutions are obtained by utilizing the fixed point theorem of cone expansion and compression type. An example is given to illustrate our main result.

Solutions for higher-order dynamic equations on time scales

Let T be a time scale. We study the existence of positive solutions for the nonlinear four-point singular boundary value problem with higher-order p-Laplacian dynamic equations on time scales. By using the fixed-point index theory, the existence of positive solution and many positive solutions for nonlinear four-point singular boundary value problem with p-Laplacian operator are obtained.

Basic qualitative and quantitative results for solutions to nonlinear, dynamic equations on time scales with an application to economic modelling

This article investigates both basic qualitative and basic quantitative properties of solutions to first- and higher-order dynamic equations on time scales and thus provides a foundation and framework for future advanced nonlinear studies in the field. Particular focus lies in the: existence; uniqueness; dependency; approximation; and explicit representation, of solutions to nonlinear initial value problems. The main tools used are from modern areas of nonlinear analysis, including: the fixed-point theorems of Banach and Schäfer; the method of successive approximations; a novel definition of measuring distance in metric spaces and normed spaces; and a “separation” of variables technique is introduced to the general time scale setting. The new results compliment and extend those of Stefan Hilger’s seminal paper of 1990. As an application of the new results we present and analyse a simple model from economics, known as the Keynesian–Cross model with “lagged” income, in the general time scale environment. Ideas suggesting further applications and possible new directions for the novel results are also presented.

Consistent Higher-Order Dynamic Equations for Soft-Core Sandwich Beams

The consistent higher-order dynamic equations and the corresponding continuity and boundary conditions for sandwich beams with a transversely flexible core are derived, with nonlinear acceleration fields in the core taken into account. A discretized formulation based on an efficient implicit finite difference scheme is presented and numerically validated. The higher-order analysis reveals that the dynamic magnitudes of normal stresses between the face sheets and the core increase dramatically at the boundaries of a sandwich beam, whereas the values remain at the level of response to quasi-static loading through the span, including the points of application of localized loads. The present work provides an accurate and efficient tool for the analysis of sandwich beams with a transversely flexible core subject to dynamic loads of arbitrary type, including the practical case of localized dynamic excitation.

Sign properties of Green's functions for disconjugate dynamic equations on time scales

This paper continues the development of disconjugacy of higher order dynamic equations on time scales. Two-point conjugate type boundary value problems for general disconjugate dynamic equations on time scales are studied and the sign properties of associated Green's functions are established. As expected, the results unify known results from the theories of ordinary differential equations and finite difference equations.

Robust stability of a special class of polynomial matrices with control applications

The robust stability problem of uncertain continuous-time systems described by higher-order dynamic equations is considered in this paper. Previous results on robust stability of Metzlerian matrices are extended to matrix polynomials, with the coefficient matrices having exactly the same Metzlerian structure. After defining the structured uncertainty for this class of polynomial matrices, we provide an explicit expression for the real stability radius and derive simplified formulae for several special cases. We also report on alternative approaches for investigating robust Hurwitz stability and strong stability of polynomial matrices. Several illustrative examples throughout the paper support the theoretical development. Moreover, an application example is included to demonstrate uncertainty modeling and robust stability analysis used in control design.

Robust stability of linear systems described by higher-order dynamic equations

The stability radius of higher-order differential and difference systems with respect to various classes of complex affine perturbations of the coefficient matrices is studied. Different perturbation norms are considered. The aim is to derive robustness criteria that are expressed directly in terms of the original data. Previous results on robust stability of Hurwitz and Schur polynomials are extended to monic matrix polynomials. For disturbances acting via a uniform input matrix, computable formulas are obtained, whereas for perturbations with multiple input matrices, structured singular values are involved. >