Second-order Dynamic Equations Research Articles

On Hyers–Ulam and Hyers–Ulam–Rassias Stability of a Nonlinear Second-Order Dynamic Equation on Time Scales

In this paper, we obtain sufficient conditions for Hyers–Ulam and Hyers–Ulam–Rassias stability of an abstract second–order nonlinear dynamic equation on bounded time scales. An illustrative example is given to show the applicability of the theoretical results.

Amended Criteria of Oscillation for Nonlinear Functional Dynamic Equations of Second-Order

In this paper, the sharp Hille-type oscillation criteria are proposed for a class of second-order nonlinear functional dynamic equations on an arbitrary time scale, by using the technique of Riccati transformation and integral averaging method. The obtained results demonstrate an improvement in Hille-type compared with the results reported in the literature. Some examples are provided to illustrate the significance of the obtained results.

Oscillatory Properties for Second-Order Impulsive Neutral Dynamic Equations with Positive and Negative Coefficients on Time Scales

We study oscillatory properties for second-order impulsive neutral dynamic equations with positive and negative coefficients on time scales. By using variable substitution, we obtain sufficient conditions for several dynamic equations to be oscillatory.

Hyers-Ulam-Rassias stability of abstract second-order linear dynamic equations on time scales

In this paper, we obtain new sufficient conditions for Hyers-Ulam and Hyers-Ulam-Rassias stability of abstract second-order linear dynamic equations on time scales. Also a new sufficient condition for the existence and uniqueness of solutions is established, via Banach's fixed point theorem. Finally, two illustrative examples are given to demonstrate the applicability of the theoretical results.

Existence of positive solutions for second-order nonlinear neutral dynamic equations on time scales

AbstractIn this article we study the existence of positive solutions for second-order nonlinear neutral dynamic equations on time scales. The main tool employed here is Schauder’s fixed point theorem. The results obtained here extend the work of Culakova, Hanustiakova and Olach [12]. Two examples are also given to illustrate this work.

New Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with a Nonpositive Neutral Term on Time Scales

New Oscillation Criteria for Second-Order Neutral Delay Dynamic Equations with a Nonpositive Neutral Term on Time Scales

EXISTENCE AND EXPONENTIAL STABILITY OF POSITIVE PERIODIC SOLUTIONS FOR SECOND-ORDER DYNAMIC EQUATIONS

In this article, we establish the existence of positive periodic solutions for second-order dynamic equations on time scales. The main method used here is the Schauder fixed point theorem. The exponential stability of positive periodic solutions is also studied. The results obtained here extend some results in the literature. An example is also given to illustrate this work.

A novel approach of active control of structures based on the critically damped condition

In this research, the active control of structures based on the critically damped condition was proposed for reducing displacements and drifts of structures under seismic loading. In this approach, the second-order dynamic equation is converted into a first-order dynamic equation and a first-order derivative operator with the same constants, which satisfy critically damped conditions of the system. Based on this concept and using the defined performance index, the feedback gain matrix of the structural velocity is obtained. In the method of active control of structures based on the critically damped condition, the control force, which is the linear definition of the feedback gain matrix and the structural velocity, is exerted on the structure in each time step. Four types of numerical examples are examined to the comparison verification of active control of structures based on the critically damped condition with the classical linear optimal control, standard model predictive control, and other methods proposed in several references. Programming and calculations are done by Mathcad prime mathematical software.

Oscillatory behavior of solutions of dynamic equations of higher order on time scales

We study the n -th-order nonlinear dynamic equations x [ n ] ( t ) + p ( t ) ϕ α n − 1 [ ( x [ n − 2 ] ( t ) ) Δ σ ] + q ( t ) ϕ γ ( x ( g ( t ) ) ) = 0 on an unbounded time scale 𝕋 , where n ≥ 2 and for i = 1 , … , n − 1 x [ i ] ( t ) : = r i ( t ) ϕ α i [ ( x [ i − 1 ] ( t ) ) Δ ] , with r n = α n = 1 and x [ 0 ] = x ; here the constants α i and the functions r i , i = 1 , … , n − 1 , are positive and p , q are nonnegative functions. Criteria are established for the oscillation of solutions for both even- and odd-order cases. The results improve several known results in the literature on second-order, third-order, and higher-order linear and nonlinear dynamic equations. In particular our results can be applied when g is not (delta) differentiable and the forward jump operator σ and g do not commute.

A Note on the Oscillation of Second-Order Quasilinear Neutral Dynamic Equations on Time Scales

By using a new method, we improve some results obtained in [S. H. Saker, Nonlin. Oscillat.,13, No. 3, 407–428 (2011)].

Application of A-analytic Functions to the Investigation of the Cauchy Problem for a Stationary Poroelasticity System

In a reversible hydrodynamic approximation, a closed system of second-order dynamic equations with respect to the displacement vector of an elastic porous body and pore pressure has been obtained. The Cauchy problem for the obtained system of poroelasticity equations in the stationary case is considered. The Carleman formula for the Cauchy problem under consideration has been constructed.

A Time Integration Method Based on Galerkin Weak Form for Nonlinear Structural Dynamics

This paper presents a step-by-step time integration method for transient solutions of nonlinear structural dynamic problems. Taking the second-order nonlinear dynamic equations as the model problem, this self-starting one-step algorithm is constructed using the Galerkin finite element method (FEM) and Newton–Raphson iteration, in which it is recommended to adopt time elements of degree m = 1,2,3. Based on the mathematical and numerical analysis, it is found that the method can gain a convergence order of 2m for both displacement and velocity results when an ordinary Gauss integral is implemented. Meanwhile, with reduced Gauss integration, the method achieves unconditional stability. Furthermore, a feasible integration scheme with controllable numerical damping has been established by modifying the test function and introducing a special integral rule. Representative numerical examples show that the proposed method performs well in stability with controllable numerical dissipation, and its computational efficiency is superior as well.

Oscillation of second-order nonlinear neutral dynamic equations with “Maxima” on time scales

This paper is concerned with the oscillation of solutions of the second order nonlinear neutral delay dynamic equation with “Maxima” of the formon a time scale 𝕋. The obtained results improve and generalize some recent published results. Examples are given to illustrate the main results.

Uncountably many nonoscillatory bounded solutions to second-order nonlinear neutral dynamic equations

This work is devoted to the study of the existence of uncountably many nonoscillatory bounded solutions to second-order nonlinear neutral dynamic equations by means of the Darbo fixed point theorem. We construct assumptions without sign conditions on the nonlinear part of the equation. Moreover, we prove the necessary condition for the existence of an asymptotically zero solution to the problem under consideration.

Aging condition assessment of transformer oil-immersed cellulosic insulation based upon the average activation energy method

It is widely accepted that the service life of transformer oil–paper insulation is determined by the aging condition of transformer cellulosic insulation. Therefore, it is rather meaningful to accurately and reliably evaluate the aging condition of transformer cellulose insulation. Existing studies have shown that activation energy can eliminate the temperature effect on the frequency dielectric spectroscopy, however, the traditional activation energy might not be suitable for assessing the aging condition of transformer cellulosic insulation. The aim of the work is to assess the aging condition of cellulosic insulation by using average activation energy technique. The experiment process is firstly discussed. Then, the second-order dynamic equation formed by Arrhenius equation and Ekenstam equation is strictly investigated to accurately calculate the degree of polymerization value of transformer cellulosic insulation. The experimental results preliminarily show that the average activation energy can effectively assess aging condition of transformer cellulosic insulation.

Superadsorbent hydrogel based on lignin and montmorillonite for Cu(II) ions removal from aqueous solution.

Superadsorbent hydrogel was prepared from lignin and montmorillonite for Cu(II) ions removal, and the chemical structure and morphology of the hydrogel were characterized by FT-IR, XRD, SEM and XPS. The swelling kinetics of the prepared hydrogel was investigated, and the result showed that the swelling process fit the Schott second-order dynamic equation. The influences of pH, contact time, Na+ concentration, and initial Cu(II) ion concentration on adsorption were studied, and the maximum adsorption value was 1.17 mmol/g, and the adsorption was rapid during the initial 5 h period, and copper ions adsorption on the superadsorbent hydrogel is a favorable process. The results also indicated that the adsorption kinetics was in accordance with the pseudo-second-order kinetic model and the adsorption isotherm conformed to the Freundlich model. FT-IR and XPS analysis revealed that the adsorption behavior was mainly due to ion exchange. The desorption ratios of copper ions from the superadsorbent were >0.8, and the regeneration efficiency was >80% after five cycles reuse, and the results show the excellent desorption performance and reusability of the prepared hydrogel.

Oscillation for Forced Second-Order Impulsive Nonlinear Dynamic Equations on Time Scales

As the unification and development of impulsive differential equations and difference equations, impulsive dynamic equations on time scales are a powerful tool to simulate the natural and social phenomena. In this paper, we study the interval oscillation of a type of forced second-order nonlinear impulsive dynamic equations with changing signs coefficients. By using the Riccati transformation technique, we obtain some new interval oscillation criteria, based only on information of a sequence of subintervals of positive axis. In addition, we provide an example to illustrate the use of our oscillatory results.

Existence of a positive solution and numerical solution for some elliptic superlinear problem

In this paper, we prove the existence of positive solutions of an elliptic superlinear problem. Also, we are interested here in getting results concerning the existence of positive solutions for the discrete formulation of our problem. Therefore, in order to do it, we employ the radial solutions of the elliptic superlinear problem, obtaining a second-order dynamic equation on time scales, which encompasses discrete, continuous and hybrid formulations of our problem. This unified equation allows us to present numerical simulations, which give us a more precise analysis and description concerning the behavior of the solution according to the parameters.

Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

Numerical Studies on the Dynamic Responses of Levitated High-temperature Superconductor with a Strongly Coupled Thermo-electromagnetic Model

In this paper, we present a 2-D numerical model based on the H-formulation and nonlinear E-J relationship to study the dynamic responses of a bulk high-temperature superconductor(HTSC) levitated above two kinds of permanent magnetic guideway(PMG). Being different from the existing related models and results of this subject, the thermal effect was brought into account by a strong coupled model with electromagnetic and thermal. The levitation forces were calculated by finite element method and then the vertical motion of the levitated HTSC subject to external disturbance is characterized by a second-order dynamic equation which couples the electromagnetic model via the levitation force. We studied the thermal influence on the vertical force and dynamic characteristics of the HTSC levitated above two kinds of PMGs. The obtained results prove that this strongly coupled model can better simulate the dynamic response of HTSC levitation system.