Order Dynamic Equations Research Articles

Periodic boundary value problems for fırst order dynamic equations on time scales

In this work, the existence criteria of extremal solutions of periodic boundary value problems for the first-order dynamic equations on time scales are given by using the method of lower and upper solutions coupled with the monotone iterative technique. Our results generalize and improve some existing results. Two examples are provided to show the effectiveness and feasibility of the obtained results.

EXISTENCE OF PERIODIC SOLUTIONS FOR HIGHER ORDER DYNAMIC EQUATIONS ON TIME SCALES

In this paper, we study a sufficient condition for the existence of a periodic solution for the $n$th order dynamic equations on time scales. The results are shown by the use of coincidence degree theory. The necessary a priori bounds are based on Wirtinger type inequality established as Lemma B.

Opial's type inequalities on time scales and some applications

We prove some new Opial type inequalities on time scales and employ them to prove several results related to the spacing between consecutive zeros of a solution or between a zero of a solution and a zero of its derivative for second order dynamic equation

Release characteristics of doxycycline chitosan nanoparticles freeze-dried powderin vitroandin vivo

This paper is to study the release characteristics of doxycycline (DC) raw material drug and doxycycline chitosan nanoparticles (DC-CS-NPs) freeze-dried powder in vitro and in vivo by ultraviolet spectrophotometry and ultra performance liquid chromatography, respectively. Results showed that the release behaviors of DC raw material drug were simulated the most in the zero order dynamic equation in artificial gastric juice, artificial intestinal juice and pH 7.4 buffer phosphate. The release of DC raw material drug was fast which could completely dissolve in one hour; The DC-CS-NPs freeze-dried powder had a significant slow-release characteristic. The release behaviors were simulated the most in the biexponential equation. The drug released fast in the early stage while it released slowly later. The order of release speed of DC-CS-NPs freeze-dried powder from fast to slow was pH 2.0 artificial gastric juice, pH 3.0 artificial gastric juice, pH 4.0 artificial gastric juice, artificial intestinal juice and pH 7.4 buffer phosphate. When DC raw material drug became DC-CS-NPs freeze-dried powder, after a single oral administration at a dose of 20 mg/kg at (25±1) ℃, the double-peak phenomenon of plasma in channel catfish was turned into single-peak, and the pharmacokinetic parameters of DC-CS-NPs freeze-dried powder were changed. Compared to DC raw material drug, the peak concentration (Cmax) decreased. The peak time (Tmax) and terminal elimination half-life (T1/2β) prolonged. The area under concentration-time curve (AUC) was larger. The DC-CS-NPs freeze-dried powder will become an ideal DC new formulation.

Bond Graphs and Lagrange Equations as Aids in Analytical Studies of Electro-Mechanical Systems

AbstractA well-known advantage of bond graph models of electro-mechanical systems is that they can be manipulated manually or with the aid of computer programs to yield first order state equations useful for numerical studies. A causal analysis can show in advance whether derivative causality will cause practical difficulties in the equation formulation which might be best circumvented by modification of the model.Numerical studies are sometimes of limited use in preliminary investigation of the feasibility of a design because the large number of parameters involved precludes a general understanding of the limits of the concept being studied. Analytical studies of simplified models may find the existence of parameter combinations which are crucial to the successful operation of the system or fundamental limitations to performance not obvious in purely numerical studies.Although the first order nature of the normal bond graph state equations are advantageous for numerical simulation, they are not necessarily the most convenient form for analytic studies using pencil and paper. In many cases, the use of Lagrange equations, based on energy state functions and degrees-of-freedom rather than state variables are advantageous for analytic work. Bond graph models and a special form of causality can be used to advantage, particularly when derivative causality is present, to yield sets of second order dynamic equations based on Lagrange equations, which are often easier to work with than their first order equivalents. This will be illustrated by an example by the analysis of the use of a rotary electric motor as an actuator in semi-active or active suspension element in a vehicle suspension.

Cosmography inf(T)gravity

Being based on the only assumption that the universe is homogenous and isotropic on large scales, cosmography is an ideal tool to investigate the cosmic expansion history in a almost model-independent way. Fitting the data on the luminosity distance and Baryon Acoustic Oscillations allows to determine the confidence ranges for the cosmographic parameters hence giving some quantitative constraints that a whatever theory has to fulfill. As an application, we consider here the case of teleparallel gravity (TEGR) also referred to as f(T)-gravity. To this end, we first work out analytical expressions to express the present day values of f(T)-derivatives as a function of the cosmographic parameters which hold under quite general and physically motivated conditions. We then use the constraints coming from cosmography to find out the confidence ranges for f(T)-derivatives up to the fifth order and show how these can be used to check the viability of given TEGR models without the need to explicitly solve the second order dynamic equations.

Study of tool dynamics with a discrete model of workpiece in orthogonal turning

This paper proposes a realistic analytical stability model of regenerative chatter in orthogonal turning operation. Tool geometry is initially developed as a solid model and analysed at different tool overhang conditions. In each case, stiffness, fundamental bending mode and corresponding damping ratios of the tool are evaluated. With this data, cutting tool can be represented with a lumped parameter single-degree of freedom vibration oscillator. Workpiece dynamics on the other hand is considered independently using a discrete finite element beam model. At some contact node of the workpiece, tool mass imposes a regenerative cutting force and second order dynamic delay differential equations are formulated in terms of tool and modal parameters. Stability criterion is formulated from characteristic equation. The effects of tool-overhang and work cross-section on the stability from the proposed model are reported. Results are illustrated with a commonly used HSS cutting tool with tailstock-supported workpiece. Experimental analysis is carried out to show the effect of tool-overhang on cutting dynamics.

Fixed point theorems for a class of nonlinear operators in Hilbert spaces and applications

We discuss the existence of fixed point for a class of nonlinear operators by a combination of topological and variational method. The theoretical results are applied to second order dynamic equation with homogeneous Dirichlet boundary conditions.

Swelling properties and kinetics of CaCl 2/polyacrylamide hygroscopic hybrid hydrogels

The novel hygroscopic hybrid hydrogels were prepared from polyacrylamide and calcium chloride. The hygroscopic swelling properties and kinetics of CaCl 2/polyacrylamide hybrid gels were investigated. To optimize the experimental condition, the influence factors such as dissolution temperature of CaCl 2, CaCl 2 mass, and hygroscopic temperature relative to the hygroscopic swelling degree of hybrid desiccants were studied systematically. The research results revealed that, under the optimal experimental condition (dissolution temperature of CaCl 2, 50 °C; CaCl 2 mass, 80%; environment temperature, 30 °C), the hybrid gels displayed the highest hygroscopic swelling degree resulting from high hygroscopic ability of calcium chloride and good water-binding of polyacrylamide. Under the optimized experimental condition, the hygroscopic swelling degree of the hybrid gels increased with the increment of relative humidity. Furthermore, the hygroscopic swelling processes of hybrid gels followed the Schott second order dynamic equation, and the hygroscopic swelling rate decreased with the increment of relative humidity.

Existence of quasibounded solutions for the higher order dynamic equations on measure chains

By using the exponential dichotomy and Schauder’s fixed point theorem, some new criteria are established for the existence of quasibounded solutions of the inhomogeneous system x Δ = A( t) x + g( t, x) + h( t), which generalize the previous results in [15,19].

Oscillation of third order nonlinear functional dynamic equations on time scales

It is the purpose of this paper to give oscillation criteria for the third order nonlinear functional dynamic equation $$ \left( {a\left( t \right)\left[ {\left( {r\left( t \right)x^\Delta \left( t \right)} \right)^\Delta } \right]^\gamma } \right)^\Delta + f\left( {t,x\left( {g\left( t \right)} \right)} \right) = 0 $$ on a time scale $$ \mathbb{T} $$ , where γ is the quotient of odd positive integers, a and r are positive rd-continuous functions on $$ \mathbb{T} $$ , and the function g: $$ \mathbb{T} \to \mathbb{T} $$ satisfies limt→∞ g(t) = ∞ and f ∈ C $$ \left( {\mathbb{T} \times \mathbb{R}, \mathbb{R}} \right) $$ . Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equations. Some examples are given to illustrate the main results.

Oscillation of third order functional dynamic equations with mixed arguments on time scales

In this paper we investigate the oscillation of third order nonlinear functional dynamic equation with mixed arguments. Our results extend and improve many known results for oscillation of third order dynamic equations. Some examples are given to illustrate the main results.

Mathematical modeling of seismic and acousto-gravitational waves in a heterogeneous earth–atmosphere model

A numerical–analytical solution to problems of seismic and acoustic-gravitational wave propagation is applied to a heterogeneous Earth–Atmosphere model. The seismic wave propagation in an elastic half-space is described by a system of first order dynamic equations of the elasticity theory. The propagation of acoustic-gravitational waves in the atmosphere is described by the linearized Navier–Stokes equations. The algorithm proposed is based on the integral Laguerre transform with respect to time, the finite integral Bessel transform along the radial coordinate with a finite difference solution of the reduced problem along the vertical coordinate. The algorithm is numerically tested for the heterogeneous Earth–Atmosphere model for different source locations.

Regular variation on measure chains

In this paper we show how the recently introduced concept of regular variation on time scales (or measure chains) is related to a Karamata type definition. We also present characterization theorems and an embedding theorem for regularly varying functions defined on suitable subsets of reals. We demonstrate that for a “reasonable” theory of regular variation on time scales, certain additional condition on a graininess is needed, which cannot be omitted. We establish a number of elementary properties of regularly varying functions. As an application, we study the asymptotic properties of solution to second order dynamic equations.

Oscillation of third order nonlinear delay dynamic equations on time scales

It is the purpose of this paper to give oscillation criteria for the third order nonlinear delay dynamic equation ( a ( t ) { [ r ( t ) x Δ ( t ) ] Δ } γ ) Δ + f ( t , x ( τ ( t ) ) ) = 0 , on a time scale T , where γ ≥ 1 is the quotient of odd positive integers, a and r are positive r d -continuous functions on T , and the so-called delay function τ : T → T satisfies τ ( t ) ≤ t for t ∈ T and lim t → ∞ τ ( t ) = ∞ and f ∈ C ( T × R , R ) . Our results are new for third order delay dynamic equations and extend many known results for oscillation of third order dynamic equation. These results in the special cases when T = R and T = N involve and improve some oscillation results for third order delay differential and difference equations; when T = h N , T = q N 0 and T = N 2 our oscillation results are essentially new. Some examples are given to illustrate the main results.

Oscillation and boundedness of solutions to first and second order forced functional dynamic equations with mixed nonlinearities

We investigate the oscillation and boundedness of first and second order dynamic equations with mixed nonlinearities. Our results extend and improve known results for oscillation of first and second order dynamic equations that have been established by Agarwal and Bohner. Some examples are given to illustrate the main results.

Non-local simulation of the stable boundary layer with a third order moments closure model

A new higher order closure model for the stable boundary layer is presented and compared with Large Eddy Simulation data. The model includes numerical solutions for the mean values, second and third order moments equations. A satisfactory agreement is found between the calculated vertical profiles of the turbulent quantities with those provided by the LES. Furthermore the new model results are compared with profiles obtained with a lower order closure model in order to verify the effective importance of including third order dynamical equations in the model.

Finite element modeling of dynamics of martensitic phase transitions

A finite element approach is suggested for the modeling of the dynamics of multivariant martensitic phase transitions (PTs) in elastic materials at the nanoscale in the three dimensional (3D) case. The model consists of a coupled system of the Ginzburg–Landau equations for transformation strain-related order parameters and dynamic elasticity equations. Thermodynamic potential [V. Levitas and D. Preston, Phys. Rev. B 66, 134206 (2002)] that captures the main features of macroscopic stress-strain curves is used. The evolution of multivariant microstructure in a 3D specimen for cubic to tetragonal PT in a NiAl alloy is modeled with dynamic and static formulations. The numerical results show the significant influence of inertial forces on microstructure evolution.

Information transmittal and time uncertainty, measuring the speed of light and time of reflection, representations of Newton’s second law and related problems

Some physical and computational aspects related to the intuitive notion of time in its connection with natural and technological processes are considered. The phenomenon of finite speed of information transmittal relative to measurement and computation is analyzed. It is argued that this phenomenon creates inevitable and irreversible time delays (uncertainties) that affect all measurements and computations, makes the exact synchronization of clocks impossible, shifts our knowledge to the past, and limits the accuracy of experiments. Some past experiments for the measurement of the speed of light are revisited, the possibility of a finite time of mirror reflection is discussed, and a stand for the experimental measurement of time spent in mirror reflection is proposed by a modification of Fizeau experiments. The positive orientation of the flow of time in its relation to the mathematical concept of time derivative is considered. It is demonstrated that right time-derivatives normally used to describe physical processes actually set forth non-causal representations of physical realities and may severely restrict the possibility of control and optimization in real life systems. The use of (causal) left time-derivatives produced by measurements and computations and consideration of variable masses lead to new representations of the second Newton’s law of motion where forces may contain controls depending on accelerations and higher order left time-derivatives of velocity. Such controls are actually required in jet-propelled space vehicles with variable masses, as demonstrated in the space shuttle example. The parallelogram rule does not apply to forces depending on higher order derivatives, so the concept of effective forces is considered within original Newton’s representation in which effective forces can be recovered in the process of integration; thereby the parallelogram law stays intact for effective forces. Consideration of forces with left higher order time derivatives alters classical methods in mechanics that were developed on the basis of absolute time and assuming no higher order derivatives in the forces of Newtonian equations of motion. Inclusion of such forces and consideration of natural time delays and time orientation opens new avenues for investigation and control of processes in physics, economy, medicine and mechanics. It is demonstrated that Lagrange and Hamilton equations stay intact in some generalized forms which forms can be used to derive higher order dynamical equations that exclude geometric constraints, thus having the minimum number of independent generalized coordinates.

Oscillation and nonoscillation of forced second order dynamic equations

Oscillation and nonoscillation properties of second order Sturm?Liouville dynamic equations on time scales ? for example, second order self-adjoint differential equations and second order Sturm?Liouville difference equations ? have attracted much interest. Here we consider a given homogeneous equation and a corresponding equation with forcing term. We give new conditions implying that the latter equation inherits the oscillatory behavior of the homogeneous equation. We also give new conditions that introduce oscillation of the inhomogeneous equation while the homogeneous equation is nonoscillatory. Finally, we explain a gap in a result given in the literature for the continuous and the discrete case. A more useful result is presented, improving the theory even for the corresponding continuous and discrete cases. Examples illustrating the theoretical results are supplied.