First-order Dynamic Equations Research Articles

Ulam stability of first-order nonlinear impulsive dynamic equations

This paper is devoted to the investigation of Ulam stability of first-order nonlinear impulsive dynamic equations on finite-time scale intervals. Our main objective is to formulate sufficient conditions under which the class of first-order nonlinear impulsive dynamic equations on time scales we consider exhibits Ulam stability. Our methods rely on the extended integral inequality on time scales for piecewise-continuous functions. We provide an example to support the validity of the results obtained.

Transport model simulation prediction and environmental risk assessment of pyrethroid insecticides in urban artificial lakes caused by rainfall: A case study of Cloud Mountain Park in subhumid climate zone

Pyrethroid insecticides are among urban parks' most widely used and harmful insecticides. The advanced prediction method is the key to studying the pollution and diffusion risk of plant conservation insecticides in parks. A two-dimensional advection-dispersion model was established for the North Lake of Cloud Mountain Park in the subhumid area of Hebei Province. The temporal and spatial distribution of lambda-cyhalothrin pollution required by plant growth in artificial lakes under different rainfall intensities and the time of water renewal after rainfall was simulated and predicted. According to the model efficiency (E: 0.98), mean absolute error (MAE: 0.016–0.064 cm), and root mean square error (RMSE: 0.014–0.041 cm), the prediction results showed that the model fits well. The results showed that the concentration of lambda-cyhalothrin in the artificial lake was positively correlated with the increase in rainfall intensity. Under the three scenarios of moderate rain, heavy rain, and rainstorm, the variation of total pollutants into the lake over time conformed to the first-order dynamic equation (R2＞0.97), and the cumulative rates were 0.013 min−1, 0.019 min−1 and 0.022 min−1, respectively. Under light rain, the accumulation rate of lambda-cyhalothrin showed a double-linear relationship, which was in accordance with the second-order kinetic equation (R2＞0.97). The rapid accumulation rate of early-stage rainfall was 0.0024 min−1, and the slow accumulation rate of late-stage rainfall was 0.0019 min−1. The human health risk assessment predicted by the simulation was lower than the hazard value (Rtgn(a-1): 9.65 E−11–1.12 E−10 a-1). However, the potential risk value to aquatic species was higher (RQ: 0.33–23.05). In addition, the increase in rainfall intensity has no significant effect on the acceleration of water renewal time. The two-dimensional dispersion model of pollutants driven by water dynamics provided relevant examples for evaluating the impact of runoff on pesticide scour in parks and supplied scientific support for improving the management of artificial lakes in urban parks.

Mineralization Characteristics of Soil Organic Carbon and Its Relationship with Organic Carbon Components in Artificial Robinia pseudoacacia Forest in Loess Hilly Region

In order to explore the characteristics of organic carbon mineralization and the variation law of organic carbon components of an artificial forest in a loess hilly area, an artificial Robinia pseudoacacia forest restored for 13 years and the adjacent slope farmland were selected as the research objects, and indoor culture experiments under three different temperature treatments (15, 25, and 35℃) were carried out. The results indicated that the mineralization rate of soil organic carbon decreased sharply at first and then stabilized. The cumulative release of organic carbon increased rapidly in the initial stage of culture and gradually slowed in the later stage. Soil organic carbon mineralization in sloping farmland was more sensitive to temperature change, and its temperature sensitivity coefficient Q10 was 1.52, whereas that in R. pseudoacacia forest land was only 1.38. According to the fitting of the single reservoir first-order dynamic equation, the soil mineralization potential Cp of R. pseudoacacia forest land and slope farmland was between 2.02-4.32 g·kg-1 and 1.25-3.17 g·kg-1, respectively, that is, the mineralization potential of the R. pseudoacacia forest was higher. During the cultivation period, the content of various active organic carbon components decreased with time, and that in the R. pseudoacacia forest land was greater than that in the slope land. The cumulative carbon release of soil was significantly positively correlated with the contents of MBC and DOC (P<0.05), and Q10 (15-25℃) was negatively correlated with the contents of SOC, EOC, and SWC (P<0.05). These results could provide some reference for the study of soil carbon sequestration in loess hilly regions under climate change.

New composition theorem for weighted Stepanov-like pseudo almost periodic functions on time scales and applications

First, we show a new composition theorem for both Stepanov almost periodic functions and for weighted Stepanov-like pseudo almost periodic functions on time scales. Next, under some suitable assumptions, we study the existence and uniqueness of weighted pseudo almost periodic solutions to some first-order dynamic equations on time scales with weighted Stepanov-like pseudo almost periodic coefficients.

PySCP: A Multiple-Phase Optimal Control Software Using Sequential Convex Programming

Optimal control problems are common in aerospace engineering. A Python software program called PySCP is described for solving multiple-phase optimal control problems using sequential convex programming methods. By constructing a series of approximated second-order cone programming subproblems, PySCP approaches to the solution of the original optimal control problem in an iterative way. The key components of the software are described in detail, including convexification, discretization, and the adaptive trust region method. The convexification of the first-order differential dynamic equation is implemented using successive linearization. Six discretization methods, including zero-order hold, first-order hold, Runge-Kutta, and three hp pseudospectral collocation methods, are implemented so that different types of optimal control problems can be tackled efficiently. Adaptive trust region method is employed, and robust convergence is achieved. Both free-final-time problem and fixed-final-time problem can be solved by the software. The application of the software is demonstrated on three optimal control problems with varying complexity. PySCP provides researchers a useful toolkit to solve a wide variety of optimal control problems using sequential convex programming.

Oscillation of even order nonlinear dynamic equations on time-scales

In this paper, the authors discuss the oscillatory behavior of solutions to a class of even order nonlinear dynamic equations on time scales. The results are established by a comparison with n-th order delay dynamic inequalities or first-order delay dynamic equations whose oscillatory characters are known. Several corollaries are obtained for special cases.

On oscillatory behaviour of third-order half-linear dynamic equations on time scales

In this work, we study the oscillation and asymptotic behaviour of third-order nonlinear dynamic equations on time scales. The findings are obtained using an integral criterion as well as a comparison theorem with the oscillatory properties of a first-order dynamic equation. As a consequence, we give conditions which guarantee that all solutions to the aforementioned problem are only oscillatory, different from any other result in the literature. We propose novel oscillation criteria that improve, extend, and simplify existing ones in the literature. The results are associated with a numerical example. We point out that the results are new even for the case \(\mathbb{T}=\mathbb{R}\) or \(\mathbb{T}=\mathbb{Z}\).

Mechanism and Purification Effect of Photocatalytic Wastewater Treatment Using Graphene Oxide-Doped Titanium Dioxide Composite Nanomaterials

The present work aims to examine the mechanism and purification effect of graphene oxide (GO) and GO composite materials for photocatalysis sewage treatment. TiO2 nanoparticles were prepared using the sol-gel technique; GO was prepared using the modified Hummers technique; and finally, a new N-TiO2/GO photocatalysis composite material was prepared by hydrothermal synthesis. As a nitrogen source, urea uses non-metal doping to broaden the photoresponse range of TiO2. The prepared GO and its composite materials are characterized. Simulation experiments, using the typical water dye pollutant rhodamine B (RhB), tested and analyzed the adsorption and photocatalysis performances of the prepared GO and its composite materials. Characterization analysis demonstrates that TiO2 is distributed on the GO surface in the prepared N-TiO2/GO material. Simultaneously, nitrogen doping causes TiO2 on the GO surface to seem uniformly dispersed. X-ray Diffractometer (XRD) spectrums suggest that TiO2 on the GO surface presents an anatase crystal structure; infrared spectrums display the characteristic vibration peaks of the TiO2 and GO. Adsorption performance analysis illustrates that N-TiO2/GO can provide an adsorption amount of 167.92 mg/g in the same time frame and photocatalysis for visible lights of 57.69%. All data present an excellent linear fitting relationship to the first-order dynamic equation. Therefore, the prepared GO composite materials possess outstanding absorption and photocatalysis performances, providing an experimental basis for sewage treatment and purification using photocatalysis approaches in the future.

Hyers–Ulam and Hyers–Ulam–Rassias Stability of First-Order Nonlinear Dynamic Equations

We investigate Hyers–Ulam and Hyers–Ulam–Rassias stability of first-order nonlinear dynamic equations for functions defined on a time scale with values in a Banach space.

Oscillation criteria for first-order dynamic equations with nonmonotone delays

In this paper, we consider the first-order dynamic equation as the following:$$x^{\Delta}(t)+\sum\limits_{i=1}^m p_i(t)x(\tau_i(t))=0,\,\,t\in[t_0,\infty)_{\mathbb{T}}$$where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) $ $(i=1,2,\ldots ,m)$ and $\tau_i(t)\leq t,\,\, \lim_{t\to\infty}\tau_i(t)=\infty$. When the delay terms $\tau_{i}(t)$ $(i=1,2,\ldots ,m)$ are not necessarily monotone, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales.

Simulation of development of individual heavy industry sectors

Nowadays, in the context of the coronavirus crisis, the issue of ensuring the sustainable development of heavy industries is acute. However, theoretical and analytical researches alone are not sufficient for this, and economic science needs to develop fundamentally new approaches to the study of the development of industrial sectors. This article is devoted to the creation and testing of a simulation model for the development of individual sectors of the economy. The object of research is the metallurgical industry, as well as related ore mining, mechanical engineering and production of finished metal products. The theoretical basis of the research is a systematic approach that combines the theory of industry markets, economic growth, industrial economics, system dynamics and mathematical economics. The main research methods used are system analysis, statistical analysis to identify trends in changes in the main economic indicators, econometric modeling to build production functions, as well as mathematical modeling of macroeconomic systems. As a result, a simulation model developed in system dynamics notation is proposed, which makes it possible to evaluate the development of individual industries taking into account various changes. This model is built on the basis of the three-sector model of the national economy, where separate adjacent industries connected by dynamic feedback loops are identified as structural elements. The paper details the structure of the simulation model based on first-order dynamic equations, balance equations and nonlinear production functions. The simulation model allowed us to predict a number of scenarios for the development of metallurgical industries, taking into account changes in the labor force and investment in fixed assets. The results of the work can be used for forming proposals on industrial policy, monitoring the condition and efficiency of individual industries.

Hyers-Ulam and Hyers-Ulam-Rassias stability of first-order linear dynamic equations

We present several new sufficient conditions for Hyers-Ulam and Hyers-Ulam-Rassias stability of first-order linear dynamic equations for functions defined on a time scale with values in a Banach space.

Positivity and Periodicity of Solutions for First-Order Neutral Delay Dynamic Equations

Positivity and Periodicity of Solutions for First-Order Neutral Delay Dynamic Equations

Oscillation and nonoscillation criteria for four-dimensional advanced and delay time-scale systems

We obtain oscillation and nonoscillation criteria for solutions to four-dimensional advanced and delay systems of first-order dynamic equations on time scales. To establish oscillation criteria, we eliminate nonoscillatory solutions of the systems based on the sign of components of the solutions. Furthermore, some of our results are new in the discrete case.

Non-Relativistic Limit of Embedding Gravity as General Relativity with Dark Matter

Regge-Teitelboim embedding gravity is the modified gravity based on a simple string-inspired geometrical principle—our spacetime is considered here as a 4-dimensional surface in a flat bulk. This theory is similar to the recently popular theory of mimetic gravity—the modification of gravity appears in both theories as a result of the change of variables in the action of General Relativity. Embedding gravity, as well as mimetic gravity, can be used in explaining the dark matter mystery since, in both cases, the modified theory can be presented as General Relativity with additional fictitious matter (embedding matter or mimetic matter). For the general case, we obtain the equations of motion of embedding matter in terms of embedding function as a set of first-order dynamical equations and constraints consistent with them. Then, we construct a non-relativistic limit of these equations, in which the motion of embedding matter turns out to be slow enough so that it can play the role of cold dark matter. The non-relativistic embedding matter turns out to have a certain self-interaction, which could be useful in the context of solving the core-cusp problem that appears in the Λ-Cold Dark Matter (ΛCDM) model.

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.

Comparative Transient Stability Assessment of Droop and Dispatchable Virtual Oscillator Controlled Grid-Connected Inverters

With the increasing integration of power electronics interfaced distributed generators, transient stability assessment of grid-connected inverters subjected to large grid disturbances is of vital importance for the secure and resilient operation of the power grid. Dispatchable virtual oscillator control (dVOC) is an emerging approach to implement nonlinear control of grid-forming inverters. Through coordinate transformation, a simple first-order nonlinear power angle dynamic equation is uncovered from the complex oscillator dynamics. Furthermore, this article proposes a concise and straightforward graphical approach to assess transient stability of dVOC using vector field on the circle. To provide a more in-depth analysis, a complete large-signal model is derived and the impact of dVOC voltage amplitude dynamics is analyzed. For comparison, transient stability of the currently prevalent droop control is also assessed using phase portraits. Salient transient stability features of dVOC and droop control during grid faults are summarized and compared. The theoretical analysis is validated by controller hardware-in-the-loop testbed using industry-grade hardware.

Linear fuzzy Volterra integral equations on time scales

In this paper, we study linear fuzzy Volterra integral equations on time scales and we show that under some appropriate hypotheses, they are equivalent to the first-order linear fuzzy dynamic equations. Moreover, an existence theorem to fuzzy Volterra integral equations on time scales is presented. Some examples are provided to illustrate our results.

Decoupled nonlinear adaptive control of position and stiffness for pneumatic soft robots

This article addresses the problem of simultaneous and robust closed-loop control of joint stiffness and position, for a class of antagonistically actuated pneumatic soft robots with rigid links and compliant joints. By introducing a first-order dynamic equation for the stiffness variable and using the additional control degree of freedom, embedded in the null space of the pneumatic actuator matrix, an innovative control approach is introduced comprising an adaptive compensator and a dynamic decoupler. The proposed solution builds upon existing adaptive control theory and provides a technique for closing the loop on joint stiffness in pneumatic variable stiffness actuators. Under a very mild assumption involving the inertia and actuator matrices, the solution is able to cope with uncertainties of the model and, when the desired stiffness is constant or slowly varying, also of the pneumatic actuator. Position and stiffness decoupling is achieved by the introduction of a first-order differential equation for an internal state variable of the controller, which takes into account the time derivative of pressure in the stiffness dynamics. A formal proof of the stability of the position and stiffness tracking errors is provided. An appealing property of the approach is that it does not require higher derivatives of position or any derivatives of stiffness. The solution is validated with respect to several use-cases, first in simulation and then via a real pneumatic soft robot with McKibben muscles. A comparison with respect to existing techniques reveals a more robust position and stiffness tracking skill.

Dissipation behavior, residues distribution and dietary risk assessment of tembotrione and its metabolite in maize via QuEChERS using HPLC-MS/MS technique.

The dissipation and residues of tembotrione in corn field application were investigated using liquid chromatography tandem mass spectrometry (LC-MS/MS) method. The average recoveries of tembotrione in maize, corncob, and straw were in the ranges of 98-107% with relative standard deviations (RSDs ≤9.3%), respectively. The recoveries of M5 was in the ranges of 90-108% in all three matrices of maize, with RSDs were 3.3-12.8%. The LODs for tembotrione and M5 in maize were 0.85μg/L and 1.0μg/L, 0.84μg/L and 0.43μg/L in corncob, 0.94μg/L and 1.5μg/L in straw, respectively. The LOQs of the method in maize grain, corncob and straw were 0.01, 0.01 and 0.05mg/kg for both analytes, respectively. The dissipation of tembotrione in straw was in compliance with the first-order dynamic equation, with half-lives of 1.18-1.23 days at Beijing and Heilongjiang. Total residue of tembotrione in maize grain and corncob matrix were both below 0.02mg/kg, lower than the max residue limit (MRL) recommended by european food safety authority (EFSA). Risk quotients (RQs) of this pesticide was assessed via comparing national estimated daily intake with acceptable daily intake. The dietary intake risk of tembotrione residue in maize was very low for all groups of Chinese residents. These data could provide scientific data and strategies and facilitate Chinese government to establish the MRLs of tembotrione.