Arbitrary Time Scale Research Articles

Consensus analysis via impulsive control for nonlinear hybrid singular switched multi-agent systems with event-triggered mechanism

ABSTRACT Our research is focused on achieving consensus for non-linear singular switched multi-agent systems on arbitrary time scales. The hybrid control strategy that combines an impulsive controller with centralised and distributed event-triggered controllers is proposed. First, the Drazin-inverse approach is applied to transform the singular multi-agent switched system into a differential algebraic switched system. Then event-triggered schemes are applied to study the leader–follower consensus conditions. In the end, some simulation-based examples are given to verify the proposed results.

A novel framework for wind energy assessment at multi-time scale based on non-stationary wind speed models: A case study in China

Wind energy assessment at multi-time scale is essential for the effective utilization of offshore wind resources. Conventionally, energy assessment factors are estimated based on parametric models under the assumption of wind speed stationarity. However, given the subjection of seasonal variability and long-term trends, the hypothesis is violated, especially in the context of climate change. In this paper, to advance the conventional approach, wind speeds were modelled using the decomposition-based method to consider the non-stationarity, and a novel framework for the estimation of energy factors at multi-time scale was designed. Using long-term ERA5 data, wind energy potential across the sea area along Chinese continental coastline was investigated. It is demonstrated that the established non-stationary models are superior to commonly used stationary models, allowing the estimation of energy assessment factors at arbitrary time scales after one-time parametrization of parametric distributions. Results confirmed that wind energy exhibits apparently temporal and spatial variability. The difference between annual averaged wind power density at different nodes exceeds 1000 W/m2. Moreover, significant long-term trends of wind energy characteristics were detected. The increase can be up to 16 W/m2/year in Taiwan Strait, emphasizing that an adequate consideration of the non-stationarity of wind speed in future research is required.

Some new bounds of Chebyshev and Grüss-type functionals on time scales

In this work, Korkine and Sonin's identities are defined on arbitrary time scales. These identities are utilized to establish the Chebyshev and Grüss-type inequalities on time scales. By applying these inequalities, we determine the bounds of the remainders in Montgomery identities that incorporate Taylor's formula on time scales. Moreover, we derive discrete and quantum inequalities based on these results.

A Qualitative Study on Fractional Logistic Integro-differential Equations in an Arbitrary Time Scale

This manuscript deals with the investigation related to uniqueness and existence of solution of fractional order nonlinear pantograph integro-differential equation in arbitrary time scale. The fractional derivatives are defined in Riemann-Liouville sense, the primary tools are taken as Banach contraction principle and Schauder’s fixed point theory to establish the theoretical outcomes. Finally, we give examples to show the efficiency of our results.

On Model of Recurrent Neural Network on a Time Scale: Exponential Convergence and Stability Research.

The majority of the results on modeling recurrent neural networks (RNNs) are obtained using delayed differential equations, which imply continuous time representation. On the other hand, these models must be discrete in time, given their practical implementation in computer systems, requiring their versatile utilization across arbitrary time scales. Hence, the goal of this research is to model and investigate the architecture design of a delayed RNN using delayed differential equations on a time scale. Internal memory can be utilized to describe the calculation of the future states using discrete and distributed delays, which is a representation of the deep learning architecture for artificial RNNs. We focus on qualitative behavior and stability study of the system. Special attention is paid to taking into account the effect of the time-scale parameters on neural network dynamics. Here, we delve into the exploration of exponential stability in RNN models on a time scale that incorporates multiple discrete and distributed delays. Two approaches for constructing exponential estimates, including the Hilger and the usual exponential functions, are considered and compared. The Lyapunov-Krasovskii (L-K) functional method is employed to study stability on a time scale in both cases. The established stability criteria, resulting in an exponential-like estimate, utilizes a tuple of positive definite matrices, decay rate, and graininess of the time scale. The models of RNNs for the two-neuron network with four discrete and distributed delays, as well as the ring lattice delayed network of seven identical neurons, are numerically investigated. The results indicate how the time scale (graininess) and model characteristics (weights) influence the qualitative behavior, leading to a transition from stable focus to quasiperiodic limit cycles.

Proportional fractional Dirac dynamic system

<abstract><p>In this study, considering the proportional fractional derivative, which is a generalization of the conformable fractional derivative, we provided some important spectral properties such as the reality of eigenvalues, the orthogonality of eigenfunctions, the self-adjointness of the operator, the asymptotic estimations of eigenfunctions, and Picone's identity for a proportional Dirac system on an arbitrary time scale. We also presented graphics representing the eigenfunctions of the Dirac system on a time scale, produced by taking advantage of the proportional fractional derivative with some special cases. The main purpose of presenting these graphics was to examine the effect of the proportional fractional derivative on the Dirac system on a time scale, as well as the effect of the eigenvalues, which are meaningful for the subject we were studying for the solution functions.</p></abstract>

Self-Adjoint Sturm-Liouville Dynamic Problem via Proportional Derivative

The concept of a conformable derivative on time scales is a relatively new development in the field of fractional calculus. Traditional fractional calculus deals with derivatives and integrals of non-integer order on continuous time domains. However, time scale calculus extends these concepts to more general time domains that include both continuous and discrete points. The conformable derivative on time scales has several properties that make it advantageous in certain applications. For example, it satisfies a chain rule and has a simple relationship with the conformable integral, which facilitates the development of differential equations involving fractional order dynamics. It also allows for the analysis of systems with both continuous and discrete data points, making it suitable for modeling and control applications in various fields, including physics, engineering, and finance. In this study, the Sturm-Liouville problem and its properties are examined on an arbitrary time scale using the proportional derivative, a more general form of the fractional derivative. Important spectral properties such as self-adjointness, Green formula, Lagrange identity, Abel formula, and orthogonality of eigenfunctions for this problem are expressed in proportional derivatives on an arbitrary time scale.

Oscillation of noncanonical fourth-order dynamic equations

We present two sufficient conditions for the oscillatory behaviour of a class of noncanonical fourth-order dynamic equations on arbitrary time scales. An illustrative example is included to show that new criteria improve related results reported in the literature.

Searching for Short-Timescale Transients in Gamma-ray Telescope Data

Astrophysical sources show variability in their emissions over a range of timescales, with transients such as fast radio bursts (FRBs) and magnetar giant flares (MGFs) showing variability on timescales as short as a few milliseconds. Recent advances in gamma-ray astronomy such as telescopes’ high temporal resolution and relatively high uptime, combined with follow-up programs between different facilities, should allow serendipitous observations of burst-like phenomena. Even so, no very-high-energy gamma-ray counterparts for FRBs have been detected so far, and there is a general lack of software tools suited to search for such phenomena. We present a tool capable of searching gamma-ray telescope data for transient phenomena over arbitrary timescales—it is based on the Gammapy package and recursively scans the given field of view for clusters of events within user-defined time and angular-separation intervals. The generalized implementation allows for its application in many other cases and multiple gamma-ray telescopes. The main features and methodology of the developed tool are presented here, along with an analysis of the open gamma ray telescope data performed using it.

Inequalities for interval-valued Riemann diamond-alpha integrals

We propose the concept of Riemann diamond-alpha integrals for time scales interval-valued functions. We first give the definition and some properties of the interval Riemann diamond-alpha integral that are naturally investigated as an extension of interval Riemann nabla and delta integrals. With the help of the interval Riemann diamond-alpha integral, we present interval variants of Jensen inequalities for convex and concave interval-valued functions on an arbitrary time scale. Moreover, diamond-alpha Hölder’s and Minkowski’s interval inequalities are proved. Also, several numerical examples are provided in order to illustrate our main results.

Existence, uniqueness, and controllability for Hilfer differential equations on times scales

We introduce a new version of ‐Hilfer fractional derivative, on an arbitrary time scale. The fundamental properties of the new operator are investigated, and in particular, we prove an integration by parts formula. Using the Laplace transform and the obtained integration by parts formula, we then propose a ‐Riemann–Liouville fractional integral on times scales. The applicability of the new operators is illustrated by considering a fractional initial value problem on an arbitrary time scale, for which we prove existence, uniqueness, and controllability of solutions in a suitable Banach space. The obtained results are interesting and nontrivial even for the following particular choices: (i) of the time scale, (ii) of the order of differentiation, and/or (iii) function , opening new directions of investigation. Finally, we end the article with comments and future work.

Deferred statistical boundedness on time scales

In this paper, we first define deferred statistical boundedness of $ \Delta $-measurable real valued functions on an arbitrary time scale. Some theorems relating deferred statistical convergence and deferred statistical boundedness on time scales have been included. Furthermore, we established some inclusion theorems relating statistical boundedness and deferred statistical boundedness on time scales. Finally, we have provided a general description of inclusion between two deferred statistical methods of boundedness on time scales.

FLOQUET MULTIPLIERS AND THE STABILITY OF PERIODIC LINEAR DIFFERENTIAL EQUATIONS: A UNIFIED ALGORITHM AND ITS COMPUTER REALIZATION

Floquet multipliers (characteristic multipliers) play significant role in the stability of the periodic equations. Based on the iterative method, we provide a unified algorithm to compute the Floquet multipliers (characteristic multipliers) and determine the stability of the periodic linear differential equations on time scales unifying discrete, continuous, and hybrid dynamics. Our approach is based on calculating the value of $\mathcal{A}$ and $\mathcal{B}$ (see Theorem 3.1), which are the sum and product of all Floquet multipliers (characteristic multipliers) of the system, respectively. We obtain an explicit expression of $\mathcal{A}$ (see Theorem 4.1) by the method of variation and approximation theory (iterative method), and an explicit expression of $\mathcal{B}$ by Liouville's formula. Furthermore, a computer program is designed to realize our algorithm. Specifically, you can determine the stability of a second order periodic linear system, whether they are discrete, continuous or hybrid, as long as you enter the program codes associated with the parameters of the equation. In fact, few literatures have dealt with the algorithm to compute the Floquet multipliers, not mention to design the program for its computer realization. Our algorithm gives the explicit expressions of all Floquet multipliers and our computer program is based on the approximations of these explicit expressions. In particular, on an arbitrary discrete periodic time scale, we can do a finite number of calculations to get the explicit value of Floquet multipliers (see Theorem 4.2). Therefore, for any discrete periodic system, we can accurately determine the stability of the system by our algorithm even without computer! Finally, in Section 6, several examples are presented to illustrate the effectiveness of our algorithm.

∆λ−statistical boundedness of order α on time scales

In this study, we introduce the notions ∆λ−statistical convergence of order α (for αϵ(0,1]) and λp−summable of order α (for αϵ(0,1]) on an arbitrary time scale. Moreover, some relations about these notions are obtained. We define ∆λ− statistically boundedness of order α (for αϵ(0,1]) on a time scale. Furthermore, We give connections between S (λ,α) T (b) , S (β,θ) T (b) and ST (b) for various sequences µ∆λ(t) and µ∆β(t) are determined in class Λ.

New Hille Type and Ohriska Type Criteria for Nonlinear Third-Order Dynamic Equations

The objective of this paper is to derive new Hille type and Ohriska type criteria for third-order nonlinear dynamic functional equations in the form of a2(ζ)φα2a1ζφα1xΔ(ζ)ΔΔ+q(ζ)φαx(g(ζ))=0, on a time scale T, where Δ is the forward operator on T, α1, α2, α>0, and g, q, ai, i = 1, 2, are positive rd-continuous functions on T, and φθ(u):=uθ−1u. Our results in this paper are new and substantial for dynamic equations of the third order on arbitrary time scales. An example is included to illustrate the results.

Novel dynamic Hardy‐type inequalities on time scales

In the present article, we proved some new dynamic inequalities of Hardy‐type on an arbitrary time scale. We have used dynamic Minkowski inequality, dynamic Hölder's inequality, dynamic Jensen's inequality and Fubini theorem on time scales to prove our main results. As special case of our results, some Hardy‐type integral inequalities known in the literature have been recaptured, and some original discrete inequalities of Hardy‐type have been established.

Controllability analysis of nonlinear switched singular system on time scales

AbstractIn this manuscript, we establish the controllability results for a nonlinear singular switched system on arbitrary time scales. We used the mathematical induction to evaluate the solution of the nonlinear slow switched subsystem and nonlinear fast switched subsystem. Also, we study the necessary and sufficient conditions for the controllability of the nonlinear singular switched system followed by the controllability of both the nonlinear switched subsystems. The tools for study include Banach fixed point theorem and pseudo‐inverse of matrices. At last, we illustrate an example to show the efficacy of the proposed approach.

Weighted dynamic Hardy-type inequalities involving many functions on arbitrary time scales

The objective of this paper is to prove some new dynamic inequalities of Hardy type on time scales which generalize and improve some recent results given in the literature. Further, we derive some new weighted Hardy dynamic inequalities involving many functions on time scales. As special cases, we get continuous and discrete inequalities.

Deep neural network methods for solving forward and inverse problems of time fractional diffusion equations with conformable derivative

Fractional diffusion equations with conformable derivative have become an important research topic in Newtonian mechanics, quantum mechanics, arbitrary time scale problems, diffusion transport, neutron dynamics and other fields. In this paper, we propose to study time fractional diffusion equations with conformable derivative by using physics-informed neural networks (PINNs) for the first time. By solving the supervised learning task, we design a new spatio-temporal function approximator with high data efficiency. L-BFGS algorithm is used to optimize our loss function, and back propagation algorithm is used to update our parameters to give the numerical solutions. Since the conformable derivative satisfies the Leibniz rule and the chain rule, we can directly derive the network through automatic differentiation without any additional numerical discretization or truncation. For the forward problem, we can take IC/BCs as the data, and use PINNs to solve the corresponding partial differential equation. Three numerical examples are carried out to demonstrate the effectiveness of our methods. In particular, when the order of the conformable fractional derivative α tends to 1, a class of weighted PINNs is introduced to overcome the accuracy degradation caused by the singularity of solutions. For the inverse problem, we use the obtained data to train the neural network and specify the estimation of the parameter λ in the equation. Numerical results show that the proposed method is easy to implement and can accurately identify parameters, even when the training data is corrupted by 1% uncorrelated noise.

The generalized Fourier convolution on time scales

In this paper, we deduct some properties of the Fourier transform on arbitrary time scales. We define the generalized shifting problem and we prove the existence of solutions. We define a generalized convolution on anarbitrary time scale and we deduct and prove the generalized convolution theorem.