Narrow Results

In this paper, we investigate the existence and uniqueness of equilibrium point for fuzzy interval delayed neural networks with impulses on time scales. And we give the criteria of the global exponential stability of the unique equilibrium point for the neural networks under consideration using Lyapunov method. Finally, we present an example to illustrate that our results are effective.

This paper investigates the existence and uniqueness of solutions for singular second-order boundary value problem on time scales by using mixed monotone method. The theorems obtained are very general and complement the previous known results. When the time scale 𝕋 is chosen as ℝ or ℤ, the problem will be the corresponding continuous or discrete boundary value problem.

We study the bacterial respiration through the numerical solution of the Fairen–Velarde coupled nonlinear differential equations. The instantaneous concentrations of the oxygen and the nutrients are computed. The fixed point solution and the stable limit cycle are found in different parameter ranges as predicted by the linearized differential equations. In a specified range of parameters, it is observed that the system spends some time near the stable limit cycle and eventually reaches the stable fixed point. This metastability has been investigated systematically. Interestingly, it is observed that the system exhibits two distinctly different time scales in reaching the stable fixed points. The slow time scale of the metastable lifetime near the stable limit cycle and a fast time scale (after leaving the zone of limit cycle) in rushing towards the stable fixed point. The gross residence time, near the limit cycle (described by a slow time scale), can be reduced by varying the concentrations of nutrients. This idea can be used to control the harmful metastable lifespan of active bacteria.

The aim of this paper is the numerical treatment of some convection systems with stiff relaxation source-terms. We will first define the notion of stiffness for such systems and will select some prototypical and physical problems. We will introduce a new numerical method in order to solve accurately this type of systems. Numerical comparisons will be performed on the evoked problems.

The relative velocity correlation function of pairs of intermediate mass fragments has been studied for d+Au collisions at 4.4 GeV. Experimental correlation functions are compared to that obtained by multibody Coulomb trajectory calculations under the assumption of various decay times of the fragmenting system. The combined approach with the empirically modified intranuclear cascade code followed by the statistical multifragmentation model was used to generate the starting conditions for these calculations. The fragment emission time is found to be less than 40 fm/c.

The fractional derivative holds historical dependence or memory effects. But it also brings error accumulation of the numerical solutions as well as the theoretical analysis since many properties from the integer order case cannot hold. The fractional difference is discrete counterpart of the fractional derivative. The memory kernel is defined by applying the discrete functions on time scale and avoids the errors from the numerical discretization. It is particularly suitable for fractional modeling with computer implementations. Many applications of fractional dynamics to neural networks, signal processing, time series and big data become possible now. This issue of work presents the latest progress in the neighborhood related to fractional calculus.

The fractional derivative holds historical dependence or non-locality and it becomes a powerful tool in many real-world applications. But it also brings error accumulation of the numerical solutions as well as the theoretical analysis since many properties from the integer order case cannot hold. This paper defines the tempered fractional derivative on an isolated time scale and suggests a new method based on the time scale theory for numerical discretization. Some useful properties like composition law and equivalent fractional sum equations are derived for theoretical analysis. Finally, numerical formulas of fractional discrete systems are provided. As a special case for the step size $h=1$, a fractional logistic map with two-parameter memory effects is reported.

This paper suggests two fractional differences for aftershock modeling with heavy tails. Discrete fractional calculus is a straightforward tool on isolated time scale. On the other hand, the fractional difference also can be derived by standard finite difference method when the difference formula is convergent. The two methods are both adopted and compared in the results. The unknown parameters are determined by use of the least square method where Ya’an earthquake aftershock data is used. It is reported that the discrete fractional calculus is an exact discretization tool without any loss in the memory effects which leads to better results.

The Hermite–Hadamard (HH)-type inequality plays a very important role in the fields of basic mathematics and applied mathematics. In recent years, many scholars have expanded and improved it. Although we have achieved some research results about HH-type inequality, the research on discrete HH-type inequalities has just begun, and a lot of work needs to be improved. In this paper, we introduce $(s,m)$-convex function and present discrete HH-type inequalities on time scale with discrete substitution method. In addition, the Hermite–Hadamard–Fejér(HHF)-type inequalities on time scale will be obtained, where the integrand is $\varphi \phi$, $\varphi$ is $(s,m)$-convex function on $[a,b]$ and $\phi$ is symmetric with respect to $\frac{a+mb}{2}$, our results in some special cases yield the well-known classic HHF-type inequalities. Finally, through the discrete substitution method, we get discrete fractional HH-type inequality and discrete fractional HHF-type inequality for $(s,m)$-convex function.

The purpose of this paper is to investigate the harmonic analysis on the time scale ${𝕋}_q$, $q\in (0,1)$ to introduce $q$-weighted Besov spaces subspaces of $L^p({𝕋}_q)$ generalizing the classical one. Further, using an example of $q$-weighted $w_{\alpha ,\beta }(.;q)$ which is introduced and studied. We give a new characterization of the $q$-Besov space using $q$-Poisson kernel and the $g_1$ Littlewood–Paley operator.

In this paper, we first obtain a new identity for time scales by using a weighted kernel. Then, by using this equality, we prove a weighted Čeby šev inequality. Moreover, we establish a weighted Ostrowski-type inequality by using a method which is different from among in the literature.

This paper deals with the development of scour holes in time and space around individual offshore monopiles. It is based on physical flume tests of a model-scale pile subjected to current and/or irregular water waves. The main focus is on backfilling, i.e. the wave-induced or current-wave-induced deposition of sediment into holes that have been previously scoured. The development of the scour depth, scour volume and a so-called scour shape factor is quantified which may be useful to understand and benchmark the development of scour holes.

The paper discusses fractional integrals and derivatives appearing in the so-called (q, h)-calculus which is reduced for h = 0 to quantum calculus and for q = h = 1 to difference calculus. We introduce delta as well as nabla version of these notions and present their basic properties. Furthermore, we give comparisons with the known results and discuss possible extensions to more general settings.

We propose a fractional time scale to model the calcium ion channels model which is retarded with injection of calcium-chelator Ethylene Glycol Tetraacetic Acid (EGTA). By using this nonlinear time scale and Modified Riemann–Liouville fractional derivative, we convert the integer-order calcium ion channels model into fractional-order differential equations. We also analyze the range of order of fractional-order differential equations to ensure the equilibria of it are asymptotically stable. The simulation results are given to demonstrate the solutions of this model coincide with the real experiment data.

Some integral comparison theorems for scalar second order linear differential equations are extended to dynamic equations on time scales. Several examples and applications to difference equations are given.

A time scale 𝕋 is an arbitrary nonempty closed subset of the real numbers ℝ. We study the existence and nonexistence of positive solutions of a class of second-order superlinear equations defined on 𝕋.

In this paper we survey Reid roundabout theorems for time scale symplectic systems (). These theorems list equivalent conditions for the positivity and nonnegativity of the quadratic functional associated with (). The Reid roundabout theorems in this paper do not impose any normality assumption. We also show that Jacobi systems for nonlinear time scale control problems naturally lead to time scale symplectic systems, and that such a system consists of the Hamiltonian equations corresponding to the weak maximum principle for the quadratic functional .

Necessary and sufficient conditions in terms of the integral of the coefficient are derived under which positive decreasing solutions of a half-linear dynamic equation are (normalized) regularly varying of a known index.