This work brings a one-dimensional logistic harvesting model with Allee effect to the time-varying framework. This new framework is more sober than the autonomous version of the system because it; the framework, permits all environment-dependent coefficients to depend on time. Based on these coefficients, we derive sets of conditions that drive population to “mathematical” extinction. More precisely, we investigate various local and global stability notions including uniform stability, attractivity, asymptotic stability and the (uniform) exponential stability.

In this study, our aim is to provide a modification of the so-called Ismail-May operators that preserve exponential functions eAx, A ∈ ℝ. In consonance to this, we begin with estimating the convergence rate of the operators in terms of usual and exponential modulus of continuity. We also provide a global approximation and a quantitative Voronovskaya result. Moreover, to validate the modification, we exhibit some graphical representations using Mathematica software to compare the original operator and its modification. We conclude that the modified operators not only preserve exponential functions but also provide faster rate of convergence when A > 0.

In the paper, we consider the local-in-time and the global-in-time infinity-ion-mass convergence of bipolar Euler-Maxwell systems by setting the mass of an electron me=1 and letting the mass of an ion mi→∞. We use the method of asymptotic expansions to handle the local-in-time convergence problem and find that the limiting process from bipolar models to unipolar models is actually decoupling, but not the vanishing of equations for the corresponding the other particle. Moreover, when the initial data is sufficiently close to the constant equilibrium state, we establish the global-in-time infinity-ion-mass convergence.

In this paper, the Laplace transform combined with the local discontinuous Galerkin method is used for distributed-order time-fractional diffusion-wave equation. In this method,at first, we convert the equation to some time-independent problems by Laplace transform.Then we can solve these stationary equations by the local discontinuous Galerkin method to discretize diffusion operators at the same time. Then, by using a numerical inversion of the Laplace transform we can find the solutions of the original equation. One of the advantages of this procedure is its parallel implementation. Another advantage of this approach is that the number of stationary problems that should be solved is much less than that are needed in time-marching methods. Finally, some numerical experiments have been provided to show the accuracy and efficiency of the method.

In this paper, using the Lyapunov-like functions, the practical stability with respect to part of the variables of fractional-order nonlinear systems depending on a small parameter is studied.

Changing and moving toward online shopping has made it necessary to customize customers' needs and provide them more selective options. The buyers search the products' features before deciding to purchase items. The recommender systems facilitate the searching task for customers via narrowing down the search space within the specific products that align the customer needs. Clustering, as a typical machine learning approach, is applied in recommender systems. As an information filtering method, a recommender system clusters user's data to indicate the required factors for more accurate predictions by calculating the similarity between members of a cluster. In this study, using the Gaussian mixture model clustering and considering the scores distance and the value of scores in the Pearson correlation coefficient, a new method is introduced for predicting scores in machine learning recommender systems. To study the proposed method's performance, a Movie Lens data set is evaluated, and the results are compared to some other recommender systems, including the Pearson correlation coefficients similarity criteria, K-means, and fuzzy C-means algorithms. The simulation results indicate that our method has less error than others by increasing the number of neighbors. The results also illustrate that when the number of users increases, the proposed method's accuracy will increase. The reason is that the Gaussian mixture clustering chooses similar users and considers the scores distance in choosing similar neighbors to the active user.

This paper analyzes a SIS model for infectious diseases with two classes of individuals with different susceptibilities. It focuses in a transition function between both classes of susceptible individuals depending on the density of the infected population. A classification of all the possible bifurcation diagrams that the model can present is done. Specifically, some conditions for the simultaneous existence of backward bifurcation and multiple endemic states are shown.

The aim of this article is to study the relationships and models among the Van den Dool and Kratz equation, the gas chromatography, and the Bézier curves constructed by aid of the Bernstein polynomials. Another aim of this article is to introduce open problems that contribute to real-world problems involving mathematics, chemistry, and plant biology, including the Van den Dool and Kratz equation, the gas chromatography, and Bézier curves. Searching for the solutions of these problems may have qualities that will create the potential that can enter the field of study of many researchers. As a result of these goals, the usability of Bézier curves was investigated while determining the chemical composition of essential oil obtained from P. Aladaghensis Leblebici. By applying the retention index from the Van den Dool and Kratz equation and evaluating chemical compositions of the essential oil are characterized by gas chromatography-mass spectrometry (GC-MS). The Van den Dool and Kratz equation results have the potential to be used not only in the chemical compositions of the oils, but also in applied mathematics and other fields. Moreover, we construct a new special finite sum. A lower bound and inequality are also given for the finite special sum involving the dead time associated with the isocratic step. Some applications and criticisms are given that include this lower bound and inequality for these sums and its effects on the chemical compositions of essential oil and the Van den Dool and Kratz equation.

Using the invariance group properties of the system describing one-dimensional unsteady flow of a non-ideal dusty reacting gas, the evolutions of characteristic shock and singular surface are obtained. The influence of non-idealness, reaction mechanism and dusty gas parameters on the characteristic shock and singular surface are analyzed. Further, the amplitude of reflected wave and (or) transmitted wave, which generated from the interaction of a characteristic shock with a singular surface, and bounce in the acceleration of shock are shown.

In this article, we consider the relatively exact controllability of fractional stochastic delay system (FSDS) driven by Lévy noise. Firstly, we derive the solution of linear FSDS via delayed matrix functions of Mittag-Leffler (M-L). Subsequently, by virtue of the controllability Grammian matrix, we explore the relatively exact controllability of linear FSDS. In addition, with the aid of Jensen’s inequality, Hölder’s inequality and Itô’s isometry, the existence and uniqueness of the considered nonlinear FSDS are investigated by employing Banach contraction principle.Thereafter, the relatively exact controllability of nonlinear FSDS is discussed. Finally, the theoretical results are supported through an example.

The purpose of this paper is to analyze a new kind of Hadamard fractional boundary value problem combining integral boundary condition and multipoint fractional integral boundary condition on an infinite interval. By the help of the Bai-Ge's fixed point theorem, multiplicity results of positive solutions are derived for the Hadamard fractional boundary value problem. In the end, to illustrative the main result, an example is also presented.

We consider a 2-D study of a plate with finite thickness and infinite extent. The upper plate surface is considered traction free and subjected to an axisymmetric heating. The lower surface is thermally insulated and layed on a rigid foundation. A cylindrical heat source affects the plate. This problem is relevant to the generalized thermoelasticity theory with one relaxation time. Laplace and Hankel transforms are considered. We use Inverse Hankel and Laplace transforms numerically. All related functions are showed graphically.

Let a particle start at some point in the unit interval I := [0, 1] and undergo Brownian motion in I until it hits one of the end points. At this instant the particle stays put for a finite holding time with an exponential distribution and then jumps back to a point inside I with a probability density μ0 or μ1 parametrized by the boundary point it was from. The process starts afresh. The same evolution repeats independently each time. Many probabilistic aspects of this diffusion process are investigated in the paper [10]. The authors in the cited paper call this process diffusion with holding and jumping (DHJ). Our simple aim in this paper is to analyze the eigenvalues of a nonlocal boundary problem arising from this process.

The aim of this paper is to give an alternative technique for the derivation of a prior energy estimate. Consequently, this allows to define a natural energy norm of the long-term well-posedness result established by S. Israwi in [2] but for the original system, in which the partial operator ∇× is not involved.

The article is dedicated towards the study of fractional-order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions in Banach spaces. First, mild solution of the system is constructed and subsequently, its existence is proven using Banach's fixed point theorem. Then, results of approximate controllability are established using the concept of fractional semigroup and an iterative technique. Suitable examples are given in the end supporting the methodology along with pointing out corrections in examples presented in previous articles.

In this letter, two time delay dynamic models, TDD-NCP model and Fudan-CCDC model, are introduced to track the data of COVID-19. The TDD-NCP model is developed recently by Cheng's group group in Fudan and SUFE. The TDD-NCP model introduced the time delay process into the differential equations to describe the latent period of the epidemic. The Fudan-CDCC model is established when Wenbin Chen suggested to determine the kernel functions in the TDD-NCP model by the public data from CDCC. By the public data of the cumulative confirmed cases in different regions in China and different countries, these models can clearly illustrate that the containment of the epidemic highly depends on early and effective isolations.

The present paper studies a fractional-order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type-II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely basic reproduction number R0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model’s analysis shows that two equilibria exist, namely, disease-free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity R0. If R0 < 1, then DFE is globally stable, leading to eradicating the population’s disease. If R0 > 1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters into disease transmission. The numerical solution of the model is obtained by the generalized Adams- Bashforth-Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results.

This paper considers minimizers of the following inhomogeneous $L^2$-subcritical energy functional \[E(u):=\int_{\R^N}|\nabla u|^{2}dx-\frac{2}{p+1}\int_{\R^N}m(x)|u|^{p+1}dx,%\ u\in H^{1}(\R^N), \] under the mass constraint $\|u\|^{2}_{2}=M$. Here $N\geq1$, $p\in(1,1+\frac{4}{N})$, $M>0$ and the inhomogeneous term $m(x)$ satisfies $0