In the present paper we investigate the qualitative behaviour of a fractional SEIR model with general incidence rate function and time delay where the fractional derivative is defined in the Caputo sense. The basic reproduction number $\mathcal{R}_{0}$ is derived using the method of next generation matrix and we give a complete study of local stability of both free and endemic equilibrium. Using Liapunov method we prove the global stability of free and endemic equilibrium under some hypotheses on the parameters of the system. Finally to illustrate our results, we use the model to predict the first peak of the COVID-19 epidemic in Algeria.

In this paper we study the partial differential equation with piecewise constant argument of the form : \[ \begin{array}{lll} x_t(t,s)=&A(t)x(t,s)+B(t,s)x([t],s)+C(t,s)x(t,[s])+\\[0.5cm] &D(t,s)x([t],[s])+f(x(t,[s])),\ \ t,s\in \R^{+}=(0,\infty) \end{array} \] where $A(t)$ is a $k\times k$ invertible and continuous matrix function on $\R^{+}$, $B(t,s)$, $C(t,s)$, $D(t,s)$ are $k \times k$ continuous and bounded matrix functions on $\R^{+}\times \R^{+}$, $[t]$, $[s]$ are the integral parts of $t,s$ respectively and $f:\R^k\rightarrow \R^k$ is a continuous function. More precisely under some conditions on the matrices $A(t)$, $B(t,s)$, $C(t,s)$, $D(t,s)$ and the function $f$ we investigate the asymptotic behaviour of the solutions of the above equation. \end{abstract}

The paper is dedicated to the modeling and asymptotic investigation of a linear elasticity problem, in the form of variational inequality, for a textile structure. The textile is made of long and thin fibers crossing each others, forming a periodic squared domain. The domain is clamped only partially and an in plane sliding between the fibers is bounded by a contact function, which is chosen to be loose. We also assume a non-penetration condition for the fibers. Both partial clamp and loose contact arise a domain split, leading to different behaviors in each of the four parts. The homogenization is made via unfolding method, with an additional dimension reduction to further simplify the problem. The four cell problems are inequalities heavily coupled by the outer plane macro-micro constraints, while the macroscopic limit problem results to be an inequality of Leray-Lions type with only macro in plane constraints. On both scales, no uniqueness is expected.

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.

A spectral problem is considered in a domain $\Omega_{\varepsilon}$ that is the union of a domain $\Omega_{0}$ and a lot of thin trees situated $\varepsilon$-periodically along some manifold on the boundary of $\Omega_{0}.$ The trees have finite number of branching levels. The perturbed Robin boundary condition $\partial_{\nu}u^{\varepsilon} + \varepsilon^{\alpha_i} k_{i,m}u^{\varepsilon} = 0$ is given on the $i$th branching layer; $\{\alpha_i\}$ are real parameters. The asymptotic analysis of this problem is made as $\varepsilon\to0,$ i.e., when the number of the thin trees infinitely increases and their thickness vanishes. In particular, the Hausdorff convergence of the spectrum to the spectrum of the corresponding nonstandard homogenized spectral problem is proved, the leading terms of asymptotics are constructed, and the corresponding asymptotic estimates are justified for the eigenvalues and eigenfunctions.

In this paper, we propose a numerical scheme of the predictor-corrector type for solving nonlinear fractional initial value problems, the chosen fractional derivative is called the Atangana-Baleanu derivative defined in Caputo sense (ABC). This proposed method is based on Lagrangian quadratic polynomials to approximate the nonlinearity implied in the Volterra integral which is obtained by reducing the given fractional differential equation via the properties of the ABC-fractional derivative. Through this technique, we get corrector formula with high accuracy which is implicit as well as predictor formula which is explicit and has the same precision order as the corrective formula. On the other hand, the so-called memory term is computed only once for both prediction and correction phases, which indicates the low cost of the proposed method. Also, the error bound of the proposed numerical scheme is offered. Furthermore, numerical experiments are presented in order to assess the accuracy of the new method on two differential equations. Moreover, a case study is considered where the proposed predictor-corrector scheme is used to obtained approximate solutions of a coupled non-autonomous ABC-fractional differential system.

The nodes and their connection relationships (CRs) are the two main parts for a complex dynamical network (CDN). In existing theoretical studies about the outer synchronization, the nodes are considered as the main part of synchronization phenomena mainly associated by coupling effect of CRs between nodes. However, if the CRs between nodes are time-varying, they can also be regarded as one dynamic system coupled with the nodes, and thus their state may evolve with time and maybe assist the nodes to achieve the outer synchronization. From the angle of large-scale systems, a CDN can be regarded as two interconnected subsystems, one of which is the node subsystem (NS) and the other is the link subsystem (LS). Hence, how the whole dynamic of LS contributes to the outer synchronization of NS is one of worthy research problem. In this paper, the two CDNs are considered with the unknown interaction. In each CDN, the dynamics of NS is modelled as the vector differential equation, the LS is modelled as the Riccati matrix differential equation, and the two kinds of differential equations are coupled with each other. By employing the above dynamic models of CDNs and the synthesized coupling terms in the two LSs, the adaptive controller of NS is synthesized for the response CDN. The results show that the outer synchronization happens when the two LSs tracking the synthesized auxiliary dynamic tracking targets. Finally, the numerical simulation is given to show the effectiveness of the theoretical results in this paper.

We are concerned with the existence and qualitative properties of travelling wave solutions for a quasilinear reaction-diffusion equation on the real line. We consider a non-Lipschitz reaction term of Fisher--KPP type and a discontinuous diffusion coefficient that allows for degenerations and singularities at equilibrium points. We investigate the joint influence of the reaction and diffusion terms on the existence and nonexistence of travelling waves and, assuming these terms are of power-type near equilibria, we provide classification of solutions based on their asymptotic properties. Our approach provides a broad theoretical background for the mathematical treatment of rather general models not only in population dynamics but also in other applied sciences and engineering.

By taking Sugeno-derivative into account, firstly, we investigate the existence of solutions to the initial value problems (IVP) of first-order differential equations with respect to non-additive measure (more precisely, distorted Lebesgue measure). It particularly occurs in the mathematical modeling of biology. We begin by expressing the differential equation in terms of ordinary derivative and the derivative with respect to the distorted Lebesgue measure. Then, by using the fixed point theorem on cones, we construct an operator and prove the existence of positive increasing solutions on cones in semi-order Banach spaces. In addition, we also use Picard’s-Lindel\”of theorem to prove the existence and uniqueness of the solution of the equation. Second, we investigate the existence of a solution to the boundary value problem (BVP) on cones with integral boundary conditions of a mix-order differential equation with respect to non-additive measures. Moreover, the Krasnoselskii fixed point theorem is also applied to both BVP and IVP and obtains at least one positive increasing solution. Examples with graphs are provided to validate the results.

This paper investigates the theoretical modeling and coupled free vibration behaviors of a rotating double-bladed shaft assembly resting on elastic supports in a spacecraft system. According to the Kirchhoff plate theory and the Euler-Bernoulli beam theory, the theoretical model is established. The studied rotor is considered to be made of porous foam metal matrix and graphene nanoplatelet (GPL) reinforcement. Non-uniform distributions of porosity and graphene nanoplatelets (GPLs) are taken into account and lead to functionally graded (FG) structures. The effective material properties of the double-bladed shaft are varying along the radius and thickness direction of the shaft and blade, respectively. Moreover, the rule of mixture, the Halpin-Tsai model, and the open-cell scheme are used to determine its material properties. Considering the gyroscopic effect, the Lagrange equation is utilized to derive the coupled equations of motion. Then the traveling wave frequencies of the double-bladed shaft assembly is obtained by employing the assumed modes method and substructure modal synthesis method. A detailed parametric analysis is conducted to examine the effects of the rotating speed, GPL weight fraction, GPL distribution pattern, GPL length-to-thickness ratio, GPL length-to-width ratio, porosity coefficient, porosity distribution pattern, shaft length-to-radius ratio, blade length-to-thickness ratio, support stiffness and support location on the free vibration behaviors of the double-bladed shaft assembly.

In this paper, a deep dynamical analysis is made using tools from multidimensional real discrete dynamics of some derivative-free iterative methods with memory. They all have good qualitative properties, but one of them (due to Traub) shows the same behavior as Newton’s method on quadratic polynomials. Then, the same techniques are employed to analyze the performance of several multipoint schemes with memory, whose first step is Traub’s method, but their construction was made using different procedures. Therefore, their stability is analyzed, showing which is the best in terms of the wideness of basins of convergence or the existence of free critical points that would yield convergence towards different elements from the desired zeros of the nonlinear function. Therefore, the best stability properties are linked with the best estimations made in the iterative expressions rather than their simplicity. These results have been checked with a numerical and graphical comparison with many other known methods with and without memory, with different orders of convergence, with excellent performance.

In this paper, we consider the low Mach number limit problem for the compressible viscous micropolar fluid model based on the concept of dissipative measure-valued solutions. We prove that the dissipative measure-valued solutions of the compressible micropolar fluid model converge to the smooth solution of the incompressible micropolar system in the case of well-prepared initial data when the Mach number tends to zero.

This paper investigates the long time well-posedness of 2-D MHD boundary layer equation without resistivity. It is proved that if the initial data satisfies \begin{align*} \|(u_0,h_0-1)\|_{H_{\mu}^{3,0}}+\|(u_0,h_0-1)\|_{H_{\mu}^{1,2}}+\|(u_0,h_0-1)\|_{H_{\mu}^{2,1}}\le \varepsilon, \end{align*} then the life span of the solution is at least of order $\varepsilon^{-\f43}$.

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 paper, we are concerned with the non-relativistic limit of a class of computable approximation models for radiation hydrodynamics. The models consist of the compressible Euler equations coupled with moment closure approximations to the radiative transfer equation. They are first-order partial differential equations with source terms. As hyperbolic relaxation systems, they are showed to satisfy the structural stability condition proposed by W.-A. Yong (1999). Base on this, we verify the non-relativistic limit by combining an energy method with a formal asymptotic analysis.

In this paper we study the existence of a Nash-type equlibrium for a non-potential nonlinear system by combining variational methods with the monotonicity approach. The advance over existing research is that we can consider systems of Dirichlet problems in which the operator is not necessarily linear.

Abstract: In this paper, the nonlinear Schrödinger-type equation −(∇ + iA) ^2 u + u + λ[I_α*(K|u|^2)]Ku=af(|u|)u/|u| in R ^3 is considered in the presence of magnetic field, where A ∈ C ^1 (R ^3 ,R^ 3 ), α ∈ (0,3), I_α denotes the Riesz potential, K ∈ L^ p (R ^3 ,(0,∞)) for some p ∈ (6/(1 + α),∞], a ∈ L^ q (R 3 ,[0,∞)) \ {0} for some q ∈ (3/2,∞], and f ∈ C(R,[0,∞)) is assumed to be asymptotically linear at infinity. Under suitable assumptions regarding A, K, a, and f, variational methods are used to establish the existence of ground-state solutions of the above equation for sufficiently small values of the parameter λ.

In this paper, we derive an Euler-Maruyama (EM) method for a class of multi-term fractional stochastic nonlinear differential equations, and prove its strong convergence. The strong convergence order of this EM method is $\min\{\alpha_{m}-0.5,~\alpha_{m}-\alpha_{m-1}\}$, where $\{\alpha_{i}\}_{i=1}^{m}$ is the order of Caputo fractional derivative satisfying that $1>\alpha_{m}>\alpha_{m-1}>\cdots>\alpha_{2}>\alpha_{1}>0$, $\alpha_{m}>0.5$, and $\alpha_{m}+\alpha_{m-1}>1$. Then, a fast implementation of this proposed EM method is also presented based on the sum-of-exponentials approximation technique. Finally, some numerical experiments are given to verify the theoretical results and computational efficiency of our EM method.