We consider the equations of motion of a bar, with given density, infinite in both directions, subjected to longitudinal vibrations under the action of an external load, and a stress-strain relation represented by a fractional order operator.  Using three types of fractional operators, the initial-boundary value problems associated with the described phenomenon are posed and solved. Through the bivariate Mittag-Leffer function, which has been recently introduced, we find the fundamental solution of these problems and calculate their moments.

A novel representation of spin 1/2 combines in a single geometric object the standard Pauli spin operator and spin state. Under the spin-position decoupling approximation it consists of the sum of three orthogonal vectors comprising a gauge phase. In the one-spin case the representation: (1) is Hermitian; (2) endowed with handedness; (3) yields all standard results, including the total spin angular momentum S=(√3 ℏ)⁄2; (4) relates basis spins by proper rotations, thus preserving handedness; (5) allows formalizing irreversibility in spin measurement. In the bipartite case: (1) entangled spins have precisely related gauge phases and opposite handedness; (2) maximally entangled spins relate by one of the four improper rotations in 3D: plane-reflections (triplets) and inversion (singlet); (3) the full spin expressions yield the standard total two-spin angular momentum; (4) all standard expected values for bipartite observations follow. Depending on whether spin operations act one – or two – sided, the formalism appears in two complementary forms, the ‘spinor’ or the ‘vector’ form, respectively. The proposed scheme provides a clear geometric picture of spin transformations and correlations in the 3D physical orientation space.

In this paper, we investigate nonhomogeneous incompressible Navier-Stokes-Landau-Lifshitz system in 2-D. This system consists of Navier-Stokes equations coupled with Landau-Lifshitz-Gilbert equation, an evolutionary equation for the magnetization vector. We establish a blowup criterion for the two-dimensional incompressible Navier-Stokes-Landau-Lifshitz system with finite positive initial density.

The aim of this article is to investigate the well-posedness and stability problems of the so-called Kawahara equation under the presence of an interior delayed damping. The system is shown to be well-posed. Furthermore, we prove that the trivial solution is exponentially stable in spite of the delay effect. Specifically, local and semi-global stability results are established according to the properties of the spatial distribution of the delay term.

A document by Xueqin Peng. Click on the document to view its contents.

In this paper, we investigate a two-grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semi-linear elliptic equation on the coarse mesh with mesh size H, then, we use the coarse mesh solution as a initial guess to linearize the semilinear equation on the fine mesh, i.e., on the fine mesh (with mesh size $h$), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution u has sufficient regularity and $h=H^2$, the two-grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.

A document by Yuk Leung. Click on the document to view its contents.

In this paper, to investigate the synthetic effect of PrEP (pre-exposure prophylaxis) and ART (antiretrovial therapy) on HIV transmission among MSM (men who have sex with men) in heterogenous environment, an realistic HIV epidemic model with spatial diffusion is established. Here, HIV infectious people are divided into three immunity based compartments, i.e., CD4+ T cell count less than 350, between 350 and 500, and more than 500, respectively. The basic reproduction number $R_0$ is established and proved as a threshold parameter: The global asymptotic stability of the disease-free steady state holds for $R_0<1$, and the disease will be present if $R_0>1$. Considering the substantial advantages of PrEP and ART in controlling HIV transmissions among MSM, the optimal control problem is presented for the case of positive constant diffusion coefficients, which minimize the total population of susceptible individual and HIV infected individual, the cost of PrEP and ART thearpy. As an illustration of our theoretical results, we conduct numerical simulations. We also conduct an optimal control case study where model parameters are estimated from the demographic and epidemiological data from China. This work suggests: (1) Spatial factors cannot be ignored during the HIV intervention; (2)Taking the PrEP intervention measure for HIV transmissions among MSM as early as possible will help to improve the control efficiency and reduces its cost; (3) Reducing the PrEP drug costs will promote the efficiency of PrEP treatment in preventing the spread of HIV among MSM.

In the present paper, we study the existence of nontrivial solutions of the following one-dimensional fractional Schr\“{o}dinger equation $$ (-\Delta)^{1/2}u+V(x)u=f(x,u), \ \ x\in \R, $$ where $(-\Delta)^{1/2}$ stands for the $1/2$-Laplacian, $V(x)\in \mathcal{C}(\R, (0,+\infty))$, and $f(x,u):\R\times\R\to \R$ is a continuous function with an exponential critical growth. Comparing with the existing works in the field of exponential-critical-growth fractional Schr\”{o}dinger equations, we encounter some new challenges due to the weaker assumptions on the reaction term $f$. By using some sharp energy estimates, we present a detailed analysis of the energy level, which allows us to establish the existence of nontrivial solutions for a wider class of nonlinear terms. Furthermore, we use the non-Nehari manifold method to establish the existence of Nehari-type ground state solutions of the one-dimensional fractional Schr\”{o}dinger equations.

This paper investigates the boundary finite-time stabilization of fractional reaction-diffusion systems (FRDSs). First, a distributed controller is designed, and sufficient conditions are obtained to ensure the finite-time stability (FTS) of FRDSs under the designed controller. Then, a boundary controller is presented to achieve the FTS. By virtue of Lyapunov functional method and inequality techniques, sufficient conditions are presented to ensure the FTS of FRDSs via the designed boundary controller. The effect of diffusion term of FRDSs on the FTS is also investigated. Both Neumann and mixed boundary conditions are considered. Moreover, the robust finite-time stabilization of uncertain FRDSs is studied when there are uncertainties in the system’s coefficients. Under the designed boundary controller, sufficient conditions are presented to guarantee the robust FTS of uncertain FRDSs. Finally, numerical examples are presented to verify the effectiveness of our theoretical results.

This article proposes an adaptive neural output feedback control scheme in combination with state and disturbance observers for uncertain fractional-order nonlinear systems containing unknown external disturbance, input saturation and immeasurable state. The radial basis function neural network (RBFNN) approximation is used to estimate unknown nonlinear function, and a state observer as well as a fractional-order disturbance observer is developed simultaneously by using the approximation output of the RBFNN to estimate immeasurable states and unknown compounded disturbances, respectively. Then, a fractional-order auxiliary system is constructed to compensate the effects caused by the saturated input. In addition, by introducing a dynamic surface control strategy, the tedious analytic computation of time derivatives of virtual control laws in the conventional backstepping method is avoided. The proposed method guarantees that the boundness of all signals in the closed loop system and the tracking errors converge to a small neighbourhood around the origin. Finally, two examples are provided to verify the effectiveness of the proposed control method.

We study the connected components of the stochastic geometry model on Poisson points which is obtained by connecting points with a probability that depends on their relative position. Equivalently, we investigate the random clusters of the ran- dom connection model defined on the points of a Poisson process in d-dimensional space where the links are added with a particular probability function. We use the thermodynamicrelationsbetweenfreeenergy,entropyandinternalenergytofindthe functions of the cluster size distribution in the statistical mechanics of extensive and non-extensive. By comparing these obtained functions with the probability function predicted by Penrose, we provide a suitable approximate probability function. More- over, we relate this stochastic geometry model to the physics literature by showing how the fluctuations of the thermodynamic quantities of this model correspond to other models when a phase transition (10.1002/mma.6965, 2020) occurs. Also, we obtain the critical point using a new analytical method.

We consider the nonlinear massless wave equation belonging to some family of the Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime. We prove the global in time small data solutions for supercritical powers in the case of decelerating expansion universe.

This paper deals with a class of fractional Schr\”{o}dinger-Poisson system \[\begin{cases}\displaystyle (-\Delta )^{s}u+V(x)u-K(x)\phi |u|^{2^*_s-3}u=a (x)f(u), &x \in \R^{3}\\ (-\Delta )^{s}\phi=K(x)|u|^{2^*_s-1}, &x \in \R^{3}\end{cases} \]with a critical nonlocal term and multiple competing potentials, which may decay and vanish at infinity, where $s \in (\frac{3}{4},1)$, $ 2^*_s = \frac{6}{3-2s}$ is the fractional critical exponent. The problem is set on the whole space and compactness issues have to be tackled. By employing the mountain pass theorem, concentration-compactness principle and approximation method, the existence of a positive ground state solution is obtained under appropriate assumptions imposed on $V, K, a$ and $f$.

In this paper, we establish a new phytoplankton-zooplankton model by considering the effects of plankton body size and stochastic environmental fluctuations. Mathematical theory work mainly gives the existence of boundary and positive equilibria, and shows their local as well as global stability in the deterministic model. Additionally, we explore the dynamics of V-geometric ergodicity, stochastic ultimate boundedness, stochastic permanence, persistence in the mean, stochastic extinction and the existence of a unique ergodic stationary distribution in the corresponding stochastic version. Numerical simulation work mainly reveals that plankton body size can generate great influences on the interactions between phytoplankton and zooplankton, which in turn proves the effectiveness of mathematical theory analysis. It is worth emphasizing that for the small value of phytoplankton cell size, the increase of zooplankton body size can not change the phytoplankton density or zooplankton density; for the middle value of phytoplankton cell size, the increase of zooplankton body size can decrease zooplankton density or phytoplankton density; for the large value of phytoplankton body size, the increase of zooplankton body size can increase zooplankton density but decrease phytoplankton density. Besides, it should be noted that the increase of zooplankton body size can not affect the effect of random environmental disturbance, while the increase of phytoplankton cell size can weaken its effect. There results may enrich the dynamics of phytoplankton-zooplankton models.

In this paper, we consider the stabilization of the generalized Rao-Nakra beam equation, which consists of four wave equations for the longitudinal displacements and the shear angle of the top and bottom layers and one Euler-Bernoulli beam equation for the transversal displacement. Dissipative mechanism are provided through viscous damping for two displacements. The location of the viscous damping are divided into two groups, characterized by whether both of the top and bottom layers are directly damped or otherwise. Each group consists of three cases. We obtain the necessary and sufficient conditions for the cases in group two to be strongly stable. Furthermore, polynomial stability of certain orders are proved. The cases in group one are left for future study.

In this article, we study the global well-posedness and large-time behaviors of solutions to the two-dimensional tropical climate system with zero thermal diffusion for a small initial data in the whole space. The main approaches include high and low frequency decomposition method and exploiting the structure of system (1) to obtain the estimates of thermal dissipation. We utilize the time decay properties of the kernels to a linear differential equation to obtain the decay rates of solutions of the low frequency part and the decay property of exponential operator for the high frequency part. The key ingredient here is the explicit large-time decay rate of solutions.

We are concerned with the global existence and decay rates of large solutions for the Poisson--Nernst--Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin--Lerner type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species is small enough. Moreover, the large solution is obtained for initial data belonging to the low regularity Besov spaces with different regularity and integral indices for the different charged species, which indicates more specific coupling relations between the negatively and positively charged species.