In this work, we demonstrate uniqueness of the weak solution to the fractional anisotropic Navier-Stokes system with only horizontal dissipation.

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.

A bosonic Laplacian is a conformally invariant second order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita-Schwinger type operators and bosonic Laplacians, we solve Poisson’s equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson’s equation in Euclidean space is also provided. In the end, we provide Green’s formulas for bosonic Laplacians in scalar-valued and Clifford-valued cases, respectively. These formulas reveal that bosonic Laplacians are self-adjoint with respect to a given L2 inner product on certain compact supported function spaces.

This paper is concerned with an initial and boundary value problem of the Navier-Stokes equations for compressible viscous barotropic flow subject to large external potential forces in a half space $R^3_+$ with Navier’s boundary conditions. The global well-posedness of strongsolutions with large oscillations and vacuum is established, provided that the initial energy is suitably small and that the unique steady state is strictly away from vacuum. As a by-product,the stability of stationary solution is obtained.

In this article, we study the exponential behavior of 3D stochastic primitive equations driven by fractional noise. Since fractional Brownian motion is essentially different from Brownian motion, the standard method via classic stochastic analysis tools is not available. Here, we develop a method which is close to the method from dynamic system to show that the weak solutions to 3D stochastic primitive equations driven by fractional noise converge exponentially to the unique stationary solution of primitive equations. This method may be applied to other stochastic hydrodynamic equations and other noises including Brownian motion and Lévy noise.

In this paper, we consider the Cauchy problem for three-dimensional isentropic compressible radiation hydrodynamic equations with density-dependent viscosity coefficients. When the viscosity coefficients are given as power of density ($\rho^\delta$ with $\delta>1$), we establish the local-in-time existence of classical solutions containing a vacuum for large initial data. Here, we point out that the initial layer compatibility conditions are not necessary.

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 this paper, we employ three integration algorithms namely, the well known Kudryashove method, the new Kudryashov method and the unified Riccati equation expansion method to extract optical soliton solutions for the generalized Kudryashov's equation with power nonlinearities. Straddled soliton, bright solitons, dark solitons and singular solitons have been found

Recent researches show that virus-to-cell infection and cell-to-cell transmission are two HIV infection modes. In this paper, we propose a delayed HIV infection model including both virus-to-cell infection and cell-to-cell transmission and CTL immune response. The time delay describes the phenomenon between viral entry and viral production. We show the nonnegativity and boundedness of solution, obtain the equilibrium points and prove local asymptotic stability of the equilibrium points. Then the optimal control problem with antiretroviral therapy and pharmacological delay is posed. We establish and analyze two types of objective functions, one is linear control and the other is quadratic control. Numerical simulations have been performed to verify the stability of equilibrium points and show the optimal control strategies and the effects of control on cells concentration by Matlab and Lingo.

The aim of this work is to investigate the exponential mean-square stability for neutral stochastic differential equations with time-varying delay and Poisson jumps. We give some conditions that all the drift, diffusion and jumps coefficients can be nonlinear, to obtain the stability of the analytic solution. It is revealed that the implicit backward Euler-Maruyama numerical solution can reproduce the corresponding stability of the analytic solution under these nonlinear conditions. This is different from the explicit Euler-Maruyama numerical solution whose stability depends on the linear growth condition. With some requirements related to the delay function and the property of compensated Poisson process, we deal with time-varying delay and Poisson jumps. One highly nonlinear example is provided to confirm the effectiveness of our theory.

The nano/microelectromechanical systems (N/MEMS) have been caught much attention in the past few decades for their attractive properties such as small size, high reliability, batch fabrication, and low power consumption. The dynamic oscillatory behavior of these systems is very complex due to strong nonlinearities in these systems. The basic aim of this manuscript is to investigate the nonlinear vibration property of N/MEMS oscillators by the homotopy perturbation method coupled with Laplace transform (also called as He-Laplace method in literature). A generalized N/MEMS oscillator is systematically studied, and a fairly accurate analytic solution is obtained. Three special cases for electrically actuated MEMS, multi-walled Carbon nanotubes-based MEMS, and MEMS subjected to van der Waals attraction are considered for comparison, and a good agreement with exact solutions is observed.

The spatiotemporal dynamics for general reaction-diffusion systems of Brusselator type under the homogeneous Neumann boundary condition is considered. It is shown that the reaction-diffusion system has a unique steady state solution. For some suitable ranges of the parameters, we prove that the steady state solution can be a codimension-2 Turing-Hopf point. To understand the spatiotemporal dynamics in the vicinity of the Turing-Hopf bifurcation point, we calculate and analyze the normal form on the center manifold by analytical methods. A wealth of complex spatiotemporal dynamics near the degenerate point are obtained. It is proved that the system undergoes a codimension-2 Turing-Hopf bifurcation. Moreover, several numerical simulations are carried out to illustrate the validity of our theoretical results.

There are a few purely periodic phenomena in nature, which allows one to consider several other generalizations, such as almost automorphic and measure pseudo almost automorphic oscillations. In this paper, by developing important properties on the composition of functions with reflection, using some exponential dichotomy properties and an application of the fixed-point theorem, several new sufficient conditions for the existence and the uniqueness of an pseudo almost automorphic solutions with measure for some general type reflection integro-differential equations. We suppose that the nonlinear part is measure pseudo almost automorphic and in which we distinguish the two constant and variable cases for the lipschitz coefficients of the functions associated with this part. It is assumed that the linear part of the equation considered admits an exponential dichotomy. Finally, an application is given on the very interesting model of Markus and Yamabe.

The main purpose of this paper is to study the following semi-linear structurally damped wave equation with nonlinearity of derivative type: $$u_{tt}- \Delta u+ \mu(-\Delta)^{\sigma/2} u_t= |u_t|^p,\quad u(0,x)= u_0(x),\quad u_t(0,x)=u_1(x),$$ with $\mu>0$, $n\geq1$, $\sigma \in (0,2]$ and $p>1$. In particular, we are going to prove the non-existence of global weak solutions by using a new test function and suitable sign assumptions on the initial data in both the subcritical case and the critical case.

The paper discusses the numerical investigation involving forced convective heat transfer (HT) in the laminar flow regime is carried out for nanofluid (NF) and hybrid nanofluid (HNF) in a microtube and wavy microchannel. Water-based Al2O3 nanofluid and water-based Al2O3-Ag hybrid nanofluid is studied for this purpose. Reynolds Number (Re), temperature, volume fraction, and nanoparticle (NP) size are varied for the analysis at a constant heat transfer rate. Numerical results characterizing the performances of nanofluid and hybrid nanofluid are presented in terms of the local heat transfer coefficient. It is found that with the increase in Reynolds number, volume fraction and temperature, local heat transfer coefficient is increased. Comparison of nanofluid and hybrid nanofluid reveals superior heat transfer property of the later. However, microtube exhibits better heat transfer coefficient than the wavy channel at constant heat flux, length and area.

This article concerns with the thermoelastic corner-edge type system with singular potential function on a wedge manifold with corner singularities. First, we introduce weighted $p-$Sobolev spaces on manifolds with corner-edge singularities. Then, we prove the corner-edge type Sobolev inequality , Poincar$\acute{e}$ inequality and Hardy inequality and obtain some results about the compactness of embedding maps on the weighted corner-edge Sobolev spaces. Finally, as an application of these results, we apply the potential well theory and the Faedo-Galerkin approximations to obtain the global weak solutions for the thermoelastic corner-edge type system.

Compartmental model is the classic and widely used approach in pharmacokinetics. Recently, fractional calculus is introduced to describe the time course of drugs in human body which follow the anomalous diffusion mechanism. To consider the different fractional order transmission process, the two compartmental fractional model has been proposed and studied. And, it will be of great significance to find out a simple and efficient numerical method to solve the model and estimate model parameters. This work investigates two numerical methods of the two compartmental fractional model using finite difference schemes, which are based on the shifted Grünwald-Letnikov approximate formula and L1 formula, respectively. The hybrid Nelder-Mead simplex search and particle swarm optimization, denoted as NMSS-PSO, is provided to estimate the fractional model parameters. Comparison between the numerical solution and the solution by the numerical inverse Laplace transform method establishes the validity of the developed numerical algorithms. Then, the influence of the order of fractional derivative on the drug amount in human body is also discussed. Finally, the two compartmental fractional model is applied to fit the amiodarone pharmacokinetics data based on the NMSS-PSO algorithms. This work will be of importance for the development of fractional pharmacokinetics.

This paper is concerned with the bounded fractal and Hausdorff dimension of the pullback attractors for 2D non-autonomous incompressible Navier-Stokes equations with constant delay terms. Using the construction of trace formula with two bases for phase spaces of product flow, the upper boundedness of fractal dimension has been achieved.