In most HIV models, the emergence of backward bifurcation means that the control for basic reproduction number less than one is no longer effective for HIV treatment. In this paper, we study an HIV model with CTL response and cell-to-cell transmission by using the dynamical approach. The local and global stability of equilibria is investigated, the relations of subcritical Hopf bifurcation and supercritical bifurcation points are revealed, especially, the so-called new type bifurcation is also found with two Hopf bifurcation curves meeting at the same Bogdanov-Takens bifurcation point. Forward and backward bifurcation, Hopf bifurcation, saddle-node bifurcation, Bogdanov-Takens bifurcation are investigated analytically and numerically. Two limit cycles are also found numerically, which indicates that the complex behavior of HIV dynamics. Interestingly, the role of cell-to-cell interaction is fully uncovered, it may cause the oscillations to disappear and keep the so-called new type bifurcation persist. Finally, some conclusions and discussions are also given.

In this paper we study exact boundary controllability for a linear wave equation with strong and weak interior degeneration of the coefficients in the principle part of the elliptic operator. The objective is to provide a well-posedness analysis of the corresponding system and derive conditions for its controllability through boundary actions. Passing to a relaxed version of the original problem, we discuss existence and uniqueness of solutions, and using the HUM method we derive conditions on the rate of degeneracy for both exact boundary controllability and the lack thereof.

In this article, the generalized (3+1)-dimensional Korteweg-de Vries-Zakharov-Kuznetsov equation is investigated, which describes the influence of magnetic fields on weak ion-acoustic waves in plasma made up of cool and hot electrons and may be regarded as a nonlinear complicated physical model. To find out some new traveling waves solutions and other exact solutions, the improved $F$-expansion approach and the $\exp(-\phi (\zeta))$-expansion approach is applied to above mentioned nonlinear higher dimensional model. Several solutions have been found, including dark soliton, periodic type solitons, bell shaped solitons, single bell shaped solitons. We also show a graphical representation of a number of exact solutions to the equation, together with a description of their behaviour. The proposed techniques can also be used to solve a range of nonlinear evolution problems in mathematical physics and plasma physics.

In order to provide a significantly richer representation of non-stationary signals appearing in various disciplines of science and engineering, we introduce here a novel fractional nonuniform multiresolution analysis (FrNUMRA) on the spectrum $\Lambda$ given by $\Lambda = \left\{0,\frac{r}{N}\right\}+2\mathbb{Z}$, where $N \geqq 1$ is an integer and $r$ is an odd integer with $ 1 \leqq r \leqq 2N-1,$ such that $r$ and $N$ are relatively prime. The necessary and sufficient condition for the existence of nonuniform wavelets of fractional order is derived and an algorithm is also presented for the construction of fractional NUMRA starting from a fractional low-pass filter $h_{0}^{\alpha}$ with appropriate conditions. Moreover, we provide a complete characterization for the biorthogonality of the translates of the scaling functions of two fractional nonuniform multiresolution analyses and the associated fractional biorthogonal wavelet families.

The present work is concerned with the well-posedness and efficient numerical algorithm for a terminal value problem with a generalized Caputo fractional derivative. We investigate the existence and uniqueness of the solution of the terminal value problem, and consider the continuous dependence of the solutions on the given data. To illustrate our theoretical results, we present a one step algorithm for solving the considered problems. Some numerical examples are shown to illustrate the theoretical results and the efficiency of the numerical method.

In this paper, we consider the generalized approximate boundary synchronization for a coupled system of wave equations with Dirichlet boundary controls. We analyse the relationship between the generalized approximate boundary synchronization and the generalized exact boundary synchronization, give a sufficient condition to realize the generalized approximate boundary synchronization and a necessary condition in terms of Kalman's matrix, and show the meaning of the number of total controls. Besides, by the generalized synchronization decomposition, we define the generalized approximately synchronizable state, and obtain its properties and a sufficient condition for it to be independent of applied boundary controls.

A unified mathematical model of three-phase-lag in a compressed rotating isotropic homogeneous micropolar thermo-viscoelastic medium based on a ramp type thermal shock is developed. An application of this model is carried out to resolve the problem of a perfectly conducting half-space subjected to certain boundary conditions in the presence of an electromagnetic field. Lame's potentials and normal mode analysis techniques are employed to get the general analytical solutions. Specific attention is paid to explore the impact of the rotation, magnetic field, ramp time, as well as initial stress on the distributions of temperature, displacement, stress, and induced electric and magnetic distribution. The findings show that the impact of the rotation, magnetic field, viscous, ramp parameter, initial stress, and phase-lag on the micropolar thermo-viscoelastic medium is noticeable.

Current study investigates thermal-hydraulic performances of turbulent forced hybrid nanofluid flow and heat transfer inside a parabolic through solar collector equipped with turbulators. The main aim of present work is to simulate and investigate the effects of different Reynolds numbers and geometrical parameters on thermal and hydraulic characteristics of the system to achieve the maximum PEC value. The heat transfer fluid is water-based MWCNT-Al2O3 (80%:20%) hybrid nanofluid which makes a Newtonian nanofluid. The RANS equations with the shear-stress (SST) k–ω turbulence model have been employed for modeling the turbulence regime. Based on obtained results, usage of turbulators leads to higher thermal diffusion because of destroying the laminar sub layer, increasing heat transfer surface and convection heat transfer. For θ=180°, the maximum values of average Nusselt number (Nuave) are achieved during all studied range of Reynolds numbers followed by θ=90° and θ=0°, respectively. For HO=15mm, the maximum values of Nuave are achieved followed by HO=10mm and HO=5mm, respectively. The model with HO=5mm has not the maximum Nuave; but, it has lower pressure drop than HO=15mm. For BO=1000mm, the maximum values of Nuave are achieved followed by BO=800mm and smooth channel, respectively. Furthermore, for δ=60°, the maximum Nuave are achieved followed by δ=40° and δ=20°, respectively. For model δ=40° the maximum values of PEC are achieved followed by smooth channel, δ=20° and δ=60°, respectively. The parabolic through solar collector model with θ=180°, HO=15, BO=1000mm, δ=40° and bO=20mm filled with hybrid nanofluid suggested as the optimum model in the present investigation.

In present study, we investigate steady MHD Casson fluid flow effect with stretching porous walls together with thermal radiation amid two equidistant sheets. The heat and mass transport experience on dimensionless parameters Soret and Dufour effect on squeezing flow are interpreted along with the impression of thermal radiation also with heat source/sink. We change the partial differential equations (PDEs) into ordinary differential equations (ODEs) dealing convenient proportionality transformations & after then solve ODEs by Runge-Kutta 4th-order scheme and shooting technology. The sequel of different parameters demonstrated by graph and table of temperature, velocity, and concentration impact. We studied the efficacy of Grashof numbers, Reynolds number, dimensionless Casson fluid parameter, and Hartman/magnetic quantity on velocity, temperature, and on concentration. The skin friction coefficient (Cf), Nusselt number (Nu), and Sherwood number (Sh) are reported and calculated at the end.

This paper introduces advances in the use of Bessel functions (cylinder functions) in achieving optimal sharpness of shapes delineated by collections of holes (dark valley regions) and bright regions in different areas of a video frame. An image hole is an island of low voxel intensities surrounded by varying voxel intensity peaks. The basic approach is to identify dominant collections of lights (high voxel intensity peaks) and darks (low voxel intensity clusters) in video frames. A main finding in this paper is that surface objects (recorded in images) are sharper wherever there are preponderant contrasting differences between image lights and darks. These contrasting differences lead to the highly accurate detection of shape holes (dark valleys) that delineate surface shapes recorded in sequences of video frames). With the use of cylindrical Bessel functions, contrasting differences between peaks and valleys can be measured in terms of voxel intensities and voxel indices.

In this paper, we investigate the decay properties of suspension bridge with memories in one dimension. To prove our results, we use the energy method to build some very delicate Lyapunov functionals that give the desired results.

By using some real analysis techniques, We study the structural characteristics of a multi-parameter Hilbert-type integral inequality with the hybrid kernel, and obtain some equivalent conditions for this inequality. We also consider the operator expression of the equivalent inequalities. The conclusions not only integrates some results of references, but also nds some new Hilbert-type integral inequalities with simple form by choosing suitable parameter values.

In this paper, we study a fractional order time delay for nonlinear financial system. By using Laplace transformation, stability and Hopf bifurcation analysis have been done for the model. Furthermore, numerical simulation has been carried out for better understanding of our results.

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

We consider the generalized Stokes resolvent problem in an infinite layer with Neumann boundary conditions. This problem arises from a free boundary problem describing the motion of incompressible viscous one-phase fluid flow without surface tension in an infinite layer bounded both from above and from below by free surfaces. We derive a new exact solution formula to the generalized Stokes resolvent problem and prove the $$-boundedness of the solution operator families with resolvent parameter $\lambda$ varying in a sector $$ for any $\gamma_0>0$ and $0<\varepsilon<\pi/2$, where $ =\{ \lambda\in\setminus\{0\} \mid \arg\lambda\leq\pi-\varepsilon, \ \lambda>\gamma_0 \}$. As applications, we obtain the maximal $L_p$-$L_q$ regularity for the nonstationary Stokes problem and then establish the well-posedness locally in time of the nonlinear free boundary problem mentioned above in $L_p$-$L_q$ setting. We make full use of the solution formula to take $\gamma_0>0$ arbitrarily, while in general domains we only know the $$-boundedness for $\gamma_0\gg1$ from the result by Shibata. As compared with the case of Neumann-Dirichlet boundary condition studied by Saito, analysis is even harder on account of higher singularity of the symbols in the solution formula.

This note presents corrections to our paper [1]. We also give remarks and additions related to other papers [2]-[15].

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.

This paper studies the non-autonomous non-Newtonian micropolar fluids in two-dimensional bounded domains. We first establish that the generated continuous process of the solutions operator possesses a pullback attractor. Then we verify the existence of statistical solutions by constructing the invariant Borel probability measures. Further, we prove that the statistical solutions possess the degenerated regularity of Lusin's type provided that the Grasshof number associated to the external forces is small enough.