The work reported here concerns with study a generalized nonlinear fractional boundary value problems involving $ \vartheta $- fractional derivative in the Riemann-Liouville sense. The existence and uniqueness of positive solutions to the problem at hand are proved. Our discussion relies on the properties of the Green’s function, the upper and lower solutions method, and the classical fixed point theorems in a cone. Moreover, building upper and lower control functions have an effective role in the analysis. Some examples are offered to justify the validity of theoretical findings.

This article is concerned with an initial-boundary value problem (IBVP) for a new phase-field model describing the evolution of structural phase transition in elastically deformable solid materials. The model consists of an elliptic-parabolic system in which the displacement field and the order parameter both satisfy periodic boundary conditions. We prove the existence of global solutions to this IBVP by applying the method of continuation of local solutions and perform numerical simulations to investigate the microstructure evolution of MnNi alloys by using this new phase-field model.

The objective of the present paper is to investigate the dynamics of Hepatitis B-type virus (HBV) infection through mathematical model. Distinct to the existing mathematical models on HBV, the present model considers the various factors such as immune impairment, the maximum number of T-cells (total carrying capacity), logistic growth term. Besides, for more accuracy, the role of antiretroviral therapies are also involved in the analysis. In addition, time delays are inevitable during the activation of immune response and during the antiretroviral therapy. Considering these factors while formulating the mathematical model which helps to gain insights into the disease progression. With the derived model, the qualitative analysis such as stability analysis, bifurcation analysis and stabilization analysis can be performed to investigate the performance of the model over the period of time. The significance of the model parameters are revealed through Hopf-type bifurcation analysis and the global stability analysis of the proposed model. With the help of dataset values that are extracted from the literature the efficiency of the derived theoretical results are explored.

In this article, we consider the zero dissipation limit of the micropolar equations with temperature-dependent viscosity and heat-conduction coefficient. If the given rarefaction wave to the corresponding conservation systems connects to vacuum at one side, we can construct a sequence of solutions to the micropolar equations which converge to the given rarefaction wave with vacuum as the transport coefficients tend to zero. And the uniform convergence rate can be obtained. The key point in our analysis is how to control the degeneracies of the density, the temperature and the temperature-dependent viscosities at the vacuum region in the zero dissipation limit process.

Quantum Fisher information (QFI) and geometric phase have recently been performed different tasks in quantum information technology. We investigate the QFI and entanglement of a three-level atom in $\Lambda $ configuration interacting with a quantized field mode by using linear entropy. We study the dynamical behavior of the geometric phase based on the engineering of a three-level atomic configuration. We analyze the effect of energy dissipation of the dynamical properties of the geometric phase and the QFI as an entanglement quantifier between the three-level atom and field. We explore the correlation between the engineering geometric phase and QFI in the absence and presence of energy dissipation effect. We have found that the $\mathrm{QFI}$ is very sensitive to the effect of the time dependent coupling and energy dissipation.

We consider the reaction-diffusion equation for fractional Dirichlet-to-Neumann operator with subcritical exponent motivated by electrical impedance tomography (EIT) and a need to overcome the Non-locality of a fractional differential equation for modeling anomalous diffusion. We mainly deal with the asymptotic behavior of global solution and the boundedness of global orbit which allows us to show that any global solution is classical solution using Moser iteration technique.

The substantial, tempered, and shifted fractional derivatives are introduced in a unified framework. Their properties are studied and, in the light of the strict sense criterion for derivative definitions, they are characterized and assessed. In the scope of the framework, new tempered linear sytems and transfer functions are introduced.

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.