Geometric approach to global asymptotic stability for the SEIRS models in epidemiology

In this paper, we present a more general criterion for the global asymptotic stability of equilibria for nonlinear autonomous differential equations based on the geometric criterion developed by Li and Muldowney. By applying this criterion, we obtain some results for the global asymptotic stability of SEIRS models with constant recruitment and varying total population size. Based on these results, we give a complete affirmative answer to Liu–Hethcote–Levin conjecture. Furthermore, an affirmative answer to Li–Graef–Wang–Karsai’s problem for SEIR model with permanent immunity and varying total population size is given.     

Modelling of two-phase incompressible flows in ducts

An improved level-set method for capturing interfaces combined with a second-order projection method for solving the time-dependent incompressible Navier–Stokes equations is implemented to compute two-phase viscous flows of Newtonian fluids on Cartesian staggered meshes. Various arrays of deformable droplets are suspended in an ambient fluid of different viscosity in ducts and the deformation and migration of droplets as they move along the duct are computed and discussed.     

Effect of inhomogeneity in energy transfer through alpha helical proteins with interspine coupling

The dynamics of homogeneous and inhomogeneous alpha helical proteins with interspine coupling is under investigation in this paper by proposing a suitable model Hamiltonian. For specific choice of parameters, the dynamics of homogeneous alpha helical proteins is found to be governed by a set of completely integrable three coupled derivative nonlinear Schrödinger (NLS) equations (Chen–Lee–Liu equations). The effect of inhomogeneity is understood by performing a perturbation analysis on the resulting perturbed three coupled NLS equation. An equivalent set of integrable discrete three coupled derivative NLS equations is derived through an appropriate generalization of the Lax pair of the original Ablowitz–Ladik lattice and the nature of the energy transfer along the lattice is studied.     

A note on exact travelling wave solutions for the modified Camassa-Holm and Degasperis-Procesi equations

Exact solutions for the modified Camassa-Holm and Degasperis-Procesi equations by Liu et al. (2010) [Y.F. Liu, X.Y. Zhu, J.X. He, Factorization technique and new exact solutions for the modified Camassa-Holm and Degasperis-Procesi equations, Appl. Math. Comput. 217 (2010) 1658-1665] are investigated. Liu et al. has used the factorization technique to reduce the modified Camassa-Holm and Degasperis-Procesi equations to first-order ordinary differential equations, and then derived some exact travelling wave solutions by direct integral method. In this note, we will explain that the implementation of the so-called factorization technique is completely unnecessary. Moreover, based on the method of complete discrimination system for polynomial, we shall demonstrate that the general explicit exact solution and its classification for the above two types of equations can be obtained directly and many exact solutions by Liu et al. are our special cases. Besides, some known results in previously relevant literatures are extended and some simple remarks are also made.     

Differential equation of a fractal electrode–electrolyte interface

Understanding an electrode–electrolyte interface (EEI) behavior is a valuable tool in several areas of science. There are models based on discrete fractal structures, which explain the measurements of linear and non-linear impedance at fixed frequencies, or at determined ranges of high and low current densities. A level by level discrete calculation is needed to evaluate these models, or the use of black-box models, which affect the good understanding of the phenomenon. A continuous model based on a differential equation of an EEI is presented in this paper. It includes an electrical circuit similar to a long transmission line. It has been deduced from the discrete Liu model.     

A remark for “Oscillation theorems for second-order nonlinear partial difference equations”

In this remark, we shall show a counter example for the main result of the paper [S.T. Liu, Y.Q. Liu, Oscillation theorems for second-order nonlinear partial difference equations, J. Comput. Appl. Math. 132 (2001) 479-482].     

Exponential stability for stochastic neutral partial functional differential equations

In this paper, we consider a class of stochastic neutral partial functional differential equations in a real separable Hilbert space. Some conditions on the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths are obtained. The known results in Govindan [T.E. Govindan, Almost sure exponential stability for stochastic neutral partial functional differential equations, Stochastics 77 (2005) 139-154], Liu and Truman [K. Liu, A. Truman, A note on almost sure exponential stability for stochastic partial functional differential equations, Statist. Probab. Lett. 50 (2000) 273-278] and Taniguchi [T. Taniguchi, Almost sure exponential stability for stochastic partial functional differential equations, Stoch. Anal. Appl. 16 (1998) 965-975; T. Taniguchi, Asymptotic stability theorems of semilinear stochastic evolution equations in Hilbert spaces, Stochastics 53 (1995) 41-52] are generalized and improved.     

Almost sure exponential stability for stochastic neutral partial functional differential equations

In this paper, we initiate a study on stochastic neutral partial functional differential equations in a real separable Hilbert space. Our goal here is to study the existence and uniqueness of a mild solution of this class of equations and also the exponential stability of the moments of a mild solution as well as its sample paths. The results obtained here generalize the main results from [Taniguchi, Stochastics and Stochastics Reports, 53, (1995) 41–52], [Taniguchi, Stochastic Analysis and Applications, 16, (1998) 965–975] and [Liu and Truman, Statistics Probability Letters, 50, (2000) 273–278]. An example is given to illustrate the theory.     

A time integration procedure for plastic deformation in elastic-viscoplastic metals

A simple unconditionally stable numerical procedure for time integration of the flow rule for large plastic deformation of an elastic-viscoplastic metal is developed. Specific attention is focused on a unified set of constitutive equations which represents a generalization (for large deformation and thermomechanical response) of the Bodner-Partom model [6, 7]. An analytical solution is obtained for large deformation simple shear at constant shear rate. Numerical examples of simple shear, a corner test exhibiting the transition from uniaxial compression to shear, and simple tension are considered which demonstrate the stability and accuracy of the procedure. It is shown that the same procedure can be used for a rate insensitive metal characterized by a yield function as well as for a rate sensitive metal characterized by an overstress model. Finally, an appendix is provided which records the basic equations associated with the small deformation theory.     

HOPF BIFURCATION AND OTHER DYNAMICAL BEHAVIORS FOR A FOURTH ORDER DIFFERENTIAL EQUATION IN MODELS OF INFECTIOUS DISEASE

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Internal Flow Analysis for Slow Moving Small Droplets in Contact with Hydrophobic Surfaces

Internal fluid flow behavior for slow moving small droplets in contact with hydrophobic surfaces is analyzed. The shape of the droplet is first computed using the Young-Laplace equation. For this purpose a Finite Element (FE) model [1], in which contact constraints are enforced through Penalty and Augmented Lagrange Multiplier methods, is used. The flow field within the droplet is then analyzed using the Stokes flow model, considering a de-coupled approach. Similar to the membrane deformation model, the formulation for the flow analysis is also expressed in the framework of FE analysis. Both, stabilized (Pressure Stabilizing/Petrov-Galerkin PSPG) and Galerkin FE formulations are considered. The motion of the fluid inside the droplet is governed by the slip condition enforced on the membrane of the droplet. Numerical examples for droplets rolling steadily are presented. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic dynamics of the one‐dimensional attraction–repulsion Keller–Segel model

The asymptotic behavior of the attraction–repulsion Keller–Segel model in one dimension is studied in this paper. The global existence of classical solutions and nonconstant stationary solutions of the attraction–repulsion Keller–Segel model in one dimension were previously established by Liu and Wang (2012), which, however, only provided a time‐dependent bound for solutions. In this paper, we improve the results of Liu and Wang (2012) by deriving a uniform‐in‐time bound for solutions and furthermore prove that the model possesses a global attractor. For a special case where the attractive and repulsive chemical signals have the same degradation rate, we show that the solution converges to a stationary solution algebraically as time tends to infinity if the attraction dominates. Copyright © 2014 John Wiley & Sons, Ltd.     

Hydraulic hysteresis effects on the coupled flow–deformation processes in unsaturated soils: Numerical formulation and slope stability analysis

The hysteresis of water retention curve has a profound influence on the coupled hydro-mechanical behaviors in unsaturated soils, but numerical implementation with consideration of this property was rarely reported due to the difficulties in the integration of the coupled constitutive models. In this study, a numerical formulation is proposed for modeling the coupled flow–deformation processes with hydraulic hysteresis. A return mapping scheme is developed to integrate the water retention curve model with hydraulic hysteresis and the elasto-plastic model simultaneously within a time step, and the deformation-dependent nature of the water retention curve is considered rigorously by modifying the coefficient matrices in the discretized governing equations. The performance and efficiency of the proposed numerical formulation is validated by two existing laboratory tests and a computational example, demonstrating better performance and convergence of the proposed formulation. The proposed procedure is then applied for modeling the coupled flow–deformation processes in a soil slope under rain infiltration. The simulated results reveal the significant effects of hydraulic hysteresis on the coupled water–air two-phase flow and elasto-plastic deformation processes. The solid deformation and the evolution of the shear band would be remarkably overestimated, and the slope failure would be early predicted when neglecting hydraulic hysteresis.     

Stochastic partial functional differential equations with locally monotone coefficients,locally Lipschitz non-linearity and delay

In this paper, we study the existence and uniqueness of strong solutions for stochastic partial functional differential equations with locally monotone coefficients, locally Lipschitz non-linearity, and time delay. Our results extend previous results obtained by Liu–Röckner, Caraballo et al. and Taniguchi et al. Examples are given to illustrate the wide applicability of our results.     

A phenomenological constitutive model for magnetostrictive materials and ferroelectric ceramics

The contribution is concerned with a thermodynamic consistent constitutive model for magnetostrictive materials and ferroelectric ceramics. It captures the nonlinear phenomenological behavior which is described by hysteresis effects. Magnetostrictive alloys and ferroelectric ceramics belong to the multifunctional materials. In recent years these materials have become widely–used in actor and sensor applications. They characterize an inherent coupling between deformation and magnetic or electric field. Due to the similarities of the coupled differential equations a uniform approach is applied for both phenomena. The presented three–dimensional material model is thermodynamically motivated. It is based on the definition of a specific free energy function and a switching criterion. Furthermore an additive split of strain and the magnetic or electric field in a reversible and an irreversible part is suggested. The irreversible quantities serve as internal variables, which is analog to plasticity theory. A one–to–one–relation between the two internal variables provides conservation of volume for the irreversible strains. The presented material model can approximate the ferromagnetic or ferroelectric hysteresis curve and the related butterfly hysteresis. Furthermore an extended approach for ferrimagnetic behavior, which occurs in magnetostrictive materials, is presented. Some numerical simulations demonstrate the capability of the presented model. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A mathematical model for horizontal axis wind turbine blades

A mathematical model describing the nonlinear vibration of horizontal axis wind turbine (HAWT) blades is proposed in this paper. The system consists of a rotating blade and four components of deformation including longitudinal vibration (named axial extension), out-of-plane bend (named flap), in-plane/edgewise bend (named lead/lag) and torsion (named feather). It is assumed that the center of mass, shear center and aerodynamic center of a cross section all lie on the chord line, and do not coincide with each other. The structural damping of the blade, which is brought about by materials and fillers is taken into account based on the Kelvin–Voigt theory of composite materials approximately. The equivalent viscosity factor can be determined from empirical data, theoretical computation and experimental test. Gravitational loading and aerodynamic loading are considered as distributed forces and moments acting on blade sections. A set of partial differential equations governing the coupled, nonlinear vibration is established by applying the generalized Hamiltonian principle, and the current model is verified by previous models. The solution of equations is discussed, and examples concerning the static deformation, aeroelastic stability and dynamics of the blade are given.     

A derivative-free three-term projection algorithm involving spectral quotient for solving nonlinear monotone equations

ABSTRACT

In this paper, we develop a three-term conjugate gradient method involving spectral quotient, which always satisfies the famous Dai-Liao conjugacy condition and quasi-Newton secant equation, independently of any line search. This new three-term conjugate gradient method can be regarded as a variant of the memoryless Broyden-Fletcher-Goldfarb-Shanno quasi-Newton method with regard to spectral quotient. By combining this method with the projection technique proposed by Solodov and Svaiter in 1998, we establish a derivative-free three-term projection algorithm for dealing with large-scale nonlinear monotone system of equations. We prove the global convergence of the algorithm and obtain the R-linear convergence rate under some mild conditions. Numerical results show that our projection algorithm is effective and robust, and is more competitive with the TTDFP algorithm proposed Liu and Li [A three-term derivative-free projection method for nonlinear monotone system of equations. Calcolo. 2016;53:427–450].     

Modeling of aerosol spray characteristics for synthesis of sensor thin film from solution

A model is developed of aerosol spray for synthesis of sensor film from solution. The synthesis technique considered involves atomization of a solution of mixed salts in methanol, spraying of solution droplets, droplet deposition on a heated substrate, evaporation and chemical reaction to produce mixed oxides, and subsequent film growth. The precise control of oxide nanoparticle size distribution and inter-particle spacing in the film is crucial to achieving high sensitivity. These in turn largely depend on the droplet characteristics prior to impingement on the substrate. This paper focuses on the development of a model to describe the atomization and spray processes prior to the film growth. Specifically, a mathematical model is developed utilizing computational fluid dynamics solution of the equations governing the transport of atomized droplets from the nozzle to the substrate in order to predict droplet characteristics in flight. The predictions include spatial distribution of droplet size and concentration, and the effect on these characteristics of swirling inlet flow at the spray nozzle.     

Discretisation procedures for multi-physics phenomena

Procedures are described for solving the equations governing a multi-physics process. Finite volume techniques are used to discretise, using the same unstructured mesh, the equations of fluid flow, heat transfer with solidification, and solid deformation. These discretised equations ofe then solved in an integrated manner. The computational mechanics environment, PHYSICA, which facilitates the building of multi-physics models, is described. Comparisons between model predictions and experimental data are presented for the casting of metal components.     

On numerical schemes for phase-field models for electrowetting with electrolyte solutions

We present an energy-stable, decoupled discrete scheme for a recent model (see [1]) supposed to describe electrokinetic phenomena in two-phase flow with general mass densities. This model couples momentum and Cahn–Hilliard type phase-field equations with Nernst–Planck equations for ion density evolution and an elliptic transmission problem for the electrostatic potential. The transport velocities in our scheme are based on the old velocity field updated via a discrete time integration of the force densities. This allows to split the equations into three blocks which can be treated sequentially: The phase-field equation, the equations for ion transport and electrostatic potential, and the Navier–Stokes type equations. By establishing a discrete counterpart of the continuous energy estimate, we are able to prove the stability of the scheme. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)