Application of a Mickens finite‐difference scheme to the cylindrical Bratu‐Gelfand problem

The boundary value problem Δu + λe^u = 0 where u = 0 on the boundary is often referred to as “the Bratu problem.” The Bratu problem with cylindrical radial operators, also known as the cylindrical Bratu‐Gelfand problem, is considered here. It is a nonlinear eigenvalue problem with two known bifurcated solutions for λ < λ_c, no solutions for λ > λ_c and a unique solution when λ = λ_c. Numerical solutions to the Bratu‐Gelfand problem at the critical value of λ_c = 2 are computed using nonstandard finite‐difference schemes known as Mickens finite differences. Comparison of numerical results obtained by solving the Bratu‐Gelfand problem using a Mickens discretization with results obtained using standard finite differences for λ < 2 are given, which illustrate the superiority of the nonstandard scheme. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 327–337, 2004     

Mathematical model for the dynamics of visceral leishmaniasis–malaria co‐infection

A mathematical model to understand the dynamics of malaria–visceral leishmaniasis co‐infection is proposed and analyzed. Results show that both diseases can be eliminated if R₀, the basic reproduction number of the co‐infection, is less than unity, and the system undergoes a backward bifurcation where an endemic equilibrium co‐exists with the disease‐free equilibrium when one of R_m or R_l, the basic reproduction numbers of malaria‐only and visceral leishmaniasis‐only, is precisely less than unity. Results also show that in the case of maximum protection against visceral leishmaniasis (VL), the disease‐free equilibrium is globally asymptotically stable if malaria patients are protected from VL infection; similarly, in the case of maximum protection against malaria, the disease‐free equilibrium is globally asymptotically stable if VL and post‐kala‐azar dermal leishmaniasis patients and the recovered humans after VL are protected from malaria infection. Numerical results show that if R_m and R_l are greater than unity, then we have co‐existence of both disease at an endemic equilibrium, and malaria incidence is higher than visceral leishmaniasis incidence at steady state. Copyright © 2016 John Wiley & Sons, Ltd.     

On the existence of steady flows of a Navier–Stokes liquid around a moving rigid body

We prove the existence of a strong solution to the three‐dimensional steady Navier–Stokes equations in the exterior of an obstacle undergoing a rigid motion. Unlike the classical exterior problem for the Navier–Stokes equations, that only takes into account the translational motion of the obstacle, is this case, the obstacle can also rotate. Assuming the total flux of the velocity field through the boundary to be sufficiently small, we first construct approximating solutions in bounded regions Ω_R = Ω∩ {x ∈ ?³:∣x∣< R} invading the liquid domain Ω. A set of estimates independent of R are shown to hold for the approximating solutions which allows to obtain a strong solution by taking the limit R→∞. Copyright © 2004 John Wiley & Sons, Ltd.     

Well‐posedness for compressible Euler equations with physical vacuum singularity

An important problem in the theory of compressible gas flows is to understand the singular behavior of vacuum states. The main difficulty lies in the fact that the system becomes degenerate at the vacuum boundary, where the characteristic speeds u ± c coincide and have unbounded spatial derivative since c behaves like x^1/2 close to the boundary. In this paper, we overcome this difficulty by presenting a new formulation and new energy spaces. We establish the local‐in‐time well‐posedness of one‐dimensional compressible Euler equations for isentropic flows with the physical vacuum singularity in some spaces adapted to the singularity. © 2009 Wiley Periodicals, Inc.     

A new approach to the nonlinear stability of viscous flow in a coplanar magnetic field

We present a new Lyapunov function for laminar flow, in the x‐direction, between two parallel planes in the presence of a coplanar magnetic field for three‐dimensional perturbations with stress‐free boundary planes that provides conditional nonlinear stability for all Reynolds numbers(R_e) and magnetic Reynolds numbers(R_m) below π²/2M. Compared with previous results on the nonlinear stability of this problem, the radius of stability ball and the energy decay rate obtained in this paper are independent of the magnetic field. Copyright © 2016 John Wiley & Sons, Ltd.     

Bifurcation analysis of HIV infection model with antibody and cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV‐1 infection with antibody, cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response is investigated. The stability of the infection‐free and infected steady states is investigated. The basic reproduction number R₀ is identified for the proposed system. If R₀ < 1, then there is an infection‐free steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R₀ > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Copyright © 2014 John Wiley & Sons, Ltd.     

Arcwise connectedness of the boundary of connected self-similar sets

Summary Let Tbe the connected attractor of injective contractions f₁,... f_mon R^d that satisfy the Open Set Condition. We show that ∂Tis arcwise connected. In particular, the boundary of the Levy dragon and those of the fundamental domains of canonical number systems are arcwise connected.     

On the longtime behavior of a 2D hydrodynamic model for chemically reacting binary fluid mixtures

Two dimensional diffuse interface model for a chemically reacting incompressible binary fluid in a bounded domain is considered. The corresponding evolution system consists of the Navier–Stokes equations for the (averaged) fluid velocity that are nonlinearly coupled with a convective Cahn–Hilliard–Oono type equation for the difference ψ of two fluid concentrations. The effects of a (reversible) chemical reaction is represented in the latter equation by an additional term of the form ε(ψ ? c₀), ε > 0. Here, c₀ is the stationary spatial average of ψ, provided that, for example, no‐slip and no‐flux boundary conditions are considered. The mass is not necessarily conserved unless the spatial average of the initial datum for ψ coincides with c₀. When ε = 0 (i.e., no chemical reaction), the model reduces to the well‐known Cahn–Hilliard–Navier–Stokes system, which has been investigated by several authors. Here, we want to show that the global dynamic behavior of the system is robust with respect to ε. More precisely, we construct a family of exponential attractors, which is continuous with respect to ε. Copyright © 2013 John Wiley & Sons, Ltd.     

On the Rayleigh wave on a curvilinear boundary between elastic and fluid media

Along the boundary between elastic and fluid media, the surface Rayleigh wave propagates. The velocity of this wave v _R0 in the case of a plane boundary is less than the velocity of the Rayleigh wave v _R on a free plane boundary of an elastic medium and less than the velocity v _P0 in a fluid medium. To investigate the velocity v _R0 in the case of curvilinear boundaries, the propagation of Rayleigh waves under consideration along cylindrical and spherical surfaces is studied. The velocity of the Rayleigh wave depends on the curvature of the wave trajectory and the curvature in the direction perpendicular to the trajectory. Furthermore this velocity depends on the presence or absence of a fluid medium. Bibliography: 5 titles.     

Pullback attractors for nonautonomous 2D Bénard problem in some unbounded domains

In this paper, we study the 2D Bénard problem, a system with the Navier–Stokes equations for the velocity field coupled with a convection–diffusion equation for the temperature, in an arbitrary domain (bounded or unbounded) satisfying the Poincaré inequality with nonhomogeneous boundary conditions and nonautonomous external force and heat source. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite‐dimensional pullback D_σ‐attractor for the process associated to the problem. Copyright © 2013 John Wiley & Sons, Ltd.     

Computable analysis of a boundary‐value problem for the generalized KdV–Burgers equation

In this paper, we investigate the computability of the solution operator of the generalized KdV‐Burgers equation with initial‐boundary value problem. Here, the solution operator is a nonlinear map H^{3m ? 1}(R⁺) × H^m(0,T)→C([0,T];H^{3m ? 1}(R⁺)) from the initial‐boundary value data to the solution of the equation. By a technique that is widely used for the study of nonlinear dispersive equation, and using the type 2 theory of effectivity as computable model, we prove that the solution map is Turing computable, for any integer m ≥ 2, and computable real number T > 0. Copyright © 2014 John Wiley & Sons, Ltd.     

A note on a third‐order multi‐point boundary value problem at resonance

Based on the coincidence degree theory of Mawhin, we prove some existence results for the following third‐order multi‐point boundary value problem at resonance where f: [0, 1] × R³ → R is a continuous function, 0 < ξ₁ < ??? < ξ_m < 1, α_i ∈ R, i = 1, …, m, m ≥ 1 and 0 < η₁ < η₂ < ??? < η_n < 1, β_j ∈ R, j = 1, 2, …, n, n ≥ 2. In this paper, the dimension of the linear space Ker L (linear operator L is defined by Lx = x′^′′) is equal to 2. Since all the existence results for third‐order differential equations obtained in previous papers are for the case dim Ker L = 1, our work is new.     

Qualitative analysis of the SICR epidemic model with impulsive vaccinations

Control of epidemic infections is a very urgent issue today. To develop an appropriate strategy for vaccinations and effectively prevent the disease from arising and spreading, we proposed a modified Susceptible‐Infected‐Removed model with impulsive vaccinations. For the model without vaccinations, we proved global stability of one of the steady states depending on the basic reproduction number R₀. As typically in the epidemic models, the threshold value of R₀ is 1. If R₀ is greater than 1, then the positive steady state called endemic equilibrium exists and is globally stable, whereas for smaller values of R₀, it does not exist, and the semi‐trivial steady state called disease‐free equilibrium is globally stable. Using impulsive differential equation comparison theorem, we derived sufficient conditions under which the infectious disease described by the considered model disappears ultimately. The analytical results are illustrated by numerical simulations for Hepatitis B virus infection that confirm the theoretical possibility of the infection elimination because of the proper vaccinations policy. Copyright © 2012 John Wiley & Sons, Ltd.     

Attractors for a plate equation with nonlocal nonlinearities

This paper contains results on well‐posedness, stability, and long‐time behavior of solutions to a class of plate models subject to damping and source terms given by the product of two nonlinear components [EQUATION1] where Ω is a bounded open set of R ⁿ with smooth boundary, γ ,ρ ?0 and are nonlocal functions. The main result states that the dynamical system {S (t )}_{t ?0} associated with this problem has a compact global attractor. In addition, in the limit case γ  = 0, it is also shown that {S (t )}_{t ?0} has a finite dimensional global attractor by using an approach on quasi‐stability because of Chueshov–Lasiecka (2010). Copyright © 2017 John Wiley & Sons, Ltd.     

Homological connectivity of random k‐dimensional complexes

Let Δ_n?1 denote the (n ? 1)‐dimensional simplex. Let Y be a random k‐dimensional subcomplex of Δ_n?1 obtained by starting with the full (k ? 1)‐dimensional skeleton of Δ_n?1 and then adding each k‐simplex independently with probability p. Let H_k?1(Y; R) denote the (k ? 1)‐dimensional reduced homology group of Y with coefficients in a finite abelian group R. It is shown that for any fixed R and k ≥ 1 and for any function ω(n) that tends to infinity © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Exponential attractor for a planar shear‐thinning flow

We study the dynamics of an incompressible, homogeneous fluid of a power‐law type, with the stress tensor T = ν(1 + µ|Dv|)^p?2Dv, where Dv is a symmetric velocity gradient. We consider the two‐dimensional problem with periodic boundary conditions and p ∈ (1, 2). Under these assumptions, we estimate the fractal dimension of the exponential attractor, using the so‐called method of ??‐trajectories. Copyright © 2007 John Wiley & Sons, Ltd.     

BIFURCATION OF TWO-DIMENSIONAL KURAMOTO-SIVASHINSKY EQUATION

This paper deals with the steady state bifurcation of the K-S equation in two spatial dimensions with periodic boundary value condition and of zero mean. With the increase of parameter a, the steady state bifurcation behaviour can be very complicated. For convenience, only the cases a=2 and a=5 witl be discussed. The asymptotic expressions of the steady state solutions bifurcated from the trivial solution near a=2 and a=5 are given. And the stability of thenontriviat sotutions bifurcated from a=2 is studied. Of course, the cases a=n^2 m^2,n,m∈N(a≠2,5) can be similarly discussed by the same method which is used to discussing the cases a=2 and a= 5.     

Partial Fourier approximation of the Lamé equations in axisymmetric domains

In this paper, we study the partial Fourier method for treating the Lamé equations in three‐dimensional axisymmetric domains subjected to non‐axisymmetric loads. We consider the mixed boundary value problem of the linear theory of elasticity with the displacement û , the body force f̂ ϵ (L₂)³ and homogeneous Dirichlet and Neumann boundary conditions. The partial Fourier decomposition reduces, without any error, the three‐dimensional boundary value problem to an infinite sequence of two‐dimensional boundary value problems, whose solutions û _n (n = 0, 1, 2,…) are the Fourier coefficients of û . This process of dimension reduction is described, and appropriate function spaces are given to characterize the reduced problems in two dimensions. The trace properties of these spaces on the rotational axis and some properties of the Fourier coefficients û _n are proved, which are important for further numerical treatment, e.g. by the finite‐element method. Moreover, generalized completeness relations are described for the variational equation, the stresses and the strains. The properties of the resulting system of two‐dimensional problems are characterized. Particularly, a priori estimates of the Fourier coefficients û _n and of the error of the partial Fourier approximation are given. Copyright © 1999 John Wiley & Sons, Ltd.     

Two-Dimensional Infinite Prandtl Number Convection: Structure of Bifurcated Solutions

This paper examines the bifurcation and structure of the bifurcated solutions of the two-dimensional infinite Prandtl number convection problem. The existence of a bifurcation from the trivial solution to an attractor Σ _R was proved by Park (Disc. Cont. Dynam. Syst. B [2005]). We prove in this paper that the bifurcated attractor Σ _R consists of only one cycle of steady-state solutions and that it is homeomorphic to S¹. By thoroughly investigating the structure and transitions of the solutions of the infinite randtl number convection problem in physical space, we confirm that the bifurcated solutions are indeed structurally stable. In turn, this will corroborate and justify the suggested results with the physical findings about the presence of the roll structure. This bifurcation analysis is based on a new notion of bifurcation, called attractor bifurcation, and structural stability is derived using a new geometric theory of incompressible flows. Both theories were developed by Ma and Wang; see Bifurcation Theory and Applications (World Scientific, 2005) and Geometric Theory of Incompressible Flows with Applications to Fluid Dynamics (American Mathematical Society, 2005).     

Global properties of a class of HIV infection models with Beddington–DeAngelis functional response

In this paper, the global properties of a class of human immunodeficiency virus (HIV) models with Beddington–DeAngelis functional response are investigated. Lyapunov functions are constructed to establish the global asymptotic stability of the uninfected and infected steady states of three HIV infection models. The first model considers the interaction process of the HIV and the CD4 ⁺ T cells and takes into account the latently and actively infected cells. The second model describes two co‐circulation populations of target cells, representing CD4 ⁺ T cells and macrophages. The third model is a two‐target‐cell model taking into account the latently and actively infected cells. We have proven that if the basic reproduction number R₀ is less than unity, then the uninfected steady state is globally asymptotically stable, and if R₀ > 1, then the infected steady state is globally asymptotically stable. Copyright © 2012 John Wiley & Sons, Ltd.