Nonhomogeneous Coefficients Research Articles

Hopf bifurcation of a diffusive SIS epidemic system with delay in heterogeneous environment

This paper performs an in-depth qualitative analysis of the dynamic behavior of a diffusive SIS epidemic system with delay in heterogeneous environment subject to homogeneous Neumann boundary condition. Firstly, we explore the principal eigenvalue to obtain the stability of the disease-free equilibrium (DFE) and the effect of the nonhomogeneous coefficients on the stable region of the DFE. Secondly, we obtain the existence, multiplicity and explicit structure of the endemic equilibrium (EE), i.e., spatially nonhomogeneous steady-state solutions, by using the implicit function theorem and Lyapunov-Schmidt reduction method. Furthermore, by analyzing the distribution of eigenvalues of infinitesimal generators, the stability of EE and the existence of Hopf bifurcations at EE are given. Finally, the direction of Hopf bifurcation and stability of the bifurcating periodic solution are obtained by virtue of normal form theory and center manifold reduction.

The effect of quadrature rules on finite element solutions of Maxwell variational problems

We study the effects of numerical quadrature rules on error convergence rates when solving Maxwell-type variational problems via the curl-conforming or edge finite element method. A complete a priori error analysis for the case of bounded polygonal and curved domains with non-homogeneous coefficients is provided. We detail sufficient conditions with respect to mesh refinement and precision for the quadrature rules so as to guarantee convergence rates following that of exact numerical integration. On curved domains, we isolate the error contribution of numerical quadrature rules.

Complex Rayleigh Waves in Nonhomogeneous Magneto-Electro-Elastic Half-Spaces.

The complex Rayleigh waves play an important role in the energy conversion efficiency of magneto-electro-elastic devices, so it is necessary to explore the wave propagation characteristics for the better applications in engineering. This paper modifies the Laguerre orthogonal polynomial to investigate the complex Rayleigh waves propagating in nonhomogeneous magneto-electro-elastic half-spaces. The improved method simplifies the calculation process by incorporating boundary conditions into the constitutive relations, shortens the solving time by transforming the solution of wave equation to an eigenvalue problem, and obtains all wave modes, including real and imaginary and complex wavenumbers. The three-dimensional curves of full frequency spectrum and phase velocities are presented for the better description of the conversion from complex Rayleigh wave modes to real wave ones; besides, the displacement distributions, electric and magnetic potential curves are obtained in thickness and propagation directions, respectively. Numerical results are analyzed and discussed elaborately in three cases: variation of nonhomogeneous coefficients, absence of magnetism, and absence of electricity. The results can be used to optimize and fabricate the acoustic surface wave devices of the nonhomogeneous magneto-electro-elastic materials.

Simplified estimation of anisotropic non-homogeneous extinction coefficients in porous solids considering spherical and cylindrical pore networks

Abstract This manuscript presents a simplified analysis for estimating anisotropic, non-homogeneous extinction coefficients in porous solids as a function of the pore morphology and size distribution. The model contains two main simplifications: (1) the pore network consists of hollow interspersed spheres and cylinders, and (2) each surface exchanges radiation (through diffuse emission) only with itself and adjacent surfaces. The model yields photon flight length (PFL) probability distribution functions (PDFs), describing Beerian and non-Beerian anisotropic extinction processes. The method is three orders of magnitude faster than the Monte Carlo Ray Tracing Method (MCRT), yielding Root Mean Square Errors between 0 and 35% for the PFL–PDFs and 0–25% for the anisotropic extinction coefficients. Finally, this work presents a local geometric criterium determining whether the model applies to some given area of the porous solid, so it remains useful even when not applicable to the whole domain.

Субдифференциальное разложение одномерной регуляризованной полной вариации с неоднородными коэффициентами

In this paper, we consider a convex function defined as a 1D-regularized total variation with nonhomogeneous coefficients, and prove the Main Theorem concerned with the decomposition of the subdifferential of this convex function to a weighted singular diffusion and a linear regular diffusion. The Main Theorem will be to enhance the previous regularity result for quasilinear equation with singularity, and moreover, it will be to provide some useful information in the advanced mathematical studies of grain boundary motion, based on KWC type energy.

Numerical analytic method for solving a large class of generalized non-homogeneous variable coefficients KdV problems based on lie symmetries

In this study, we use Lie symmetry method to reduce a generalized KdV equation with initial and boundary conditions with non-homogeneous variable coefficients to initial value problem. In particular, we concentrate on the cases that the reduced IVP cannot be solved analytically. We also compare the approximated solutions of IVP using numerical methods with IBVP, and note that they are more efficient than doing the same procedure for IBVP. In fact, it is shown that reducing IBVP to IVP and solving the reduced problem numerically will lead to more accurate solutions.

Exact superregular breather solutions to the generalized nonlinear Schrödinger equation with nonhomogeneous coefficients and dissipative effects.

Superregular breathers are peculiar solutions to the integrable nonlinear Schrödinger equation that constitute the building blocks for analysis of the nonlinear stage of modulation instability developing from a localized perturbation on the nonvanishing condensate background. Here superregular breather solutions are extended to the generalized nonlinear Schrödinger equation with nonhomogeneous coefficients and in the presence of dissipation. Concrete examples are shown that may allow observation of new solutions in fiber optics where dissipation is unavoidable, nonhomogeneous spatial distribution of the amplification profile can be controlled, and current technology allows design of the longitudinal dispersion profile.

A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients

<p style='text-indent:20px;'>This paper is devoted to a spatial heterogeneous SIS model with the infected group equipped with a free boundary. Our main aim is to determine whether the disease is spreading forever or extinct eventually, and to illustrate, under the nonhomogeneous spatial environment, free boundaries can have a large influence on the infected behavior at the large time. For this purpose, we first introduce a basic reproduction number and then establish a spreading-vanishing dichotomy. Then by investigating the effect of the diffusion rate, initial domain and spreading speed on the asymptotic behavior of the infected group, we establish some sufficient conditions and even necessary and sufficient conditions for disease spreading or vanishing.

Non-Homogeneous Thermoelastic Timoshenko Systems

The well-established Timoshenko system is characterized by a particular relation between shear stress and bending moment from its constitutive equations. Accordingly, a (thermal) dissipation added on the bending moment produces exponential stability if and only if the so called “equal wave speeds” condition is satisfied. This remarkable property extends to the case of non-homogeneous coefficients. In this paper, we consider a non-homogeneous thermoelastic system with dissipation restricted to the shear stress. To this new problem, by means of a delicate control observability analysis, we prove that a local version of the equal wave speeds condition is sufficient for the exponential stability of the system. Otherwise, we study the polynomial stability of the system with decay rate depending on the regularity of initial data.

Finite Elements with mesh refinement for wave equations in polygons

Error estimates for the space semi-discrete approximation of solutions of the Wave equation in polygons G⊂R2 are presented. Based on corner asymptotics of the solution, it is shown that for continuous, simplicial Lagrangian Finite Elements of polynomial degree p≥1 with either suitably graded mesh refinement or with bisection tree mesh refinement towards the corners of G, the maximal rate of convergence O(N−p/2) which is afforded by the Lagrangian Finite Element approximations on quasiuniform meshes for smooth solutions is restored. Dirichlet, Neumann and mixed boundary conditions are considered. Numerical experiments which confirm the theoretical results are presented. Generalizations to nonhomogeneous coefficients and elasticity and electromagnetics are indicated.

Diagrammatic analysis of nonhomogeneous diffusion.

By virtue of its complexity, realistic approaches to describe diffusion in cellular media require the employment of computational methods. Among others, this type of studies has shown that the apparent diffusion coefficient of a macromolecular solute through a cytoplasmic-like medium exhibits a power-law dependence with the excluded volume. Power laws are ubiquitous findings in diverse systems, such as metabolic processes, population dynamics, and communication networks, and have been the object of many interpretative formal approaches. This study introduces a diagrammatic algorithm, inspired in previous ones employed to analyze multicyclic chemical systems, to derive expressions for nonhomogeneous diffusion coefficients and to study the effects of volume exclusion. A most noteworthy result of this work is that midsize diagrams of nonhomogeneous diffusion are already able to exhibit an approximate power-law dependence of the diffusion coefficient with the excluded volume. The employment of the diagrammatic method for the analysis of simple situations may thus prove useful to interpret some properties of larger network systems.

Large deviation principle of stochastic differential equations with non-Lipschitzian coefficients

We study the large deviation principle of stochastic differential equations with non-Lipschitzian and non-homogeneous coefficients. We consider at first the large deviation principle when the coefficients σ and b are bounded, then we generalize the conclusion to unbounded case by using bounded approximation program. Our results are generalization of S. Fang-T. Zhang’s results.

Finite elements with mesh refinement for wave equations in polygons

Error estimates for the space semi-discrete approximation of solutions of the Wave equation in polygons G ? R 2 are presented. Based on corner asymptotics of the solution, it is shown that for continuous, simplicial Lagrangian Finite Elements of polynomial degree p ? 1 with either suitably graded mesh refinement or with bisection tree mesh refinement towards the corners of G , the maximal rate of convergence O ( N - p / 2 ) which is afforded by the Lagrangian Finite Element approximations on quasiuniform meshes for smooth solutions is restored. Dirichlet, Neumann and mixed boundary conditions are considered. Numerical experiments which confirm the theoretical results are presented. Generalizations to nonhomogeneous coefficients and elasticity and electromagnetics are indicated.

Stochastic differential equations driven by a Wiener process and fractional Brownian motion: Convergence in Besov space with respect to a parameter

A stochastic differential equation involving both a Wiener process and fractional Brownian motion, with nonhomogeneous coefficients and random initial condition, is considered. The coefficients and initial condition depend on a parameter. The assumptions on the coefficients and the initial condition supplying continuous dependence of the solution on a parameter, with respect to the Besov space norm, are established.

The rate of convergence of the Euler scheme to the solution of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion

We study one-dimensional stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion. We prove some properties of the solutions of such equations and of the corresponding Euler scheme. We obtain the convergence rate of the Euler scheme for diffusions with weak singularity at zero.

Properties of solutions of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion

Properties of solutions of stochastic differential equations with nonhomogeneous coefficients and non-Lipschitz diffusion

Mode III Yoffe Dynamic Crack in an Infinite Length Strip for Orthotropic Anisotropy Functionally Graded Material

A moving crack in an infinite strip of orthotropic anisotropy functionally graded material (FGM) with free boundary subjected to anti-plane shear loading is considered. The shear moduli in two directions of FGM are assumed to be of exponential form. The dynamic stress intensity factor is obtained by utilizing integral transforms and dual-integral equations. The numerical results show the relationships among the dynamic stress intensity factor and crack velocity, the height of the strip, gradient parameters and nonhomogeneous coefficients.

Gültigkeit und Reproduzierbarkeit radiologischer Klassifikationskriterien des knöchernen Substanzdefekts vor Revisionshüftendoprothetik

It was the purpose of this prospective study to analyze validity and reliability of three different radiographic classification systems, which are used to assess bone stock loss prior to revision total hip arthroplasty. Investigation was performed using the classification systems according to 1) Paprosky et al., 2) Hungerford et al., and 3) the German Society of Orthopaedics and Traumatology (DGOT). Twenty-eight consecutive patients with diagnosis of aseptic failure of total hip arthroplasty were enrolled in the study. Interrater agreement was analyzed by evaluating the classifications preoperatively by three investigators with different level of clinical experience using a blinded observer technique. Validity was analyzed by comparison of preoperative radiographic findings (n = 3) and an intraoperative control by visualization and palpation (n = 1). Spearman coefficient of correlation (r(s)) was used to establish levels of agreement among multiple ordinal variables. Interrater reliability testing using paired comparison between the three investigators revealed non-homogeneous coefficients of correlation (r(s): Paprosky femur: 0.45 - 0.67, acetabulum: 0.38 - 0.63; Hungerford: 0.46 - 0.66; DGOT femur: 0.38 - 0.59, acetabulum: 0.42 - 0.76). Paired analysis of correlation between preoperative and intraoperative findings again showed non-homogeneous coefficients of correlation (r(s): Paprosky femur: 0.59 - 0.68, acetabulum: 0.39 - 0.70; Hungerford: 0.39 - 0.74; DGOT femur: 0.44 - 0.60, acetabulum: 0.36 - 0.76). In general, defects were grossly underestimated preoperatively. The level of experience did not influence the validity of measurements. Our results indicate that these classifications do not provide valid and reliable assessment of femoral and acetabular bone stock loss prior to revision total hip arthroplasty.

Conservation laws in non-homogeneous electro-magneto-elastic materials

Noether's theorem on invariant variational principles is applied for linear transversely isotropic non-homogeneous electro-magneto-elastic material. Invariant condition of the electromagnetic enthalpy is derived and expressed in original manner. Usually, inserting infinitesimal symmetry-transformations relating to material space of homogeneous materials into the invariant condition, one can obtain some expressions for the non-homogeneous material. These expressions indicate the relations between material inhomogeneity force and energy-momentum tensor. In order to obtain conservation laws of the non-homogeneous material, it follows from the invariant condition that the material coefficients have to satisfy a set of first-order linear partial differential equations with which there is a possible mathematical form of infinitesimal symmetry-transformation relating to material space. Satisfaction of these partial differential equations not only assures the existence of infinitesimal symmetry-transformations but also determines the number of them. Several independent and non-trivial conservation laws relating to material space with different non-homogeneous material coefficients are given. Some results of the path-independent integrals emanating from the conservation laws calculated around a crack tip are presented.

Geometric properties of root vectors for equation of nonhomogeneous damped string: transformation operators method

The spectral decomposition theorem for a class of nonselfadjoint operators in a Hilbert space is obtained in the paper. These operators are the dynamics generators for the systems governed by 1–dim hyperbolic equations with spatially nonhomogeneous coefficients containing first order damping terms and subject to linear nonselfadjoint boundary conditions. These equations and boundary conditions describe, in particular, a spatially nonhomogeneous string subject to a distributed viscous damping and also damped at the boundary points. The main result leading to the spectral decomposition is the fact that the generalized eigenvectors (root vectors) of the above operators form Riesz bases in the corresponding energy spaces. The proofs are based on the transformation operators method. The classical concept of transformation operators is extended to the equation of damped string. Originally, this concept was developed by I. M. Gelfand, B. M. Levitan and V. A. Marchenko for 1–dim Schrödinger equation in connection with the inverse scattering problem. In the classical case, the transformation operator maps the exponential function (stationary wave function of the free particle) into the Jost solution of the perturbed Schrödinger equation. For the equation of a nonhomogeneous damped string, it is natural to introduce two transformation operators (outgoing and incoming transformation operators). The terminology is motivated by an analog with the Lax—Phillips scattering theory. The transformation operators method is used to reduce the Riesz bases property problem for the generalized eigenvectors to the similar problem for a system of nonharmonic exponentials whose complex frequencies are precisely the eigenvalues of our operators. The latter problem is solved based on the spectral asymptotics and known facts about exponential families. The main result presented in the paper means that the generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of non—selfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems.