Dirichlet‐to‐Neumann Map for a Nonlinear Diffusion Equation

A nonlinear diffusive equation with moving boundaries is analyzed by constructing the corresponding Dirichlet‐to‐Neumann map. In particular, the Dirichlet boundary value and the initial condition are used to derive the unknown Neumann boundary value. Then, a contraction‐mapping technique is used to prove existence and uniqueness of the solution for small times.     

On the Global Relation and the Dirichlet‐to‐Neumann Correspondence

The recently developed Fokas method for solving two‐dimensional Boundary Value Problems (BVP) via the use of global relations is utilized to solve axisymmetric problems in three dimensions. In particular, novel integral representations for the interior and exterior Dirichlet and Neumann problems for the sphere are derived, which recover and improve the already known solutions of these problems. The BVPs considered in this paper can be classically solved using either the finite Legendre transform or the Mellin‐sine transform (which can be derived from the classical Mellin transform in a way similar to the way that the sine transform can be derived from the Fourier transform). The Legendre transform representation is uniformly convergent at the boundary, but it involves a series that is not useful for many applications. The Mellin‐sine transform involves of course an integral but it is not uniformly convergent at the boundary. In this paper: (a) The Legendre transform representation is rederived in a simpler approach using algebraic manipulations instead of solving ODEs. (b) An integral representation, different that the Mellin‐sine transform representation is derived which is uniformly convergent at the boundary. Furthermore, the derivation of the Fokas approach involves only algebraic manipulations, instead of solving an ordinary differential equation.     

The Klein–Gordon Equation in a Domain with Time‐Dependent Boundary

We solve a Dirichlet boundary value problem for the Klein–Gordon equation posed in a time‐dependent domain. Our approach is based on a general transform method for solving boundary value problems for linear and integrable nonlinear PDE in two variables. Our results consist of the inversion formula for a generalized Fourier transform, and of the application of this generalized transform to the solution of the boundary value problem.     

The Generalized Dirichlet to Neumann Map for the KdV Equation on the Half-Line

For the two versions of the KdV equation on the positive half-line an initial-boundary value problem is well posed if one prescribes an initial condition plus either one boundary condition if q _t and q _xxx have the same sign (KdVI) or two boundary conditions if q _t and q _xxx have opposite sign (KdVII). Constructing the generalized Dirichlet to Neumann map for the above problems means characterizing the unknown boundary values in terms of the given initial and boundary conditions. For example, if {q(x,0),q(0,t)} and {q(x,0),q(0,t),q _x (0,t)} are given for the KdVI and KdVII equations, respectively, then one must construct the unknown boundary values {q _x (0,t),q _xx (0,t)} and {q _xx (0,t)}, respectively. We show that this can be achieved without solving for q(x,t) by analysing a certain “global relation” which couples the given initial and boundary conditions with the unknown boundary values, as well as with the function Φ ^(t)(t,k), where Φ ^(t) satisfies the t-part of the associated Lax pair evaluated at x=0. The analysis of the global relation requires the construction of the so-called Gelfand–Levitan–Marchenko triangular representation for Φ ^(t). In spite of the efforts of several investigators, this problem has remained open. In this paper, we construct the representation for Φ ^(t) for the first time and then, by employing this representation, we solve explicitly the global relation for the unknown boundary values in terms of the given initial and boundary conditions and the function Φ ^(t). This yields the unknown boundary values in terms of a nonlinear Volterra integral equation. We also discuss the implications of this result for the analysis of the long t-asymptotics, as well as for the numerical integration of the KdV equation.     

The Inhomogeneous Neumann Problem in Lipschitz Domains

Abstract

Existence and uniqueness is established for the solution to the inhomogeneous Neumann problem for Laplace's equation in Lipschitz domains with data in L ^P Sobolev spaces.     

Remarks on Nonlinear Neumann Problems in Periodic Domains

We study a semilinear elliptic equation Au = f(x, u) with nonlinear Neumann boundary condition Bu = φ(ξ, u) in an unbounded domain Ω ? ?ⁿ, the boundary of which is defined by periodic functions. We assume that f and φ and the coefficients of the operators are asymptotically periodic in the space variables. Our main result states the existence of an asymptotically decaying, nontrivial solution of this problem with minimal energy. The proof is based on the concentration-compactness principle.     

Neumann Problem in Domains with Outlets of Bounded Diameter

We study the Neumann problem for the Laplace equation in domains having quasicylindrical outlets. The aim is to collect and sort known results spread over the literature and to fill in the gaps of the theory which still exist. Namley, solutions with finite Dirichlet integral are found applying on the one hand the Helmholtz decomposition and on the other hand the Riesz representation theorem. We investigate generalizations for the non-Hilbert space setting and collect several types of Saint Venant estimates.     

Nucleation Barriers for the Cubic‐to‐Tetragonal Phase Transformation

We are interested in the phase transformation from austenite to martensite. This transformation is typically accompanied by the generation and growth of small inclusions of martensite. We consider a model from geometrically linear elasticity with sharp energy penalization for phase boundaries. Focusing on a cubic‐to‐tetragonal phase transformation, we show that the minimal energy for an inclusion of martensite scales like $\max \{ V^{2/3}, V^{9/11} \}$ in terms of the volume V. Moreover, our arguments illustrate the importance of self‐accommodation for achieving the minimal scaling of the energy. The analysis is based on Fourier representation of the elastic energy. © 2012 Wiley Periodicals, Inc.     

Generalized Wave Operators in Von Neumann Algebras

Let M?B(H)be a countable decomposable properly infinite von Neumann algebra with a faithful normal semifinite tracial weight r where B(H)is the set of all bound...     

Stability Analysis of Full Annular Rub in Rotor‐to‐Stator Systems

This paper outlines the analytical study of the stability of a more realistic rotor‐to‐stator contact model, which considers the stator as an elastic body having finite mass and being elastically supported. For this model, no report on the stability analysis is available in the literature by our knowledge. In this study, the analytical solution, corresponding to the synchronous full annular rub motion, is first solved, then the stability condition of it is derived. Based on the systematic parametric studies, one gains a deeper insight into the nature of the dynamical behavior of the rotor‐to‐stator contact system. Since the previously investigated simpler rotor‐to‐stator contact models are only the special cases of the present model, the limitation of the simpler models and the influence of the system parameters, which were omitted or assigned to specific values in the simpler models, have been revealed through the present investigation.     

Science Self‐Efficacy of Preservice Teachers in Face‐to‐Face versus Blended Environments

Using a quasi‐experimental mixed methods concurrent design, this study measured the science self‐efficacy of pre‐service elementary teachers before and after a survey of science content course. Further, this course was delivered in two different formats: face‐to‐face and hybrid (approximately 50% online), and compared pre‐and post‐science self‐efficacy of students in the two different course formats. Our quantitative results showed increases in personal efficacy, but not outcome expectancy for both formats, and no significant differences between the increases for either format. Our qualitative data showed that participants attributed their increased levels of personal efficacy to the hands‐on components of the course, as well as perceived teacher attitudes toward science, both of which would be challenging to replicate in a purely online format, as opposed to the hybrid format included in this study.     

Approximation of Solutions of the Neumann Problem in Disintegrating Domains

This article deals with approximation of solutions of the Neumann problem in domains, where small tubes are cut out. With an increasing number of tubes some kind of a porous layer inside the domain is approximated. Our aim is to find an asymptotic solution for the separated limit domains. We show that this asymptotics is described by a boundary value problem for the two limit domains, where the solutions for each domain are connected by the boundary conditions.     

Boundary Sensitivities for Diffusion Processes in Time Dependent Domains

We study the sensitivity, with respect to a time dependent domain of expectations of functionals of a diffusion process stopped at the exit from or normally reflected at the boundary of We establish a differentiability result and give an explicit expression for the gradient that allows the gradient to be computed by Monte Carlo methods. Applications to optimal stopping problems and pricing of American options, to singular stochastic control and others are discussed.     

Time‐harmonic and asymptotically linear Maxwell equations in anisotropic media

This paper is focused on following time‐harmonic Maxwell equation: where is a bounded Lipschitz domain, is the exterior normal, and ω is the frequency. The boundary condition holds when Ω is surrounded by a perfect conductor. Assuming that f is asymptotically linear as , we study the above equation by improving the generalized Nehari manifold method. For an anisotropic material with magnetic permeability tensor and permittivity tensor , ground state solutions are established in this paper. Applying the principle of symmetric criticality, we find 2 types of solutions with cylindrical symmetries in particular for the uniaxial material.     

Input‐to‐Output Linearization of Explicit Control Systems with Constrained Measurements

Many design methods for non‐linear control problems require the measurement of the whole state of the plant. This contribution deals with a refinement of the well established exact input‐to‐output linearization such that the control laws depend only on a predefined set of measurements. More precisely, we derive the determining equations for all static control laws, such that the input‐to‐output map from the new input to the new output is linear and such that the constraints concerning the measurements are met.     

John and Uniform Domains in Generalized Siegel Boundaries

Potential Analysis - Given the pair of vector fields X = ?x + |z|2my?t and Y = ?y ?|z|2mx?t,where (x,y,t) = , we give a condition on a bounded domain which ensures...     

Asymptotics for mixed Dirichlet–Robin problems in irregular domains

We study the asymptotic behaviour of the solutions of two-dimensional elliptic problems with Robin boundary conditions on the “prefractal” curves approximating the Koch curve type fractals.     

Time–accurate Modular CFD‐CSD Coupling for Aeroelastic Rotor Simulations

This paper addresses the timewise accuracy of different coupling approaches applied to instationary aeroelastic simulations of rotors in forward flight. Two different approaches which are widely discussed in literature are examined: the tight or strong coupling, and the fully integrated or monolithic coupling. Strong coupling means an exchange of fluid loads and structural deformations at each time step which is effectuated in a fully modular manner. We will address aspects of conservativity and time‐accuracy, and will present results for a helicopter forward flight scenario. However, objections concerning the correct solution of the global non‐linear three field problem – structure, grid deformation, aerodynamics – remain. These objections are normally rejected by the monolithic approach. Here, a common set of partial differential equations is derived and solved in a single code. However, a truly monolithic system of equations is only needed for stability analysis, and it can be decomposed in a three field problem respecting appropriate boundary conditions for each domain. Thus, modularity can be maintained, conceiving a quasi‐monolithic procedure, when both domains are simultaneously solved in a common non‐linear iteration loop on a per time‐step basis. First results will be shown for a 2D flutter testcase.     

二维Lipschitz区域上一类带L^p边值的非齐次多调和Neumann问题

该文运用层位势方法研究了二维Lipschitz区域上一类带L^p边值的非齐次多调和Neumann问题.利用多层S位势,给出了该类问题的惟一积分表示解,其中,多层S位势是经典单层位势的高阶类似物,通过多调和基本解加以定义.