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Showing papers on "Boundary value problem published in 2016"


Book
20 Oct 2016
TL;DR: Fractional Functional Differential Equations Fractional Abstract Differentials Equations (ADE) as mentioned in this paper fractional Hamiltonian Systems Existence and uniqueness Continuation Mild Solutions C0-semigroup Almost Sectorial Operators Multiplicity Variational Approach Critical Point Theory.
Abstract: Fractional Functional Differential Equations Fractional Abstract Differential Equations Fractional Evolution Equations Fractional Boundary Value Problems Fractional Schrodinger Equations Fractional Euler - Lagrange Equations Time-Fractional Diffusion Equations Fractional Hamiltonian Systems Existence and Uniqueness Continuation Mild Solutions C0-semigroup Almost Sectorial Operators Multiplicity Variational Approach Critical Point Theory.

970 citations


Journal ArticleDOI
TL;DR: In this article, a complete treatment of boundary terms in general relativity to include cases with lightlike boundary segments along with the usual spacelike and timelike ones is provided.
Abstract: The present paper provides a complete treatment of boundary terms in general relativity to include cases with lightlike boundary segments along with the usual spacelike and timelike ones. Applications of this exhaustive treatment includes a recent conjecture on computational complexity in the context of AdS/CFT.

453 citations


Book
09 Aug 2016
TL;DR: In this paper, the authors explore the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations and demonstrate the use of Lyapunov functions in this type of analysis.
Abstract: This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices. The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control. Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.

393 citations


Journal ArticleDOI
TL;DR: In this paper, a general formalism to associate a gauge-invariant classical phase space to a spatial slice with boundary by introducing new degrees of freedom on the boundary is presented.
Abstract: We consider the problem of defining localized subsystems in gauge theory and gravity. Such systems are associated to spacelike hypersurfaces with boundaries and provide the natural setting for studying entanglement entropy of localized subsystems. We present a general formalism to associate a gauge-invariant classical phase space to a spatial slice with boundary by introducing new degrees of freedom on the boundary. In Yang-Mills theory the new degrees of freedom are a choice of gauge on the boundary, transformations of which are generated by the normal component of the nonabelian electric field. In general relativity the new degrees of freedom are the location of a codimension-2 surface and a choice of conformal normal frame. These degrees of freedom transform under a group of surface symmetries, consisting of diffeomorphisms of the codimension-2 boundary, and position-dependent linear deformations of its normal plane. We find the observables which generate these symmetries, consisting of the conformal normal metric and curvature of the normal connection. We discuss the implications for the problem of defining entanglement entropy in quantum gravity. Our work suggests that the Bekenstein-Hawking entropy may arise from the different ways of gluing together two partial Cauchy surfaces at a cross-section of the horizon.

384 citations


Journal ArticleDOI
TL;DR: In this paper, a size-dependent beam model is proposed for nonlinear free vibration of a functionally graded (FG) nanobeam with immovable ends based on the nonlocal strain gradient theory (NLSGT) and Euler-Bernoulli beam theory in conjunction with the von-Karman's geometric nonlinearity.

313 citations


Journal ArticleDOI
TL;DR: In this paper, the free and forced vibration characteristics of functionally graded (FG) porous beams with non-uniform porosity distribution whose elastic moduli and mass density are nonlinearly graded along the thickness direction were investigated.

305 citations


Journal ArticleDOI
TL;DR: In this paper, the authors enumerate the cases in 2D conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular Hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight.
Abstract: We enumerate the cases in 2d conformal field theory where the logarithm of the reduced density matrix (the entanglement or modular Hamiltonian) may be written as an integral over the energy-momentum tensor times a local weight. These include known examples and new ones corresponding to the time-dependent scenarios of a global and local quench. In these latter cases the entanglement Hamiltonian depends on the momentum density as well as the energy density. In all cases the entanglement spectrum is that of the appropriate boundary CFT. We emphasize the role of boundary conditions at the entangling surface and the appearance of boundary entropies as universal O(1) terms in the entanglement entropy.

286 citations


Journal ArticleDOI
TL;DR: This paper presents a solution to the fully general case, in which the number of PDEs in either direction is arbitrary, and where actuation is applied on only one boundary, and solves the problem of control of coupled “homodirectional” hyperbolic linear PDE s, where multiple transport PDES convect in the same direction with arbitrary local coupling.
Abstract: Research on stabilization of coupled hyperbolic PDEs has been dominated by the focus on pairs of counter-convecting (“heterodirectional”) transport PDEs with distributed local coupling and with controls at one or both boundaries. A recent extension allows stabilization using only one control for a system containing an arbitrary number of coupled transport PDEs that convect at different speeds against the direction of the PDE whose boundary is actuated. In this paper we present a solution to the fully general case, in which the number of PDEs in either direction is arbitrary, and where actuation is applied on only one boundary (to all the PDEs that convect downstream from that boundary). To solve this general problem, we solve, as a special case, the problem of control of coupled “homodirectional” hyperbolic linear PDEs, where multiple transport PDEs convect in the same direction with arbitrary local coupling. Our approach is based on PDE backstepping and yields solutions to stabilization, by both full-state and observer-based output feedback, and trajectory tracking problems.

251 citations


Journal ArticleDOI
TL;DR: In this paper, the buckling analysis of two-directional functionally graded materials (FGM) nano-beams with small scale effects is carried out based on the nonlocal elasticity theory and the governing equations are obtained, employing the principle of minimum potential energy.

244 citations


Journal ArticleDOI
TL;DR: Two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method are developed and establish error estimates optimal with respect to the regularity of problem data.
Abstract: We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.

240 citations


Journal ArticleDOI
TL;DR: In this paper, the mathematical foundations and a practical guide for the numerical solution of gravitational boundary value problems are explained and several tools and tricks that have been useful throughout the literature are presented.
Abstract: The wide applications of higher dimensional gravity and gauge/gravity duality have fuelled the search for new stationary solutions of the Einstein equation (possibly coupled to matter). In this topical review, we explain the mathematical foundations and give a practical guide for the numerical solution of gravitational boundary value problems. We present these methods by way of example: resolving asymptotically flat black rings, singly-spinning lumpy black holes in anti-de Sitter (AdS), and the Gregory-Laflamme zero modes of small rotating black holes in AdS5 × S5. We also include several tools and tricks that have been useful throughout the literature.

Journal ArticleDOI
TL;DR: In this paper, the boundary conditions needed for drift-diffusion models to treat interfaces with spin-orbit coupling are presented, together with the driftdiffusion equations, which give an analytical model of spinorbit torques caused by both the spin Hall and Rashba-Edelstein effects.
Abstract: Most spintronic devices share two features: they utilize spin-orbit coupling and they contain interfaces. While bulk spin-orbit effects are thought to be well described by phenomenological theories, interfacial spin-orbit effects are not. A theory that could describe interfacial spin-orbit effects would be useful in analyzing experiments on heavy-metal ferromagnet bilayers, which are a key feature of potential energy-efficient implementations of MRAM. To develop such a theory, the authors present the boundary conditions needed for drift-diffusion models to treat interfaces with spin-orbit coupling. Together with the drift-diffusion equations, these boundary conditions give an analytical model of spin-orbit torques caused by both the spin Hall and Rashba-Edelstein effects. A key feature of these boundary conditions is that they capture spin currents generated by interfacial spin-orbit scattering. The authors validate this phenomenological approach by comparing the results with those obtained by solving the spin-dependent Boltzmann equation. They discuss the interpretation of current experiments, and describe in particular how interfacial effects give rise to torques on a nearby ferromagnetic layer even through a nonmagnetic spacer layer.

Journal ArticleDOI
TL;DR: In this article, the longitudinal vibration analysis of small-scaled rods is studied in the framework of the nonlocal strain gradient theory and the equations of motion and boundary conditions are derived by employing the Hamilton principle.

Journal ArticleDOI
TL;DR: In this paper, the authors prove Cα-regularity up to the boundary for weak solutions of a non-local, non-linear problem driven by the fractional p-Laplacian operator.
Abstract: By virtue of barrier arguments we prove Cα-regularity up to the boundary for the weak solutions of a non-local, non-linear problem driven by the fractional p-Laplacian operator. The equation is boundedly inhomogeneous and the boundary conditions are of Dirichlet type. We employ different methods according to the singular (p 2) case.

Journal ArticleDOI
Abstract: We prove that non-extremal black holes in four-dimensional general relativity exhibit an infinite-dimensional symmetry in their near horizon region. By prescribing a physically sensible set of boundary conditions at the horizon, we derive the algebra of asymptotic Killing vectors, which is shown to be infinite-dimensional and includes, in particular, two sets of supertranslations and two mutually commuting copies of the Witt algebra. We define the surface charges associated to the asymptotic diffeomorphisms that preserve the boundary conditions and discuss the subtleties of this definition, such as the integrability conditions and the correct definition of the Dirac brackets. When evaluated on the stationary solutions, the only non-vanishing charges are the zero-modes. One of them reproduces the Bekenstein-Hawking entropy of Kerr black holes. We also study the extremal limit, recovering the NHEK geometry. In this singular case, where the algebra of charges and the integrability conditions get modified, we find that the computation of the zero-modes correctly reproduces the black hole entropy. Furthermore, we analyze the case of three spacetime dimensions, in which the integrability conditions notably simplify and the field equations can be solved analytically to produce a family of exact solutions that realize the boundary conditions explicitly. We examine other features, such as the form of the algebra in the extremal limit and the relation to other works in the literature.

Journal ArticleDOI
TL;DR: In this paper, the steady boundary layer flow of MHD Williamson fluid through porous medium toward a horizontal linearly stretching sheet in the presence of nanoparticles is investigated numerically, and the resultant non-dimensionalized boundary value problem is solved numerically by Runge-Kutta-Fehlberg fourth-fifth order method with shooting technique.

Journal ArticleDOI
TL;DR: This work investigates the uniqueness of solutions for a class of nonlinear boundary value problems for fractional differential equations with Lipschitz constant related to the first eigenvalues corresponding to the relevant operators.

Journal ArticleDOI
TL;DR: In this article, a neural network approach is used for the thermal modeling and validation of temperature rise in a prismatic lithium-ion battery with LiFePO 4 (also known as LFP) cathode material.

Journal ArticleDOI
TL;DR: In this article, the authors investigated the effect of partial slip on the velocity at the boundary, convective thermal boundary condition, Brownian and thermophoresis diffusion coefficients on the concentration boundary condition.
Abstract: In this paper we report on combined Dufour and Soret effects on the heat and mass transfer in a Casson nanofluid flow over an unsteady stretching sheet with thermal radiation and heat generation. The effects of partial slip on the velocity at the boundary, convective thermal boundary condition, Brownian and thermophoresis diffusion coefficients on the concentration boundary condition are investigated. The model equations are solved using the spectral relaxation method. The results indicate that the fluid flow, temperature and concentration profiles are significantly influenced by the fluid unsteadiness, the Casson parameter, magnetic parameter and the velocity slip. The effect of increasing the Casson parameter is to suppress the velocity and temperature growth. An increase in the Dufour parameter reduces the flow temperature, while an increase in the value of the Soret parameter causes increase in the concentration of the fluid. Again, increasing the velocity slip parameter reduces the velocity profile whereas increasing the heat generation parameter increases the temperature profile. A validation of the work is presented by comparing the current results with existing literature.

Journal ArticleDOI
TL;DR: In this article, a nonlocal higher-order refined magneto-electro-viscoelastic beam model for vibration analysis of smart nanostructures under different boundary conditions is presented.

Journal ArticleDOI
TL;DR: In this article, the development of continuum models to describe processes in gases in which the particle collisions cannot maintain thermal equilibrium is discussed, and typical results are reviewed for channel flow, cavity flow, and flow past a sphere in the low-Mach number setting for which both evolution equations and boundary conditions are well established.
Abstract: This article discusses the development of continuum models to describe processes in gases in which the particle collisions cannot maintain thermal equilibrium. Such a situation typically is present in rarefied or diluted gases, for flows in microscopic settings, or in general whenever the Knudsen number—the ratio between the mean free path of the particles and a macroscopic length scale—becomes significant. The continuum models are based on the stochastic description of the gas by Boltzmann's equation in kinetic gas theory. With moment approximations, extended fluid dynamic equations can be derived, such as the regularized 13-moment equations. Moment equations are introduced in detail, and typical results are reviewed for channel flow, cavity flow, and flow past a sphere in the low–Mach number setting for which both evolution equations and boundary conditions are well established. Conversely, nonlinear, high-speed processes require special closures that are still under development. Current approaches are ...

Journal ArticleDOI
TL;DR: In this paper, a nonlocal continuum model is developed for the nonlinear free vibration of size-dependent magneto-electro-elastic nanoplates subjected to external electric and magnetic potentials.

Journal ArticleDOI
TL;DR: In this paper, the authors explored the magnetohydrodynamic (MHD) three-dimensional flow of viscous nanofluid subject to convective surface boundary condition, which is generated by an impermeable surface which is stretched nonlinearly.

Journal ArticleDOI
TL;DR: In this article, the authors considered the most general asymptotically anti-de Sitter boundary conditions in three-dimensional Einstein gravity with negative cosmological constant, which modifies the usual Fefferman-Graham expansion.
Abstract: We consider the most general asymptotically anti-de Sitter boundary conditions in three-dimensional Einstein gravity with negative cosmological constant. The metric contains in total twelve independent functions, six of which are interpreted as chemical potentials (or non-normalizable fluctuations) and the other half as canonical boundary charges (or normalizable fluctuations). Their presence modifies the usual Fefferman-Graham expansion. The asymptotic symmetry algebra consists of two $$ \mathfrak{s}\mathfrak{l}{(2)}_k $$ current algebras, the levels of which are given by k = l/(4G N ), where l is the AdS radius and G N the three-dimensional Newton constant.

Journal ArticleDOI
TL;DR: In this paper, the buckling load of two-dimensional functionally graded materials (2D-FGMs) was investigated for the first time to investigate the bucking of beams with different boundary conditions, assuming that the material properties of the beam vary in both axial and thickness directions according to the power-law form.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the most general asymptotically anti-de Sitter boundary conditions in three-dimensional Einstein gravity with negative cosmological constant, which modifies the usual Fefferman-Graham expansion.
Abstract: We consider the most general asymptotically anti-de Sitter boundary conditions in three-dimensional Einstein gravity with negative cosmological constant. The metric contains in total twelve independent functions, six of which are interpreted as chemical potentials (or non-normalizable fluctuations) and the other half as canonical boundary charges (or normalizable fluctuations). Their presence modifies the usual Fefferman-Graham expansion. The asymptotic symmetry algebra consists of two sl(2)_k current algebras, the levels of which are given by k=l/(4G_N), where l is the AdS radius and G_N the three-dimensional Newton constant.

Journal ArticleDOI
TL;DR: In this article, an exact solution of the general problem of an ocean on a rotating sphere is presented; this describes a steady flow that is moving only in the azimuthal direction, with no variation in this direction.
Abstract: The general problem of an ocean on a rotating sphere is considered. The governing equations for an inviscid, incompressible fluid, written in spherical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions, are introduced. An exact solution of this system is presented; this describes a steady flow that is moving only in the azimuthal direction, with no variation in this direction. However, this azimuthal velocity component has an arbitrary variation with depth (i.e., radius), and so, for example, an Equatorial Undercurrent (EUC) can be accommodated. The pressure boundary condition at the free surface relates this pressure to the shape of the surface via a Bernoulli relation; this provides the constraint on the existence of a solution, although the restrictions are somewhat involved in spherical coordinates. To examine this constraint in more detail, the corresponding problems in model cylindrical coordinates (with the equat...

Journal ArticleDOI
TL;DR: In this article, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen's two-phase local/nonlocal model.
Abstract: Though widely used in modelling nano- and micro- structures, Eringen’s differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings are considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein h...

Journal ArticleDOI
TL;DR: In this paper, the authors studied SO(d+1) invariant solutions of the classical vacuum Einstein equations in p + d + 3 dimensions and showed that these solutions can be corrected to nonsingular solutions at first sub-leading order in 1 d if and only if the membrane shape and velocity field obey equations of motion which they determine.
Abstract: We study SO(d+1) invariant solutions of the classical vacuum Einstein equations in p + d + 3 dimensions. In the limit d → ∞ with p held fixed we con- struct a class of solutions labelled by the shape of a membrane (the event horizon), together with a 'velocity' field that lives on this membrane. We demonstrate that our metrics can be corrected to nonsingular solutions at first sub-leading order in 1 d if and only if the membrane shape and 'velocity' field obey equations of motion which we determine. These equations define a well posed initial value problem for the membrane shape and this 'velocity' and so completely determine the dynamics of the black hole. They may be viewed as governing the non-linear dynamics of the light quasi normal modes of Emparan, Suzuki and Tanabe.

Journal ArticleDOI
TL;DR: In this article, the authors derived the closed-form analytical solutions of original integral model for static bending of Euler Bernoulli and Timoshenko beams, in a simple manner, for different loading and boundary conditions.