Showing papers on "Boundary value problem published in 2018"
TL;DR: A deep learning algorithm similar in spirit to Galerkin methods, using a deep neural network instead of linear combinations of basis functions is proposed, and is implemented for American options in up to 100 dimensions.
1,290 citations
28 Mar 2018
TL;DR: The bulk-boundary correspondence does not hold in non-hermitian topological insulators as mentioned in this paper, and the existence of boundary states (with zero energy and possessing chiral or helical properties) with the topological numbers defined in bulk is not held any longer.
Abstract: The bulk-boundary correspondence is crucial to topological insulators. It associates the existence of boundary states (with zero energy and possessing chiral or helical properties) with the topological numbers defined in bulk. In recent years, topology has been extended to non-hermitian systems, opening a new research area called non-hermitian topological insulator. In this paper, however, we will illustrate that the bulk-boundary correspondence does not hold in these new models. This is because a prerequisite condition: 'the boundaries cannot alter most of the bulk states, so as to the topological numbers defined on them' does not hold any longer. This cuts out the correspondence between the topological numbers and the boundary states. We will illustrate that, as approaching the open boundary condition by eliminating the strength of the hopping between the two ends of a chain, a new series of exceptional points must be passed through and the topological structure of the spectrum in the complex plane has been changed. This makes the spectrum topology different for the chains with and without boundaries. We also discuss that such exotic behavior does not emerge when the open boundary is replaced by a domain-wall. So the index theorem can be applied to the systems with domain-walls but cannot be further used to those with open boundaries.
437 citations
TL;DR: This work reviews recent investigations in wall-modeled LES, including the development of novel approximate boundary conditions and the application of wall models to complex flows (e.g., boundary-layer separation, shock/boundary-layer interactions, transition).
Abstract: Large-eddy simulation (LES) has proven to be a computationally tractable approach to simulate unsteady turbulent flows. However, prohibitive resolution requirements induced by near-wall eddies in high–Reynolds number boundary layers necessitate the use of wall models or approximate wall boundary conditions. We review recent investigations in wall-modeled LES, including the development of novel approximate boundary conditions and the application of wall models to complex flows (e.g., boundary-layer separation, shock/boundary-layer interactions, transition). We also assess the validity of underlying assumptions in wall-model derivations to elucidate the accuracy of these investigations, and offer suggestions for future studies.
325 citations
02 May 2018
TL;DR: The Navier-Stokes Equations and Prandtl Derivation of the Boundary Layer System as the first approximation to Asymptotic Solution of the Navier Stokes Equation near the boundary separation of the boundary layer as mentioned in this paper.
Abstract: The Navier-Stokes Equations and Prandtl Derivation of the Prandtl System Solution of the Boundary Layer System as the First Approximation to Asymptotic Solution of the Navier-Stokes Equations near the Boundary Separation of the Boundary Layer Setting of the Main Problems for the Equations of Boundary Layer Boundary Layer Equations for Non-Newtonian Fluids Boundary Layers in Magnetohydrodynamics Stationary Boundary Layer: von Mises Variables Continuation of Two-Dimensional Boundary Layer Asymptotic Behavior of the Velocity Component along the Boundary Layer Conditions for Boundary Layer Separation Self-Similar Solutions of the Boundary Layer Equations Solving the Continuation Problem by the Line Method On Three-Dimensional Boundary Layer Equations Comments Stationary Boundary Layer: Crocco Variables Axially Symmetric Stationary Boundary Layer Symmetric Boundary Layer The Problem of Continuation of the Boundary Layer Weak Solutions of the Boundary Layer System Nonstationary Boundary Layer Axially Symmetric Boundary Layer The Continuation Problem for a Nonstationary Axially Symmetric Boundary Layer Continuation of the Boundary Layer: Successive Approximations On t-Global Solutions of the Prandtl System for Axially Symmetric Flows Stability of Solutions of the Prandtl System Time-Periodic Solutions of the Nonstationary Boundary Layer System Solving the Nonstationary Prandtl System by the Line Method in the Time Variable Formation of the Boundary Layer Solutions and Asymptotic Expansions for the Problem of Boundary Layer formation: The Case of Gradual Acceleration Formation of the Boundary Layer about a Body that Suddenly Starts to Move Comments Finite-Difference Method Solving the Boundary Layer Continuation Problem by the Finite Difference Method Solving the Prandtl System for Axially Symmetric Flows by the Finite Difference Method Comments Diffraction Problems for the Prandtl System Boundary Layer with Unknown Border between Two media Mixing of Two Fluids with Distinct Properties at the Interface between Two Flows Comments Boundary Layer in Non-Newtonian Flows Symmetric Boundary Layer in Pseudo-Plastic Fluids Weak Solutions of the Boundary Layer Continuation Problem for Pseudo-Plastic Fluids Nonstationary Boundary Layer for Pseudo-Plastic Fluids Continuation of the Boundary Layer in Dilatable Media Symmetric Boundary Layer in Dilatable Media Comments Boundary Layer in Magnetic Hydrodynamics Continuation of the MHD Boundary Layer in Ordinary Fluids Solving the Equations of the MHD Boundary Layer in Pseudo-Plastic Fluids Self-Similar Solutions of the MHD Boundary Layer System for a Dilatable Fluid Solving the Equations of Boundary Layer for Dilatable Conducting Fluids in a Transversal Magnetic Field Comments Homogenization of Boundary Layer Equations Homogenization of the Prandtl System with Rapidly Oscillating Injection and Suction Homogenization of the Equations of the MHD Boundary Layer in a Rapidly Oscillating Magnetic Field Comments Some Open Problems References Index
293 citations
TL;DR: This work discusses how to impose boundary conditions at irregular domains and free boundaries, as well as the extension of level-set methods to adaptive Cartesian grids and parallel architectures.
289 citations
TL;DR: In this article, the longitudinal free vibration behaviors of one-dimensional nanostructures with various boundary conditions are investigated based on Eringen's nonlocal theory and the governing differential equation of motion is analytically solved for a number of different boundary conditions like clamped, free, attached mass and/or spring.
172 citations
TL;DR: The Spectral Domain Theory of Higher Order Impedance Boundary Conditions Planar Higher Order Impendance Boundaries condition Boundary conditions for Curved Dielectric and Chiral Coatings Scattering by a dielectric-filled Groove in a Ground Plan Scatter by Two-dimensional Dielectrics-Coated Cylinders Scattering of Bodies of revolution.
172 citations
TL;DR: In this article, a general approach for the vibration analysis of a rotating cylindrical shell coupled with an annular plate is proposed, where the Sanders shell theory and Mindlin plate theory are employed to calculate the strain energy of the shell and plate, respectively.
164 citations
TL;DR: In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes.
Abstract: In 2013, a new nonlocal symmetry reduction of the well-known AKNS (an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, and Alan C. Newell et al. (1974)) scattering problem was found. It was shown to give rise to a new nonlocal PT symmetric and integrable Hamiltonian nonlinear Schrodinger (NLS) equation. Subsequently, the inverse scattering transform was constructed for the case of rapidly decaying initial data and a family of spatially localized, time periodic one-soliton solutions was found. In this paper, the inverse scattering transform for the nonlocal NLS equation with nonzero boundary conditions at infinity is presented in four different cases when the data at infinity have constant amplitudes. The direct and inverse scattering problems are analyzed. Specifically, the direct problem is formulated, the analytic properties of the eigenfunctions and scattering data and their symmetries are obtained. The inverse scattering problem, which...
156 citations
29 Mar 2018
TL;DR: In this article, the influence of dipole-dipole inter-atomic coupling on super-radiance was discussed and the effect of diffraction on superradiance on the boundary of a resonant medium was discussed.
Abstract: Preface. Introduction. The elementary theory of super-radiance. The observation of super-radiance. Quantum electrodynamical approach. Quantum fluctuations and self-organization in super-radiance. The semiclassical theory. The influence of dipole-dipole inter-atomic coupling upon super-radiance. Super-radiance of multi-spin systems. Effects of diffraction upon super-radiance. Reflection and transmission on the boundary of a resonant medium. Resonant boundary value problem with local field effects. New sources and applications of super-radiance. Super-radiance references and further reading. Other references. Index.
142 citations
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TL;DR: This work provides a quantitative assessment of new numerical methods as well as available state-of-the-art methods for discretizing the fractional Laplacian, and presents new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions.
Abstract: The fractional Laplacian in R^d has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function u(x) must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. In contrast, the spectral definition requires only the standard local boundary condition. These differences, among others, lead us to ask the question: "What is the fractional Laplacian?" We compare several commonly used definitions of the fractional Laplacian (the Riesz, spectral, directional, and horizon-based nonlocal definitions), and we use a joint theoretical and computational approach to examining their different characteristics by studying solutions of related fractional Poisson equations formulated on bounded domains.
In this work, we provide new numerical methods as well as a self-contained discussion of state-of-the-art methods for discretizing the fractional Laplacian, and we present new results on the differences in features, regularity, and boundary behaviors of solutions to equations posed with these different definitions. We present stochastic interpretations and demonstrate the equivalence between some recent formulations. Through our efforts, we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications.
12 Mar 2018
TL;DR: In this article, a review of the key challenges relevant to the understanding of exchange processes in the mountain boundary layer and outlines possible research priorities for the future is presented. But the authors do not consider the impact of slope and valley breezes on the structure of the convective boundary layer, and the role of intermittent mixing and wave-turbulence interaction in the stable boundary layer.
Abstract: The exchange of heat, momentum, and mass in the atmosphere over mountainous terrain is controlled by synoptic-scale dynamics, thermally driven mesoscale circulations, and turbulence. This article reviews the key challenges relevant to the understanding of exchange processes in the mountain boundary layer and outlines possible research priorities for the future. The review describes the limitations of the experimental study of turbulent exchange over complex terrain, the impact of slope and valley breezes on the structure of the convective boundary layer, and the role of intermittent mixing and wave–turbulence interaction in the stable boundary layer. The interplay between exchange processes at different spatial scales is discussed in depth, emphasizing the role of elevated and ground-based stable layers in controlling multi-scale interactions in the atmosphere over and near mountains. Implications of the current understanding of exchange processes over mountains towards the improvement of numerical weather prediction and climate models are discussed, considering in particular the representation of surface boundary conditions, the parameterization of sub-grid-scale exchange, and the development of stochastic perturbation schemes.
TL;DR: The free vibration analysis of a circular plate made up of a porous material integrated by piezoelectric actuator patches has been studied in this article, where the plate is assumed to be thin and its shear deformations have been neglected.
Abstract: The free vibration analysis of a circular plate made up of a porous material integrated by piezoelectric actuator patches has been studied. The plate is assumed to be thin and its shear deformations have been neglected. The porous material properties vary through the plate thickness according to some given functions. Using Hamilton's variational principle and the classical plate theory (CPT) the governing motion equations have been obtained. Simple and clamped supports have been considered for the boundary conditions. The differential quadrature method (DQM) has been used for the discretizations required for numerical analysis. The effect of some parameters such as thickness ratio, porosity, piezoelectric actuators, variation of piezoelectric actuators-to-porous plate thickness ratio, pores distribution and pores compressibility on the natural frequency, radial and circumferential stresses has been illustrated. The results have been compared with the similar ones in the literature.
TL;DR: In this article, the authors prove the equivalence between the nonlocal strain gradient integral model of elasticity and the differential problem with boundary conditions and provide a viable approach to study size-dependent phenomena in nano-beams of applicative interest.
TL;DR: In this article, the effects of Rayleigh number, oscillating frequency, cavity inclination angle and nanoparticles volume fraction on fluid flow and heat transfer have been analyzed in an inclined square cavity having isothermal wall at x¯ = L, while temperature of the wall x ¯ = 0 is changed as a sinusoidal function of time, other walls are adiabatic.
TL;DR: In this article, general soliton solutions to nonlinear Schrodinger (NLS) with Parity (PT)-symmetry for both zero and nonzero boundary conditions are obtained.
Abstract: General soliton solutions to a nonlocal nonlinear Schrodinger (NLS) equation with PT-symmetry for both zero and nonzero boundary conditions are considered via the combination of Hirota's bilinear method and the Kadomtsev–Petviashvili (KP) hierarchy reduction method. First, general N-soliton solutions with zero boundary conditions are constructed. Starting from the tau functions of the two-component KP hierarchy, it is shown that they can be expressed in terms of either Gramian or double Wronskian determinants. On the contrary, from the tau functions of single component KP hierarchy, general soliton solutions to the nonlocal NLS equation with nonzero boundary conditions are obtained. All possible soliton solutions to nonlocal NLS with Parity (PT)-symmetry for both zero and nonzero boundary conditions are found in the present paper.
TL;DR: In this paper, a geometrically regularized gradient-damage model with energetic equivalence for cracking evolution is proposed for modeling localized failure in solids, which is applied to several benchmark tests of concrete under mode I and mixed-mode failure.
TL;DR: In this article, the linear and nonlinear vibration behaviors of the smart piezoelectric composite plate reinforced by uniformly and non-uniformly dispersing graphene platelets (GPLs) were investigated.
TL;DR: In this paper, the size-dependent axial and flexural free vibrations of Bernoulli-Euler nano-beams are investigated by the modified nonlocal strain gradient elasticity model presented in (Barretta & Marotti de Sciarra, 2018).
TL;DR: In this article, boundary conditions for asymptotically flat spacetimes are given at spatial infinity, and the boundary conditions are invariant under the BMS group, which acts non trivially.
Abstract: New boundary conditions for asymptotically flat spacetimes are given at spatial infinity. These boundary conditions are invariant under the BMS group, which acts non trivially. The boundary conditions fulfill all standard consistency requirements: (i) they make the symplectic form finite; (ii) they contain the Schwarzchild solution, the Kerr solution and their Poincare transforms, (iii) they make the Hamiltonian generators of the asymptotic symmetries integrable and well-defined (finite). The boundary conditions differ from the ones given earlier in the literature in the choice of the parity conditions. It is this different choice of parity conditions that makes the action of the BMS group non trivial. Our approach is purely Hamiltonian and off-shell throughout.
TL;DR: In this article, an analytical method is proposed for the nonlinear primary resonance analysis of cylindrical shells made of functionally graded (FG) porous materials subjected to a uniformly distributed harmonic load including the damping effect.
Abstract: An analytical method is proposed for the nonlinear primary resonance analysis of cylindrical shells made of functionally graded (FG) porous materials subjected to a uniformly distributed harmonic load including the damping effect. The Young's modulus, shear modulus and density of porous materials are assumed to vary through the thickness direction based on the assumption of a common mechanical feature of the open-cell foam. Three types of FG porous distributions, namely symmetric porosity distribution, non-symmetric porosity stiff or soft distribution and uniform porosity distribution are considered in this paper. Theoretical formulations are derived based on Donnell shell theory (DST) and accounting for von-Karman strain-displacement relation and damping effect. The first mode of deflection function that satisfies the boundary conditions is introduced into this nonlinear governing partial differential equation and then a Galerkin-based procedure is utilized to obtain a Duffing-type nonlinear ordinary differential equation with a cubic nonlinear term. Finally, the governing equation is solved analytically by conducting the method of multiple scales (MMS) which results in frequency-response curves of FG porous cylindrical shells in the presence of damping effect. The detailed parametric studies on porosity distribution, porosity coefficient, damping ratio, amplitude and frequency of the external harmonic excitation, aspect ratio and thickness ratio, shown that the distribution type of FG porous cylindrical shells significantly affects primary resonance behavior and the response presents a hardening-type nonlinearity, which provides a useful help for the design and optimize of FG porous shell-type devices working under external harmonic excitation.
TL;DR: In this paper, the authors considered the well-posedness of the boundary value problems for second gradient elasticity in planar pantographic lattices and showed that the considered strain energy density is coercive and positive definite.
Abstract: The well-posedness of the boundary value problems for second gradient elasticity has been studied under the assumption of strong ellipticity of the dependence on the second placement gradients (see, e.g., Chambon and Moullet in Comput. Methods Appl. Mech. Eng. 193:2771–2796, 2004 and Mareno and Healey in SIAM J. Math. Anal. 38:103–115, 2006). The study of the equilibrium of planar pantographic lattices has been approached in two different ways: in dell’Isola et al. (Proc. R. Soc. Lond. Ser. A 472:20150, 2016) a discrete model was introduced involving extensional and rotational springs which is also valid in large deformations regimes while in Boutin et al. (Math. Mech. Complex Syst. 5:127–162, 2017) the lattice has been modelled as a set of beam elements interconnected by internal pivots, but the analysis was restricted to the linear case. In both papers a homogenized second gradient deformation energy, quadratic in the neighbourhood of non deformed configuration, is obtained via perturbative methods and the predictions obtained with the obtained continuum model are successfully compared with experiments. This energy is not strongly elliptic in its dependence on second gradients. We consider in this paper also the important particular case of pantographic lattices whose first gradient energy does not depend on shear deformation: this could be considered either a pathological case or an important exceptional case (see Stillwell et al. in Am. Math. Mon. 105:850–858, 1998 and Turro in Angew. Chem., Int. Ed. Engl. 39:2255–2259, 2000). In both cases we believe that such a particular case deserves some attention because of what we can understand by studying it (see Dyson in Science 200:677–678, 1978). This circumstance motivates the present paper, where we address the well-posedness of the planar linearized equilibrium problem for homogenized pantographic lattices. To do so: (i) we introduce a class of subsets of anisotropic Sobolev’s space as the most suitable energy space $E$
relative to assigned boundary conditions; (ii) we prove that the considered strain energy density is coercive and positive definite in $E$
; (iii) we prove that the set of placements for which the strain energy is vanishing (the so-called floppy modes) must strictly include rigid motions; (iv) we determine the restrictions on displacement boundary conditions which assure existence and uniqueness of linear static problems. The presented results represent one of the first mechanical applications of the concept of Anisotropic Sobolev space, initially introduced only on the basis of purely abstract mathematical considerations.
TL;DR: It is found that velocity of the fluid diminishes for large values of magnetic parameter and porosity parameter, whereas thermal slip parameter shows converse effect and chemical reaction parameter significantly enhances the nanoparticle concentration profile.
Abstract: In this article, we have examined three-dimensional unsteady MHD boundary layer flow of viscous nanofluid having gyrotactic microorganisms through a stretching porous cylinder. Simultaneous effects of nonlinear thermal radiation and chemical reaction are taken into account. Moreover, the effects of velocity slip and thermal slip are also considered. The governing flow problem is modelled by means of similarity transformation variables with their relevant boundary conditions. The obtained reduced highly nonlinear coupled ordinary differential equations are solved numerically by means of nonlinear shooting technique. The effects of all the governing parameters are discussed for velocity profile, temperature profile, nanoparticle concentration profile and motile microorganisms' density function presented with the help of tables and graphs. The numerical comparison is also presented with the existing published results as a special case of our study. It is found that velocity of the fluid diminishes for large values of magnetic parameter and porosity parameter. Radiation effects show an increment in the temperature profile, whereas thermal slip parameter shows converse effect. Furthermore, it is also observed that chemical reaction parameter significantly enhances the nanoparticle concentration profile. The present study is also applicable in bio-nano-polymer process and in different industrial process.
TL;DR: A new finite element method for embedded domain computations, which falls in the category of surrogate/approximate boundary algorithms, is proposed, which is simple, efficient, and also robust, since it is not affected by the small-cut-cell problem.
TL;DR: This article investigates a new class of boundary value problems of one-dimensional lower-order nonlinear Hadamard fractional differential equations and nonlocal multi-point discrete and hadamard integral boundary conditions and shows that the monotone iterative schemes converge to a unique positive solution of the problem.
TL;DR: In this article, a simple first-order shear deformation shell theory (S-FSDST) was proposed for free and transient vibration analysis of composite laminated open cylindrical shells with general boundary conditions.
TL;DR: In this article, uniqueness results for boundary value problem of fractional differential equation are obtained using both the Banach's contraction mapping principle and the theory of linear operator, and a comparison between the obtained results is provided.
Abstract: In this paper, uniqueness results for boundary value problem of fractional differential equation are obtained. Both the Banach's contraction mapping principle and the theory of linear operator are used, and a comparison between the obtained results is provided.
TL;DR: In this article, the entanglement hamiltonian and the spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints are studied.
Abstract: Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.
TL;DR: In this paper, the authors show how states with conserved numbers of dynamical defects (strings, domain walls, etc.) can be understood as possessing generalized global symmetries even when the microscopic origins of these symmetry properties are unknown, and build an effective theory of a $2+1$-dimensional fluid state with two perpendicular sets of immersed elastic line defects.
Abstract: In this work, we show how states with conserved numbers of dynamical defects (strings, domain walls, etc.) can be understood as possessing generalized global symmetries even when the microscopic origins of these symmetries are unknown. Using this philosophy, we build an effective theory of a $2+1$-dimensional fluid state with two perpendicular sets of immersed elastic line defects. When the number of defects is independently conserved in each set, then the state possesses two one-form symmetries. Normally, such viscoelastic states are described as fluids coupled to Goldstone bosons associated with spontaneous breaking of translational symmetry caused by the underlying microscopic structure---the principle feature of which is a transverse sound mode. At the linear, nondissipative level, we verify that our theory, based entirely on symmetry principles, is equivalent to a viscoelastic theory. We then build a simple holographic dual of such a state containing dynamical gravity and two two-form gauge fields, and use it to study its hydrodynamic and higher-energy spectral properties characterized by nonhydrodynamic, gapped modes. Based on the holographic analysis of transverse two-point functions, we study consistency between low-energy predictions of the bulk theory and the effective boundary theory. Various new features of the holographic dictionary are explained in theories with higher-form symmetries, such as the mixed-boundary-condition modification of the quasinormal mode prescription that depends on the running coupling of the boundary double-trace deformations. Furthermore, we examine details of low- and high-energy parts of the spectrum that depend on temperature, line defect densities and the renormalization group scale.
TL;DR: Computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed, reproducing kernel Hilbert space.
Abstract: Many problems arising in different fields of sciences and engineering can be reduced, by applying some appropriate discretization, either to a system of integrodifferential algebraic equations or to a sequence of such systems. The aim of the present analysis is to implement a relatively recent computational method, reproducing kernel Hilbert space, for obtaining the solutions of integrodifferential algebraic systems of temporal two-point boundary value problems. Two extended inner product spaces W[0, 1] and H[0, 1] are constructed in which the boundary conditions of the systems are satisfied, while two smooth kernel functions R
t
(s) and r
t
(s) are used throughout the evolution of the algorithm in order to obtain the required grid points. An efficient construction is given to obtain the numerical solutions for the systems together with an existence proof of the exact solutions based upon the reproducing kernel theory. In this approach, computational results of some numerical examples are presented to illustrate the viability, simplicity, and applicability of the algorithm developed.