Dirichlet Boundary Conditions Research Articles

Entanglement entropies of an interval for the massless scalar field in the presence of a boundary

We study the entanglement entropies of an interval for the massless compact boson either on the half line or on a finite segment, when either Dirichlet or Neumann boundary conditions are imposed. In these boundary conformal field theory models, the method of the branch point twist fields is employed to obtain analytic expressions for the two-point functions of twist operators. In the decompactification regime, these analytic predictions in the continuum are compared with the lattice numerical results in massless harmonic chains for the corresponding entanglement entropies, finding good agreement. The application of these analytic results in the context of quantum quenches is also discussed.

Hopf bifurcation in a delay-diffusion housefly equation with Dirichlet boundary conditions

The dynamical behaviors for a delay-diffusion housefly equation with two kinds of Dirichlet boundary conditions are considered in this paper. The existence and uniqueness of the steady state solutions are investigated, and the stability of the constant steady state solutions is obtained by using qualitative theory. The existence of Hopf bifurcation near the positive constant steady state solution is discussed and the expressions which can identify the bifurcation properties, including the stability of the bifurcating periodic solution and the bifurcation direction, are presented.

Stability estimates for some parabolic inverse problems with the final overdetermination via a new Carleman estimate

This paper is about Hölder and Lipschitz stability estimates and uniqueness theorems for some coefficient inverse problems and associated inverse source problems for a general linear parabolic equation of the second order with variable coefficients. The data for the inverse problem are given at the final moment of time . In addition, both Dirichlet and Neumann boundary conditions are given either on a part or on the entire lateral boundary. Thus, if these boundary conditions are given only at a part of the boundary, then even if the target coefficient is known, still the forward problem is not a classical initial boundary value problem.

A note on the persistence of multiplicity of eigenvalues of fractional Laplacian under perturbations

We consider the eigenvalue problem for the fractional Laplacian (−Δ)s, s∈(0,1), in a bounded domain Ω with Dirichlet boundary condition. A recent result (see Fall et al., 2023) states that, under generic small perturbations of the coefficient of the equation or of the domain Ω, all the eigenvalues are simple. In this paper we give a condition for which a perturbation of the coefficient or of the domain preserves the multiplicity of a given eigenvalue. Also, in the case of an eigenvalue of multiplicity ν=2 we prove that the set of perturbations of the coefficients which preserve the multiplicity is a smooth manifold of codimension 2 in C1(Rn).

Numerical Reconstruction of Time-Dependent Boundary Conditions to 2D Heat Equation on Disjoint Rectangles at Integral Observations

In this paper, two-dimensional (2D) heat equations on disjoint rectangles are considered. The solutions are connected by interface Robin’s-type internal conditions. The problem has external Dirichlet boundary conditions that, in the forward (direct) formulation, are given functions. In the inverse problem formulation, the Dirichlet conditions are unknown functions, and the aim is to be reconstructed upon integral observations. Well-posedness both for direct and inverse problems is established. Using the given 2D integrals of the unknown solution on each of the domains and the specific interface boundary conditions, we reduce the 2D inverse problem to a forward heat 1D one. The resulting 1D problem is solved using the explicit Saul’yev finite difference method. Numerical test examples are discussed to illustrate the efficiency of the approach.

Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method’s key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete H1 and L2 norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.

Fermions, quantum gravity, and holography in two dimensions

We study a model comprising N flavors of Kähler Dirac fermion propagating on a triangulated two-dimensional disk which is constrained to have a negative average bulk curvature. Dirichlet boundary conditions are chosen for the fermions. Quantum fluctuations of the geometry are included by summing over all possible triangulations consistent with these constraints. We show in the limit N→∞ that the partition function is dominated by a regular triangulation of two-dimensional hyperbolic space. We use strong coupling expansions and Monte Carlo simulation to show that in this limit boundary correlators of the fermions have a power law dependence on boundary separation as one expects from holography. However, we argue that this behavior breaks down for any finite number of massive fields in the thermodynamic limit and quantum fluctuations of the bulk geometry drive the theory into a nonholographic phase. In contrast, for massless fermions, we find evidence that the boundary is conformal even for finite N. This is consistent with theoretical results in quantum Liouville theory. Published by the American Physical Society 2024

Two solutions for Dirichlet double phase problems with variable exponents

Abstract This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

A Study of Singular Similarity Solutions to Laplace’s Equation with Dirichlet Boundary Conditions

Maximum principles for elliptic operators in unbounded Riemannian domains

The necessity of a Maximum Principle arises naturally when one is interested in the study of qualitative properties of solutions to partial differential equations. In general, to ensure the validity of these kinds of principles one has to consider some additional assumptions on the ambient manifold or on the differential operator. The present work aims to address, using both of these approaches, the problem of proving Maximum Principles for second order, elliptic operators acting on unbounded Riemannian domains under Dirichlet boundary conditions. Hence there is a natural division of this article in two distinct and standalone sections.

Numerical simulation of Rashba spin–orbit coupling effect on quantum Hall resistivity within tight-binding approximation

A numerical simulation of the effect of Rashba spin–orbit coupling (RSO) on quantum transport properties of electrons confined to a two-dimensional electron gas (2DEG) in the integer quantum Hall regime is presented. The mechanism of quantization and the dependence of the Hall resistivity (HR) on the system built-in and external parameters are studied using the non-equilibrium Green’s function (NEGF) technique. Relevant Green’s functions are computed numerically using spectral decomposition method in combination with recursive techniques. The tight-binding-like representation of the system Hamiltonian including magnetic field is obtained within the finite difference method, using Dirichlet boundary conditions. Corresponding formula linking the Hall resistivity with the transmission function of the system has been derived analytically within the NEGF formalism in a general form, in the sense that the formula can also be used beyond the linear response limit and for nonzero temperatures. In particular, it was shown explicitly that in the presence of the RSO coupling the even- and odd plateaus emerging in the magnetic-field dependence of the quantum Hall resistivity are related to different physical mechanisms.

Genocchi Wavelet Method for the Solution of Time-Fractional Telegraph Equations with Dirichlet Boundary Conditions

Genocchi Wavelet Method for the Solution of Time-Fractional Telegraph Equations with Dirichlet Boundary Conditions

Solving forward and inverse problems of contact mechanics using physics-informed neural networks

This paper explores the ability of physics-informed neural networks (PINNs) to solve forward and inverse problems of contact mechanics for small deformation elasticity. We deploy PINNs in a mixed-variable formulation enhanced by output transformation to enforce Dirichlet and Neumann boundary conditions as hard constraints. Inequality constraints of contact problems, namely Karush–Kuhn–Tucker (KKT) type conditions, are enforced as soft constraints by incorporating them into the loss function during network training. To formulate the loss function contribution of KKT constraints, existing approaches applied to elastoplasticity problems are investigated and we explore a nonlinear complementarity problem (NCP) function, namely Fischer–Burmeister, which possesses advantageous characteristics in terms of optimization. Based on the Hertzian contact problem, we show that PINNs can serve as pure partial differential equation (PDE) solver, as data-enhanced forward model, as inverse solver for parameter identification, and as fast-to-evaluate surrogate model. Furthermore, we demonstrate the importance of choosing proper hyperparameters, e.g. loss weights, and a combination of Adam and L-BFGS-B optimizers aiming for better results in terms of accuracy and training time.

Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry

We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues λ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (n)$$\\end{document} of shapes with n edges that are of the form λ(n)∼x∑k=0∞Ck(x)nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (n) \\sim x\\sum _{k=0}^{\\infty } \\frac{C_k(x)}{n^k}$$\\end{document} where x is the limiting eigenvalue for n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document}. Expansions of this form have previously only been known for regular polygons with Dirichlet boundary conditions and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order Ck(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_k(x)$$\\end{document} and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons, and star shapes with sinusoidal boundary).

Transient Heat Transfer Analysis in Metal Plates with Variable Thickness

Nonlinear transient heat transfer via conduction–radiation is a dynamic topic of long-standing interest with applications ranging from aeronautical and mechanical engineering to industrial and civil security. To gain a better understanding of the performances of materials having thermal proprieties that change during nonlinear heat transfer, several studies using the finite element method (FEM) have been conducted. Such studies apply nonlinear thermal material characteristics to describe the complete system under different loading conditions in each region by adjusting the temperature values for the other three edges and the thickness parameter with Dirichlet boundary conditions. As a result, while modeling and simulating temperature distributions for such situations, nonlinearities generated by temperature-dependent thermal conductivity must be considered. In this work, we focus on the analysis of coupled transient heat transfer through two metal plates with temperature-dependent thermal characteristics in which the temperature is fixed along the bottom edge and heat is transferred from both the top and bottom faces of the two plates. FEM is employed to solve the nonlinear heat equation and compute the temperature as a function of time for variable thickness. The study examines the effect of modifying the thickness parameter values on the temperature distribution over time for various edge values over 5,000 s.

Global well-posedness for 2D non-resistive MHD equations in half-space

This paper focuses on the initial boundary value problem of two-dimensional non-resistive MHD equations in a half space. We prove that the non-resistive MHD equations have a unique global strong solution around the equilibrium state (0, e1) for Dirichlet boundary condition of velocity and modified Neumann boundary condition of magnetic.

Weak Galerkin finite element methods for optimal control problems governed by second order elliptic equations

This paper is concerned with the development of weak Galerkin (WG) finite element methods for optimal control problems governed by second order elliptic partial differential equations. WG methods are advantageous over other existing methods in that the control variable along the domain boundary is a natural variable by design in the numerical scheme. A convergence theory was established for the numerical solutions in various Sobolev norms. Numerical results are presented to demonstrate the efficiency and accuracy of the new scheme.

Wavelet Potentials: An Efficient Potential Recovery Technique for Pointwise Incompressible Fluids

AbstractWe introduce an efficient technique for recovering the vector potential in wavelet space to simulate pointwise incompressible fluids. This technique ensures that fluid velocities remain divergence‐free at any point within the fluid domain and preserves local volume during the simulation. Divergence‐free wavelets are utilized to calculate the wavelet coefficients of the vector potential, resulting in a smooth vector potential with enhanced accuracy, even when the input velocities exhibit some degree of divergence. This enhanced accuracy eliminates the need for additional computational time to achieve a specific accuracy threshold, as fewer iterations are required for the pressure Poisson solver. Additionally, in 3D, since the wavelet transform is taken in‐place, only the memory for storing the vector potential is required. These two features make the method remarkably efficient for recovering vector potential for fluid simulation. Furthermore, the method can handle various boundary conditions during the wavelet transform, making it adaptable for simulating fluids with Neumann and Dirichlet boundary conditions. Our approach is highly parallelizable and features a time complexity of O(n), allowing for seamless deployment on GPUs and yielding remarkable computational efficiency. Experiments demonstrate that, taking into account the time consumed by the pressure Poisson solver, the method achieves an approximate 2x speedup on GPUs compared to state‐of‐the‐art vector potential recovery techniques while maintaining a precision level of 10−6 when single float precision is employed. The source code of ‘Wavelet Potentials’ can be found in https://github.com/yours321dog/WaveletPotentials.

Exploring Physics-Informed Neural Networks for the Generalized Nonlinear Sine-Gordon Equation

The nonlinear sine-Gordon equation is a prevalent feature in numerous scientific and engineering problems. In this paper, we propose a machine learning-based approach, physics-informed neural networks (PINNs), to investigate and explore the solution of the generalized non-linear sine-Gordon equation, encompassing Dirichlet and Neumann boundary conditions. To incorporate physical information for the sine-Gordon equation, a multiobjective loss function has been defined consisting of the residual of governing partial differential equation (PDE), initial conditions, and various boundary conditions. Using multiple densely connected independent artificial neural networks (ANNs), called feedforward deep neural networks designed to handle partial differential equations, PINNs have been trained through automatic differentiation to minimize a loss function that incorporates the given PDE that governs the physical laws of phenomena. To illustrate the effectiveness, validity, and practical implications of our proposed approach, two computational examples from the nonlinear sine-Gordon are presented. We have developed a PINN algorithm and implemented it using Python software. Various experiments were conducted to determine an optimal neural architecture. The network training was employed by using the current state-of-the-art optimization methods in machine learning known as Adam and L-BFGS-B minimization techniques. Additionally, the solutions from the proposed method are compared with the established analytical solutions found in the literature. The findings show that the proposed method is a computational machine learning approach that is accurate and efficient for solving nonlinear sine-Gordon equations with a variety of boundary conditions as well as any complex nonlinear physical problems across multiple disciplines.

Flow of Bingham fluid through a three dimensional thin layer

The paper is devoted to the study of asymptotic behavior of the solution of three dimensional steady flow of Bingham fluid through a three dimensional thin layer with Dirichlet boundary conditions. We are interested in the asymptotic behavior, to this aim we prove some convergence results concerning the velocity and pressure when the thickness tends to zero. The limit problem obtained after transforming the original problem into one posed over a fixed reference domain and the parameter representing the thickness of the layer tend to zero is studied. The lower-dimensional constitutive law and the differential equation satisfied by the limit variables in the non rigid zone are obtained. In addition, the uniqueness of limit solution has been also established. Existence and uniqueness results and a lower-dimensional constitutive law are obtained. An identical study of a two-dimensional problem yields a one dimensional model prevalent in engineering literature.