Dirichlet Boundary Conditions Research Articles

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.

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.

Global existence and blowup of solutions to a class of wave equations with Hartree type nonlinearity

Abstract In this paper, we consider a class of wave-Hartree equations on a bounded smooth convex domain with Dirichlet boundary condition. We prove the local existence of solutions in the natural energy space by using the standard Galërkin method. The results on global existence and nonexistence of solutions are obtained mainly by means of the potential well theory and concavity method.

Existence of solutions for anisotropic parabolic Ni-Serrin type equations originated from a capillary phenomena with nonstandard growth nonlinearity

We consider an initial boundary value problem for a class of anisotropic parabolic Ni-Serrin type equations with nonstandard nonlinearity in a bounded smooth domain with homogeneous Dirichlet boundary condition. Because the nonlinear perturbation leads to difficulties (it does not have a definite sign) in obtaining a priori estimates in the energy method, we had to modify the Tartar method significantly. Under suitable assumptions, we obtain the global existence, decay and extinction of solutions.

Lyapunov-type inequalities for a nonlinear system including operators

UDC 517.9 We obtain new Lyapunov-type inequalities for a nonlinear system including p -relativistic operator and q -prescribed curvature operator under the Dirichlet or antiperiodic boundary condition.

Minimal graphs in Riemannian spaces

In this treatise, we discuss existence and uniqueness questions for parametric minimal surfaces in Riemannian spaces, which represent minimal graphs. We choose the Riemannian metric suitably such that the variational solution of the Riemannian Dirichlet integral under Dirichlet boundary conditions possesses a one-to-one projection onto a plane. At first we concentrate our considerations on existence results for boundary value problems. Then we study uniqueness questions for boundary value problems and for complete minimal graphs. Here we establish curvature estimates for minimal graphs on discs, which imply Bernstein-type results.

Mathematical properties and numerical approximation of pseudo-parabolic systems

The paper is concerned with the mathematical theory and numerical approximation of systems of partial differential equations (pde) of hyperbolic, pseudo-parabolic type. Some mathematical properties of the initial-boundary-value problem (ibvp) with Dirichlet boundary conditions are first studied. They include the weak formulation, well-posedness and existence of traveling wave solutions connecting two states, when the equations are considered as a variant of a conservation law. Then, the numerical approximation consists of a spectral approximation in space based on Legendre polynomials along with a temporal discretization with strong stability preserving (SSP) property. The convergence of the semidiscrete approximation is proved under suitable regularity conditions on the data. The choice of the temporal discretization is justified in order to guarantee the stability of the full discretization when dealing with nonsmooth initial conditions. A computational study explores the performance of the fully discrete scheme with regular and nonregular data.

A Partition of Unity construction of the stabilization function in Nitsche’s method for variational problems

In this paper we develop a partition-of-unity construction of the stabilization function required in Nitsche’s method, which can be seen as a generalization of the element-wise construction that is widely used in finite element methods. This allows for the use of Nitsche’s method within the Partition of Unity Method with a stabilization function that is not simply a constant over the whole boundary. In addition to that, we introduce a patch-aggregation approach designed to avoid arbitrarily large values of the stabilization function and the associated ill-conditioned systems and deteriorated convergence rates. We present numerical results to validate the proposed methods, covering Dirichlet boundary conditions, interface constraints and higher-order problems. These results clearly show that our approach leads to optimal convergence rates.

Long time dynamics of nonequilibrium electroconvection

The Nernst-Planck-Stokes (NPS) system models electroconvection of ions in a fluid. We consider the system, for two oppositely charged ionic species, on three dimensional bounded domains with Dirichlet boundary conditions for the ionic concentrations (modelling ion selectivity), Dirichlet boundary conditions for the electrical potential (modelling an applied potential), and no-slip boundary conditions for the fluid velocity. In this paper, we obtain quantitative bounds on solutions of the NPS system in the long time limit, which we use to prove (1) the existence of a compact global attractor with finite fractal (box-counting) dimension and (2) space-time averaged electroneutrality ρ ≈ 0 \rho \approx 0 in the singular limit of Debye length going to zero, ϵ → 0 \epsilon \to 0 .

Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

A boundary condition-enhanced direct-forcing immersed boundary method for simulations of three-dimensional phoretic particles in incompressible flows

In this paper we propose an improved three-dimensional immersed boundary method coupled with a finite-difference code to simulate self-propelled phoretic particles in viscous incompressible flows. We focus on the phenomenon of diffusiophoresis which, using the driving of a concentration gradient, can generate a slip velocity on a surface. In such a system, both the Dirichlet and Neumann boundary conditions are involved. In order to enforce the boundary conditions, we propose two improvements to the basic direct-forcing immersed boundary method. The main idea is that the immersed boundary terms are corrected by adding the force of the previous time step, in contrast to the traditional method which relies only on the instantaneous forces in each time step. For the Neumann boundary condition, we add two auxiliary layers inside the body to precisely implement the desired concentration gradient. To verify the accuracy of the improved method, we present problems of different complexity: The first is the pure diffusion around a sphere with Dirichlet and Neumann boundary conditions. Then we show the flow past a fixed sphere. In addition, the motion of a self-propelled Janus particle in the bulk and the spontaneously symmetry breaking of an isotropic phoretic particle are reported. The results are in very good agreements with the data that are reported in previously published literature.