Dirichlet Boundary Conditions Research Articles

On Caputo-Hadamard fractional pantograph problem of two different orders with Dirichlet boundary conditions

This manuscript aims to study the effectiveness of two different levels of fractional orders in the frame of Caputo-Hadamard (CH)-derivatives on a special type class of delay problem supplemented by Dirichlet boundary conditions. The corresponding Hadamard fractional integral equation is derived for a proposed CH-fractional pantograph system. The Banach, Schaefer, and Krasnoselskii fixed point theorems (FPTs), are used to investigate sufficient conditions of the existence and uniqueness theorems for the proposed system. Furthermore, the Green functions properties are investigated and used to discuss the Ulam-Hyers (UH) stability and its generalized by utilizing nonlinear analysis topics. Finally, three mathematical examples are provided with numerical results and figures by using Matlab software to illustrate the validity of our findings.

Determination of Spacewise− Dependent Heat Source Term in Pseudoparabolic Equation from Overdetermination Conditions

This paper examines the finding of spacewise dependent heat source function in pseudoparabolic equation with initial and homogeneous Dirichlet boundary conditions, as well as the final time value / integral specification as additional conditions that ensure the uniqueness solvability of the inverse problem. However, the problem remains ill-posed because tiny perturbations in input data cause huge errors in outputs. Thus, we employ Tikhonov’s regularization method to restore this instability. In order to choose the best regularization parameter, we employ L-curve method. On the other hand, the direct (forward) problem is solved by a finite difference scheme while the inverse one is reformulated as an optimization problem. The later problem is accomplished by employing lsqnolin subroutine from MATLAB. Two test examples are presented to show the efficiency and accuracy of the employed method by including many noises level and various regularization parameters.

The convergence rate of p-harmonic to infinity-harmonic functions

The purpose of this paper is to prove a uniform convergence rate of the solutions of the p-Laplace equation with Dirichlet boundary conditions to the solution of the infinity-Laplace equation as . The rate scales like for general solutions of the Dirichlet problem and like for solutions with positive gradient. An explicit example shows that it cannot be better than . The proof of this result solely relies on the comparison principle with the fundamental solutions of the p-Laplace and the infinity-Laplace equation, respectively. Our argument does not use viscosity solutions, is purely metric, and is therefore generalizable to more general settings where a comparison principle with Hölder cones and Hölder regularity is available.

3d exceptional gauge theories and boundary confinement

We find boundary confining dualities of 3d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 supersymmetric gauge theories with exceptional gauge groups. The half-indices which enumerate the boundary BPS local operators in the presence of Neumann and Dirichlet boundary conditions for gauge fields are identified with the Askey-Wilson type q-beta integrals and Macdonald type sums respectively. New conjectural identities of E6 and E7 integrals and sums are derived from the boundary confining dualities. We also consider theories with a vector multiplet and adjoint chiral, which correspond to an N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 vector multiplet, with appropriate boundary conditions. We argue for the boundary confinement of the N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 vector multiplet and comment on such theories also with classical gauge groups.

LabPET II scanner performances improvement: Thermal stability control based on FPGA

The LabPET II detection module serves as the fundamental component within Positron Emission Tomography (PET) scanners designed for achieving ultra-high-resolution imaging of small to medium-sized animals as well as the human brain. To ensure peak performance, it is imperative that this module operates consistently at a stable temperature. In this paper, we present an automatic and real-time method to improve the thermal control of the scanner LabPET II by evacuating the heat to reduce the overall temperature and maintain the thermal stability of the system. Based on the Thermo-Electric Cooler (TEC) process, our method applies two numerical techniques, Computational Fluid Dynamics (CFD) and Heat Transfer Analysis (HTA). The Dirichlet Boundary Conditions (DBC) method is applied for the Application-Specific Integrated Circuit (ASIC) at 25∘ C. The real measurements confirm and validate the simulation results with the Finite Element Method (FEM) based on the software COMSOL Multiphysics™. Field-Programmable Gate Array (FPGA) implementation on the Development and Education (DE1) board is performed with comparative results study. The proposed method is both simple and practical, and ensures thermal stability at 35∘C for the duration of operation of the scanner, which has a complex structure.

Boundary conditions and infrared divergences

We review the procedure to construct quasi-free ground states, for real scalar fields whose dynamics is dictated by the Klein-Gordon equation, on standard static Lorentzian manifolds with a time-like boundary. We observe that, depending on the assigned boundary condition of Robin type, this procedure does not always lead to the existence of a suitable bi-distribution ω2∈D′(M×M) due to the presence of infrared divergences. As a concrete example we consider a Bertotti-Robinson spacetime in two different coordinate patches. In one case we show that infrared divergences do not occur only for Dirichlet boundary conditions as one might expect a priori, while, in the other case, we prove that they occur only when Neumann boundary conditions are imposed at the time-like boundary.

Local well posedness for a system of quasilinear PDEs modelling suspension bridges

In this paper we provide a local well posedness result for a quasilinear beam-wave system of equations on a one-dimensional spatial domain under periodic and Dirichlet boundary conditions. This kind of systems provides a refined model for the time-evolution of suspension bridges, where the beam and wave equations describe respectively the longitudinal and torsional motion of the deck. The quasilinearity arises when one takes into account the nonlinear restoring action of deformable cables and hangers. To obtain the a priori estimates for the solutions of the linearized equation we build a modified energy by means of paradifferential changes of variables. Then we construct the solutions of the nonlinear problem by using a quasilinear iterative scheme à la Kato.

On Lewy Stampacchia inequalities for a pseudomonotone parabolic obstacle problem with [formula omitted]-data

The aim of this paper is to prove the existence of entropy solution, and the associated Lewy-Stampacchia inequalities in a renormalized form, to some parabolic obstacle problems governed by a Leray-Lions pseudomonotone operator in the presence of L1-data and homogeneous Dirichlet boundary conditions.

Physically constrained learning of MOS capacitor electrostatics

In recent years, neural networks have achieved phenomenal success across a wide range of applications. They have also proven useful for solving differential equations. The focus of this work is on the Poisson–Boltzmann equation (PBE) that governs the electrostatics of a metal–oxide–semiconductor capacitor. We were motivated by the question of whether a neural network can effectively learn the solution of PBE using the methodology pioneered by Lagaris et al. [IEEE Trans. Neural Netw. 9 (1998)]. In this method, a neural network is used to generate a set of trial solutions that adhere to the boundary conditions, which are then optimized using the governing equation. However, the challenge with this method is the lack of a generic procedure for creating trial solutions for intricate boundary conditions. We introduce a novel method for generating trial solutions that adhere to the Robin and Dirichlet boundary conditions associated with the PBE. Remarkably, by optimizing the network parameters, we can learn an optimal trial solution that accurately captures essential physical insights, such as the depletion width, the threshold voltage, and the inversion charge. Furthermore, we show that our functional solution can extend beyond the sampling domain.

A note on the one-dimensional critical points of the Ambrosio–Tortorelli functional

This note addresses the question of convergence of critical points of the Ambrosio–Tortorelli functional in the one-dimensional case under pure Dirichlet boundary conditions. An asymptotic analysis argument shows the convergence to two possible limits points: either a globally affine function or a step function with a single jump at the middle point of the space interval, which are both critical points of the one-dimensional Mumford–Shah functional under a Dirichlet boundary condition. As a byproduct, non minimizing critical points of the Ambrosio–Tortorelli functional satisfying the energy convergence assumption as in (Babadjian, Millot and Rodiac (2022)) are proved to exist.

Precise eigenvalues in the solutions of generalized Sturm–Liouville problems

In the paper, we determine the eigenvalues and eigenfunctions of the generalized Sturm–Liouville problems, whose potentials may be nonlinear functions of eigen-parameter, by developing new iterative algorithms based on the fictitious time integration method (FTIM) and half-interval method (HIM). We derive two eigen-parameter dependent linear shape functions, from which we can transform the generalized Sturm–Liouville problem to an initial value problem for a new variable, and automatically preserve the prescribed eigen-parameter dependent Sturm–Liouville boundary conditions. A nonlinear equation in terms of the relative norm of two consecutive right-end values of the new variable is derived for iteratively determining the eigenvalue by using the FTIM. The resultant sequence of the iterated eigenvalues are monotonically convergent to the desired eigenvalue, and meanwhile the unknown initial values of the eigenfunction can be determined for computing the eigenfunction by integrating the Sturm–Liouville equation. We propose a high precision point target method by using the HIM to determine the eigenvalue and eigenfunction of the generalized Sturm–Liouville problem, which is transformed to a definite initial value problem with a simpler Dirichlet or Neumann boundary condition on the right-end as a point target equation. Several new theoretical results are proved such that more simple generalized Sturm–Liouville problem can be derived with a single point target on the right-end. Depending on different transformation techniques and the derived target equations, seven types FTIM and four types HIM are developed. Numerical examples confirm that the present FTIM and HIM are effective and accurate.

Black hole perturbation theory and multiple polylogarithms

We study black hole linear perturbation theory in a four-dimensional Schwarzschild (anti) de Sitter background. When dealing with a positive cosmological constant, the corresponding spectral problem is solved systematically via the Nekrasov-Shatashvili functions or, equivalently, classical Virasoro conformal blocks. However, this approach can be more complicated to implement for certain perturbations if the cosmological constant is negative. For these cases, we propose an alternative method to set up perturbation theory for both small and large black holes in an analytical manner. Our analysis reveals a new underlying recursive structure that involves multiple polylogarithms. We focus on gravitational, electromagnetic, and conformally coupled scalar perturbations subject to Dirichlet and Robin boundary conditions. The low-lying modes of the scalar sector of gravitational perturbations and its hydrodynamic limit are studied in detail.

Unfitted generalized finite element methods for Dirichlet problems without penalty or stabilization

AbstractUnfitted finite element methods (FEM) have attractive merits for problems with evolving or geometrically complex boundaries. Conventional unfitted FEMs incorporate penalty terms, parameters, or Lagrange multipliers to impose the Dirichlet boundary condition weakly. This to some extent increases computational complexity in implementation. In this article, we propose an unfitted generalized FEM (GFEM) for the Dirichlet problem, which is free from any penalty or stabilization. This is achieved by means of partition of unity frameworks of GFEM and designing a set of new enrichments for the Dirichlet boundary. The enrichments are divided into two groups: the one is used to impose the Dirichlet boundary condition strongly, and the other one serves as energy space of variational formulations. The shape functions in energy space vanish at the boundary so that standard variational formulae like those in the conventional fitted FEM can be applied, and thus the penalty and stabilization are not needed. The optimal convergence rate in the energy norm is proven rigorously. Numerical experiments and comparisons with other methods are executed to verify the theoretical result and effectiveness of the algorithm. The conditioning of new method is numerically shown to be of same order as that of the standard FEM.

A reduced Landau-de Gennes study for nematic equilibria in three-dimensional prisms

Abstract We model nematic liquid crystal configurations inside three-dimensional prisms, with a polygonal cross-section and Dirichlet boundary conditions on all prism surfaces. We work in a reduced Landau-de Gennes framework, and the Dirichlet conditions on the top and bottom surfaces are special in the sense that they are critical points of the reduced Landau-de Gennes energy on the polygonal cross-section. The choice of the boundary conditions allows us to make a direct correspondence between the three-dimensional Landau-de Gennes critical points and pathways on the two-dimensional Landau-de Gennes solution landscape on the polygonal cross-section. We explore this concept by means of asymptotic analysis and numerical examples, with emphasis on a cuboid and a hexagonal prism, focusing on three-dimensional multistability tailored by two-dimensional solution landscapes.

Nine-point compact sixth-order approximation for two-dimensional nonlinear elliptic partial differential equations: Application to bi- and tri-harmonic boundary value problems

Nine point sixth order compact numerical approximations are suggested to solve 2D nonlinear elliptic partial differential equations (NLEPDEs) and for the estimation of normal derivatives on a uniform rectangular grid subject to Dirichlet boundary conditions. We deliberate error analysis and reveal that, under specific conditions, our method converges to the sixth order. In addition, we extend our technique to vector form in order to solve the system of NLEPDEs. In application, we discuss nine-point compact sixth order approximations for bi- and tri-harmonic elliptic boundary value problems. Numerical experiments are carried out on several benchmark problems including bi- and tri-harmonic equations, and verified the sixth order convergence of the proposed methods.

Large time behavior of signed fractional porous media equations on bounded domains

Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.

Magnetic Skyrmions Under Confinement

We present a variational treatment of confined magnetic skyrmions in a minimal micromagnetic model of ultrathin ferromagnetic films with interfacial Dzylashinksii-Moriya interaction (DMI) in competition with the exchange energy, with a possible addition of perpendicular magnetic anisotropy. Under Dirichlet boundary conditions that are motivated by the asymptotic treatment of the stray field energy in the thin film limit we prove existence of topologically non-trivial energy minimizers that concentrate on points in the domain as the DMI strength parameter tends to zero. Furthermore, we derive the leading order non-trivial term in the Γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Gamma $$\\end{document}-expansion of the energy in the limit of vanishing DMI strength that allows us to completely characterize the limiting magnetization profiles and interpret them as particle-like states whose radius and position are determined by minimizing a renormalized energy functional. In particular, we show that in our setting the skyrmions are strongly repelled from the domain boundaries, which imparts them with stability that is highly desirable for applications. We provide explicit calculations of the renormalized energy for a number of basic domain geometries.

Orthogonal ring patterns in the plane

We introduce orthogonal ring patterns consisting of pairs of concentric circles generalizing circle patterns. We show that orthogonal ring patterns are governed by the same equation as circle patterns. For every ring pattern there exists a one parameter family of patterns that interpolates between a circle pattern and its dual. We construct ring patterns analogues of the Doyle spiral, Erf and z^alpha functions. We also derive a variational principle and compute ring patterns based on Dirichlet and Neumann boundary conditions.

Computational evaluation of flight performance of flapping-wing nano air vehicles using hierarchical coupling of nonlinear dynamic and fluid-structure interaction analyses

This study endeavours to introduce an inventive hierarchical coupling methodology for evaluating the flight capabilities of polymer micromachined flapping-wing nano air vehicles (FWNAVs) using advanced computational tools. Furthermore, our objective is to provide a practical demonstration of FWNAVs in both tethered and airborne scenarios. These dual objectives represent the distinctive and pioneering aspects of this research. Such FWNAVs, which are insect-inspired robots, have a 2.5-dimensional structure. An FWNAV comprises a micro piezoelectric drive system and a pair of micro thin flexible wings. The drive system includes a micro transmission and a piezoelectric bimorph actuator. The flight performance of the designed FWNAV is evaluated using a hierarchical coupling approach, where the whole system is decomposed into a micro wing and a micro piezoelectric drive system. The coupling between these parts is modelled as one-way coupling and that between the micro wings and the surrounding air is modelled as strong coupling. In the one-way coupling analysis, nonlinear structural dynamic analysis is conducted for the micro piezoelectric drive system, where the dynamic response is transmitted to the micro wings via the Dirichlet boundary condition. In the strong coupling analysis, strongly coupled fluid-structure interaction analysis is conducted for the micro wings and the surrounding air to consider their strong coupling. The optimisation of flight performance is conducted using fluid-structure interaction analysis to achieve sufficient lift force to support the weight of an FWNAV. The design of a tethered and flyable FWNAV with a size of 10 mm or smaller is demonstrated. This FWNAV can be easily fabricated using polymer micromachining.

Homogenization in perforated domains at the critical scale

We describe the asymptotic behaviour of the minimal heterogeneous d-capacity of a small set, which we assume to be a ball for simplicity, in a fixed bounded open set Ω⊆Rd, with d≥2. Two parameters are involved: ɛ, the radius of the ball, and δ, the length scale of the heterogeneity of the medium. We prove that this capacity behaves as C|logɛ|1−d, where C=C(λ) is an explicit constant depending on the parameter λ≔limɛ→0|logδ|/|logɛ|.We determine the Γ-limit of oscillating integral functionals subjected to Dirichlet boundary conditions on periodically perforated domains. Our first result is used to study the behaviour of the functionals near the perforations which, in this instance, are balls of radius ɛ. We prove that an additional strange term arises involving C(λ).