Neumann Boundary Conditions Research Articles

Topological wave equation eigenmodes in continuous 2D periodic geometries

Abstract In this paper, we address the topological characterization of the
wave equation solutions in continuous two-dimensional (2D) periodic geometries with
Neumann or Dirichlet boundary conditions. This characterization is
relevant in the context of 2D vibrating membranes and our approach
allows one to understand the topological behavior recently observed
in acoustic three-dimensional (3D) artificial lattices. In particular, the dependence
of the topological behavior on the experimental positioning of the
coupling channels is explained using simple arguments and a simple
method of construction of an equivalent effective tight-binding Hamiltonian
is presented.

Flux and symmetry effects on quantum tunneling

AbstractMotivated by the analysis of the tunneling effect for the magnetic Laplacian, we introduce an abstract framework for the spectral reduction of a self-adjoint operator to a hermitian matrix. We illustrate this framework by three applications, firstly the electro-magnetic Laplacian with constant magnetic field and three equidistant potential wells, secondly a pure constant magnetic field and Neumann boundary condition in a smoothed triangle, and thirdly a magnetic step where the discontinuity line is a smoothed triangle. Flux effects are visible in the three aforementioned settings through the occurrence of eigenvalue crossings. Moreover, in the electro-magnetic Laplacian setting with double well radial potential, we rule out an artificial condition on the distance of the wells and extend the range of validity for the tunneling approximation recently established in Fefferman et al. (SIAM J Math Anal 54: 1105–1130, 2022), Helffer & Kachmar (Pure Appl Anal, 2024), thereby settling the problem of electro-magnetic tunneling under constant magnetic field and a sum of translated radial electric potentials.

The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane

The initial-boundary value problem for the Schrödinger equation with the nonlinear Neumann boundary condition on the half-plane

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.

Optimal distributed control for a Cahn-Hilliard-Darcy system with mass sources, unmatched viscosities and singular potential

We study a Cahn-Hilliard-Darcy system with mass sources, which can be considered as a basic, though simplified, diffuse interface model for the evolution of tumor growth. This system is equipped with an impermeability condition for the averaged velocity u as well as homogeneous Neumann boundary conditions for the phase function ϕ and the chemical potential µ. The source term in the convective Cahn-Hilliard equation contains a control R that can be thought as a drug or a nutrient in applications. Our goal is to study a distributed optimal control problem in the two-dimensional setting with a cost functional of tracking-type. In the physically relevant case with unmatched viscosities for the binary fluid mixtures and a singular potential, we first prove the existence and uniqueness of a global strong solution with ϕ being strictly separated from the pure phases ±1. This well-posedness result enables us to characterize the control-to-state mapping S: R→ϕ. Then we obtain the existence of an optimal control, the Frechet differentiability of S and first-order necessary optimality conditions expressed through a suitable variational inequality for the adjoint variables. Finally, we prove the differentiability of the control-to-costate operator and establish a second-order sufficient condition for the strict local optimality.

A modified cell-centered nodal integral scheme for the convection-diffusion equation

The nodal integral methods (NIMs) are very efficient and accurate coarse-mesh methods for solving partial differential equations. The cell-centered NIM (CCNIM) is a simplified variant of the NIMs that has recently shown its efficiency in solving fluid flow problems but has been hampered by issues such as inapplicability to one-dimensional problems, complexities in handling Neumann boundary conditions and the formulation of a system of differential-algebraic equations (DAEs) for discrete unknowns. Here, we present a modified version of the CCNIM designed to overcome the challenges associated with its previous version. Our novel development retains the essence of CCNIM while resolving these issues. The proposed scheme, grounded in the nodal framework, achieves second-order accuracy in both spatial and temporal dimensions. Unlike its precursor, the proposed method formulates algebraic equations for discrete variables per node, eliminating the cumbersome DAE system. Neumann boundary conditions are seamlessly incorporated through a straightforward flux definition, and applicability to one-dimensional problems is now feasible. We successfully apply our approach to one and two-dimensional convection-diffusion problems with known analytical solutions to validate our approach. The simplicity and robustness of the approach lay the foundation for its seamless extension to more complex fluid flow problems.

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.

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.

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.

Asymptotic Behavior of Global Solution of an Anomalous Gray Scott Model

The object of this paper is to prove that existence global and asymptotic behavior of solutions for anomalous coupled reaction diffusion system (Gray Scott model) with homogeneous Neumann boundary conditions. The existence and uniqueness of the local solution are given by the Banach fixed point theorem. However, the asymptotic behavior is investiged by technique semi group estimates and the Sobolev embedding theorem      

Convergence of solutions of a one-phase Stefan problem with Neumann boundary data to a self-similar profile

We study a one-dimensional one-phase Stefan problem with a Neumann boundary condition on the fixed part of the boundary. We construct the unique self-similar solution, and show that starting from arbitrary initial data, solution orbits converge to the self-similar solution.

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.

Solutions to a coupled Cahn–Hilliard-phase-field-crystal model for grain boundary motion

In this paper, we consider an initial–boundary value problem (IBVP) of a coupled Cahn–Hilliard-phase-field-crystal (CH-PFC) system subject to homogeneous Neumann boundary conditions in one spatial dimension. This CH-PFC model couples the composition field of a diffusing species with the crystallographic and can be used to model the diffusion-induced grain boundary migration in crystalline materials. Under suitable assumptions on the coefficients and initial data, we prove that the IBVP possesses a global weak solution. Our existence proof, which contributes to the verification of the model, is only valid in one space dimension.

Exponential convergence to steady-states for trajectories of a damped dynamical system modeling adhesive strings

We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modeling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.

Stability analysis and numerical simulations of the infection spread of epidemics as a reaction–diffusion model

This paper presents a spatiotemporal reaction–diffusion model for epidemics to predict how the infection spreads in a given space. The model is based on a system of partial differential equations with the Neumann boundary conditions. First, we study the existence and uniqueness of the solution of the model using the semigroup theory and demonstrate the boundedness of solutions. Further, the proposed model's basic reproduction number is calculated using the eigenvalue problem. Moreover, the dynamic behavior of the disease‐free steady states of the model for is investigated. The uniform persistence of the model is also discussed. In addition, the global asymptotic stability of the endemic steady state is examined. Finally, the numerical simulations validate the theoretical results.

Positional embeddings for solving PDEs with evolutional deep neural networks

This work extends the paradigm of evolutional deep neural networks (EDNNs) to solving parametric time-dependent partial differential equations (PDEs) on domains with geometric structure. By introducing positional embeddings based on eigenfunctions of the Laplace-Beltrami operator, geometric properties are encoded intrinsically and Dirichlet, Neumann and periodic boundary conditions of the PDE solution are enforced directly through the neural network architecture. The proposed embeddings lead to improved error convergence for static PDEs and extend EDNNs towards computational domains of realistic complexity.Several steps are taken to improve performance of EDNNs: Solving the EDNN update equation with a Krylov solver avoids the explicit assembly of Jacobians and enables scaling to larger neural networks. Computational efficiency is further improved by an ad-hoc active sampling scheme that uses the PDE dynamics to effectively sample collocation points. A modified linearly implicit Rosenbrock method is proposed to alleviate the time step requirements of stiff PDEs. Lastly, a completely training-free approach, which automatically enforces initial conditions and only requires time integration, is compared against EDNNs that are trained on the initial conditions. We report results for the Korteweg-de Vries equation, a nonlinear heat equation and (nonlinear) advection-diffusion problems on domains with and without holes and various boundary conditions, to demonstrate the effectiveness of the method. The numerical results highlight EDNNs as a promising surrogate model for parametrized PDEs with slow decaying Kolmogorov n-width.

The Conservative and Efficient Numerical Method of 2-D and 3-D Fractional Nonlinear Schrödinger Equation Using Fast Cosine Transform

This paper introduces a novel approach employing the fast cosine transform to tackle the 2-D and 3-D fractional nonlinear Schrödinger equation (fNLSE). The fractional Laplace operator under homogeneous Neumann boundary conditions is first defined through spectral decomposition. The difference matrix Laplace operator is developed by the second-order central finite difference method. Then, we diagonalize the difference matrix based on the properties of Kronecker products. The time discretization employs the Crank–Nicolson method. The conservation of mass and energy is proved for the fully discrete scheme. The advantage of this method is the implementation of the Fast Discrete Cosine Transform (FDCT), which significantly improves computational efficiency. Finally, the accuracy and effectiveness of the method are verified through two-dimensional and three-dimensional numerical experiments, solitons in different dimensions are simulated, and the influence of fractional order on soliton evolution is obtained; that is, the smaller the alpha, the lower the soliton evolution.

Existence and stability of steady-state bifurcations in a prey–predator system with a nonmonotonic functional response

A cross-diffusion prey–predator system exhibiting the prey group defense under homogeneous Neumann boundary conditions is studied. By considering the diffusive rate of the prey as a bifurcation parameter, we investigate sudden changes in the population dynamics of the prey and predator which can have a substantial effect on population size of the species. First a priori estimate for positive steady states is obtained. Next we prove the existence of a pitchfork bifurcation of positive steady states at a simple eigenvalue. The structure of the global steady-state bifurcation is discussed. We also investigate the stability of the trivial solution line and nontrivial steady-state solutions via the eigenvalue perturbation theory. To illustrate our theoretical results some numerical simulations are given. Numerical examples contain a supercritical and a subcritical pitchfork bifurcation.

Well-posedness and Tikhonov regularization of an inverse source problem for a parabolic equation with an integral constraint

Abstract We investigate a time-dependent inverse source problem for a parabolic partial differential equation with an integral constraint and subject to Neumann boundary conditions in a domain of R d \mathbb{R}^{d} , d ≥ 1 d\geq 1 . We prove the well-posedness as well as higher regularity of solutions in Hölder spaces. We then develop and implement an algorithm that we use to approximate solutions of the inverse problem by means of a finite element discretization in space. Due to instability in inverse problems, we apply Tikhonov regularization combined with the discrepancy principle for selecting the regularization parameter in order to obtain a stable reconstruction. Our numerical results show that the proposed scheme is an accurate technique for approximating solutions of this inverse problem.

Space-only gradient estimates of Schrödinger equation with Neumann boundary condition under integral Ricci curvature bounds

Assume that M (M⊆N) is an n-dimensional compact Riemannian submanifold with boundary, satisfying the integral Ricci curvature assumption:D2supx∈N⁡(∮B(x,d)|Ric−|pdy)1p<κ for κ=κ(n,p) small enough, p>n/2, where diam(M)≤D. The boundary of M needs to satisfy the interior rolling R-ball condition. We prove a Hamilton type gradient estimate for positive solutions to the parabolic Schrödinger equation with Neumann boundary conditions, on a compact Riemannian submanifold M with boundary ∂M, satisfying the integral Ricci curvature assumption.