Robin Boundary Conditions Research Articles

Error analysis of deep Ritz methods for elliptic equations

Using deep neural networks to solve partial differential equations (PDEs) has attracted a lot of attention recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on the deep Ritz method (DRM) for second-order elliptic equations with Dirichlet, Neumann and Robin boundary conditions, respectively. We establish the first nonasymptotic convergence rate in [Formula: see text] norm for DRM using deep neural networks with smooth activation functions including logistic and hyperbolic tangent functions. Our results show how to set the hyper-parameter of depth and width to achieve the desired convergence rate in terms of the number of training samples.

Spectral Convergence of the Laplace Operator with Robin Boundary Conditions on a Small Hole

In this paper, we study a bounded domain with a small hole removed. Our main result concerns the spectrum of the Laplace operator with the Robin conditions imposed at the hole boundary. Moreover, we prove that under some suitable assumptions on the parameter in the boundary condition, the spectrum of the Laplacian converges in the Hausdorff distance sense to the spectrum of the Laplacian defined on the unperturbed domain.

Steady-state solutions for a reaction–diffusion equation with Robin boundary conditions: Application to the control of dengue vectors

AbstractIn this paper, we investigate an initial-boundary value problem of a reaction–diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase plane analysis, the present paper studies the existence and properties of non-constant steady-state solutions depending on several parameters. Then, we prove some sufficient conditions for their stability. We show that the long-time efficiency of this control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.

Advanced numerical scheme and its convergence analysis for a class of two-point singular boundary value problems

In the past decades, many applications related to applied physics, physiology and astrophysics have been modelled using a class of two-point singular boundary value problems (SBVPs). In this article, a novel approach based on the shooting projection method and the Legendre wavelet operational matrix formulation for approximating a class of two-point SBVPs with Dirichlet and Neumann–Robin boundary conditions is proposed. For the new approach, an initial guess is postulated in contrast to the boundary conditions in the first step. The second step deals with the usage of the Legendre wavelet operational matrix method to solve the initial value problem (IVP). Further, the resulting solution of the IVP is utilized at the second endpoint of the domain of a differential equation in a shooting projection method to improve the initial condition. These two steps are repeated until the desired accuracy of the solution is achieved. To support the mathematical formulation, a detailed convergence analysis of the new approach is conducted. The new approach is tested against some existing methods such as various types of the variational iteration method, considering several numerical examples to which it provides high-quality solutions.

Non-Darcy Newtonian Liquid Flow with Internal Heat Generation using Boundary Conditions of the Third Kind of Fully Developed Mixed Convection in a Vertical Channel

An analysis of the impact of internal heat generation and other heat parameters on the temperature, velocity and rate of heat transfer of the liquid moving upward in a channel is studied. A fully developed non-Darcy flow using boundary conditions of the third kind with internal heat generation in a vertical channel is selected as the mathematical model. By utilising the similarity transformation, the governing equations are reduced to non-linear ordinary differential equations (ODEs). The converted ODEs are numerically solved and analysed using the fourth-order Runge-Kutta (RK4) method, incorporating a shooting technique with Newton’s method. The generated numerical algorithms are programmed in MATLAB for velocity, temperature and the local Nusselt number analysis. The numerical results of the flow and temperature variables are presented graphically. The impact of the parameters on the Nusselt number is also graphed to determine which of the three types of boundary conditions is the best for allowing heat transmission. The severity of flow reversal is increased under the Robin and Dirichlet conditions by enhancing the Darcy and Forchheimer numbers while decreasing the Brinkman and internal heat generation values. The temperature profiles improved with the increase in Brinkman numbers. Both Nusselt numbers remained constant for the Neumann boundary condition for all parameters except internal heat generation and local heat exponent. The Robin boundary condition is found to be the best facilitates heat transmission, since it delivers more pleasing and realistic results than the Dirichlet and Neumann conditions

An implicit immersed boundary method for Robin boundary condition

In this work, a novel boundary condition-enforced immersed boundary method (IBM) for simulating complex thermal–fluid–structure interaction (TFSI) problems with Robin boundary conditions is proposed. In this work, two auxiliary layers of Lagrangian points are introduced and placed within the inner and outer parts of the immersed object to enable the simultaneous evaluation of the temperature and its gradient on the solid surface. The mutual thermal interactions between the immersed object and the surrounding flow are taken into account through the temperature corrections on the Lagrangian points by accurately enforcing the Robin boundary condition on the solid boundary. Hence, a system of linear equations can be formulated to compute the temperature corrections. Subsequently, the temperature corrections are biasedly distributed to the Eulerian points located in the inner auxiliary layer to eliminate the diffusion generated by the smooth delta function on the temperature field outside the solid object. The proposed IBM integrated with the reconstructed thermal lattice Boltzmann flux solver (RTLBFS) is extensively and critically assessed with classical benchmark cases ranging from two-dimensional stationary problems to three-dimensional problems involving moving boundaries to demonstrate the robustness and accuracy of the proposed method for Robin boundary conditions. Results from the present work are extensively corroborated with previous works that adopt different approaches for imposing the Robin boundary condition. It shows that the proposed method not only provides a much more accurate representation of the Robin boundary condition but also introduces a simple implementation for the complex Robin boundary condition within the diffuse interface IBM framework.

Vacuum polarization induced by a cosmic string and a brane in AdS spacetime

In this paper we investigate the vacuum polarization effects associated to a charged quantum massive scalar field on a (D+1)-dimensional anti-de Sitter background induced by a magnetic-flux-carrying cosmic string in the braneworld model context. We consider the brane parallel to the anti-de Sitter boundary and the cosmic string orthogonal to them. Moreover, we assume that the field obeys the Robin boundary condition on the brane. Because the brane divides the space into two regions with different properties of the quantum vacuum, we calculate the vacuum expectation value (VEV) of the field squared and the energy–momentum tensor (EMT) in each region. To develop these analyses, we have constructed the positive frequency Wightman function for both regions. The latter is decomposed in a part associated with the anti-de Sitter bulk in the presence of a cosmic string only, and the other part induced by the brane. The vacuum polarization effects associated with the higher-dimensional anti-de Sitter bulk in the presence of cosmic string have been developed in the literature, and here we are mainly interested in the effects induced by the brane. We show that the VEVs of the field squared and the components of the EMT induced by the cosmic string are finite on the brane. Explicitly, we compare these observables with the corresponding ones induced by the brane only, and show that near the brane the contribution induced by the latter is larger than the one induced by the string; however, for points distant from the brane the situation is reversed. Moreover, some asymptotic expressions for the VEV of the field squared and EMT are provided for specific limiting cases of the physical parameters of the model. Also, an application of our results is given for a cosmic string in the Z_2-symmetric Randall–Sundrum braneworld model with a single brane.

Reverse Isoperimetric Inequality for the Lowest Robin Eigenvalue of a Triangle

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.

Solution of conservative-form transport equations with physics-informed neural network

Physics-informed neural network (PINN) has recently gained significant attention in the field of scientific computing. This paper proposes a PINN framework for solving conservative-form transport equations (PINNforCTE). The PINNforCTE features an accelerated back-propagation in the neural network (NN) training due to low-order derivatives in the governing equations. We demonstrate the application of PINNforCTE in simulating a classical flow and heat transfer problem, i.e., the lid-driven cavity flow and heat transfer under three types of boundary conditions, and the simulation results are compared with those obtained from the traditional PINN and the finite element CFD method. The comparisons indicate that PINNforCTE can significantly accelerate the NN training and potentially shorten the solution time. For all cases considered, the training time of PINNforCTE is reduced by approximately ∼50% compared with the conventional PINN. Furthermore, PINNforCTE can accelerate the Adam optimization algorithm (L-BFGS-B optimization algorithm) by ∼100% (∼80%) compared to the traditional PINN. Additionally, PINNforCTE shows greater convenience in applying the Neumann and Robin boundary conditions directly and higher calculation accuracy than the conventional PINN. Finally, the ability of PINNforCTE to solve the inverse problems is confirmed with an example of 2D steady lid-driven cavity flow and heat transfer. PINNforCTE is an efficient NN solver for conservative-form transport equations (CTEs) that are ubiquitous in science and engineering.

Boundary output feedback stabilization of state delayed reaction–diffusion PDEs

This paper addresses the problem of output feedback stabilization of a class of 1-D reaction–diffusion PDEs in the presence of a state delay in the reaction term. The control input applies through a Robin boundary condition. The system output is selected as either a Dirichlet or Neumann boundary trace. The developed control design procedure does not aim to compensate, but rather to dominate the state delayed term. This is achieved by introducing an observer that only estimates a finite number of modes of the PDE while applying a suitable pole shifting-based procedure on a truncated model that captures the unstable part of the plant. We show that the reported output feedback control strategy always achieves, for an arbitrarily given value of the state delay, the exponential stabilization of the plant in H1-norm. A notable feature of the domination-based approach is that the tuning of the controller and observer gains, as well as the subsequent stability conditions, appears to be independent of the value of the state delay.

Improved sharp spectral inequalities for Schrödinger operators on the semi-axis

We prove a Lieb–Thirring inequality for Schrödinger operators -\frac{\mathrm{d}^2}{\mathrm{d}x^2}+V on the semi-axis with Robin boundary condition at the origin. The result improves on a bound obtained by P. Exner, A. Laptev, and M. Usman [Commun. Math. Phys. 362 (2014), 531–541] albeit under the additional assumption V\in L^1(\mathbb{R}_+) . The main difference in our proof is that we use the double commutation method in place of the single commutation method. We also establish an improved inequality in the case of a Dirichlet boundary condition.

Parameter correlation study on two new analytical solutions for radially divergent tracer tests in two-zone confined aquifers with vertical dispersion effect

This study develops two new analytical models for radially divergent tracer tests at a fully-penetrating well in two-zone confined aquifers of skin and formation zones. Mixing length in the well is below the well length for inducing vertical dispersion. One model treats skins as an adsorptive Robin boundary condition for reflecting the adsorption effect. The other model applies skin's governing equation. The analytical solutions of the models are derived. The deep neural network is applied to obtain second model's solution. Results suggest the boundary condition is applicable when a Peclet number is below 0.2 for dominant dispersion in skins. The analytical solution considering the boundary condition predicts reliable parameters by using absorbed and non-absorbed tracers to avoid parameter correlation. The other solutions demonstrate skin's width correlated to its retardation factor and skin's transverse dispersivity to formation's one. The parameter correlation results from the model formulation rather than model-solving methods.

Analysis of a radial free boundary tumor model with time-dependent absorption efficiency

We investigate an n-dimensional free boundary problem modeling tumor growth in radial form with angiogenesis process. This process is reflected in the Robin boundary condition where time-dependent absorption rate β(t) is considered. The system consists of a reaction-diffusion equation for the nutrient u(r,t) with nonlinear consumption and an integro-differential equation for the tumor radius R(t) with nonlinear cell proliferation. It is shown that the large-time behavior of the tumor is greatly tied to the properties of β(t). We prove that for any sufficiently small c>0, the tumor radius R(t) will remain bounded or shrink to zero when β(t) is uniformly bounded or tends to zero, respectively; and if β(t)→β⁎, the solution (u,R) will converge to the unique steady state (u⁎,R⁎) associating with β⁎. We also give examples of R(t) blowing up as t→∞ even if β(t)→0 when c is not small.

A space–time variational method for optimal control problems: well-posedness, stability and numerical solution

We consider an optimal control problem constrained by a parabolic partial differential equation with Robin boundary conditions. We use a space–time variational formulation in Lebesgue–Bochner spaces yielding a boundedly invertible solution operator. The abstract formulation of the optimal control problem yields the Lagrange function and Karush–Kuhn–Tucker conditions in a natural manner. This results in space–time variational formulations of the adjoint and gradient equation in Lebesgue–Bochner spaces, which are proven to be boundedly invertible. Necessary and sufficient optimality conditions are formulated and the optimality system is shown to be boundedly invertible. Next, we introduce a conforming uniformly stable simultaneous space–time (tensorproduct) discretization of the optimality system in these Lebesgue–Bochner spaces. Using finite elements of appropriate orders in space and time for trial and test spaces, this setting is known to be equivalent to a Crank–Nicolson time-stepping scheme for parabolic problems. Comparisons with existing methods are detailed. We show numerical comparisons with time-stepping methods. The space–time method shows good stability properties and requires fewer degrees of freedom in time to reach the same accuracy.

On a nonlinear transmission eigenvalue problem with a Neumann–Robin boundary condition

Let be a bounded domain in with smooth boundary and let be a subdomain of with smooth boundary such that . Denote Consider the transmission eigenvalue problem where is a real parameter, and a.e. on Under additional suitable assumptions on , we prove the existence of a sequence of eigenvalues The proof is based on Lusternik–Schnirelmann theory on ‐manifolds.

Comment on “Breakdown of electroneutrality in nanopores”

Levy et al. [1] reported the breakdown of electroneutrality in confined nanopores embedded in a dielectric medium. A Robin boundary condition was derived which eliminates the need to include the dielectric medium explicitly when solving for the electric field within the nanopore. In this comment, we point out issues related to the approximations made during the derivation of the boundary condition. The errors caused by the use of this boundary condition can be significant even for nanochannels of large aspect (length to radius) ratio, a condition on which the approximations in Levy et al. [1] are based.

Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks

We consider time-dependent convection-diffusion problems with high Péclet number of order O(ε−1) in thin three-dimensional graph-like networks consisting of cylinders that are interconnected by small domains (nodes) with diameters of order O(ε). On the lateral surfaces of the thin cylinders and the boundaries of the nodes we account for solution-dependent inhomogeneous Robin boundary conditions which render the associated initial-boundary problem to be nonlinear. The strength of the inhomogeneity is controlled by an intensity factor of order O(εα), α∈R. The asymptotic behavior of the solution is studied as ε→0, i.e., when the diffusion coefficients are eliminated and the thin three-dimensional network is shrunk into a graph. Three qualitatively different cases are discovered in the asymptotic behavior of the solution depending on the value of the intensity parameter α: α=1, α>1, and α<1. We construct the asymptotic approximation of the solution, which provides us with the hyperbolic limit model for ε→0 for the first two cases, and prove the corresponding uniform pointwise estimates and energy estimates. As the main result, we derive uniform pointwise estimates for the difference between the solutions of the convection-diffusion problem and the zero-order approximation that includes the solution of the corresponding hyperbolic limit problem.

Temperature on rods with Robin boundary conditions

We consider solutions uf to the one-dimensional Robin problem with the heat source f∈L1[−π,π] and Robin parameter α>0. For given m, M, and s, 0≤m<s<M, we identify the heat sources f0, such that uf0 maximizes the temperature gap max[−π,π]⁡uf−min[−π,π]⁡uf over all heat sources f such that m≤f≤M and ‖f‖L1=2πs. In particular, this answers a question raised by J. J. Langford and P. McDonald in [5]. We also identify heat sources, which maximize/minimize uf at a given point x0∈[−π,π] over the same class of heat sources as above and discuss a few related questions.

An improved continuum surface tension model in SPH for simulating free-surface flows and heat transfer problems

The continuum surface force (CSF) model in conjunction with the smoothed particle hydrodynamics (SPH) method is commonly used in the simulation of multi-phase flows with surface tension effect. However, it cannot be directly applied for modeling free-surface flows since the normalization requirement, which states that the integral of the surface delta function through the interface should be equal to one, is violated due to the particle deficiency issue. In this work, we propose an improved CSF model based on the SPH method with the aim of recovering the normalization requirement. The particle deficiency issue at the free surface is tackled by introducing the jump in color function into the calculation of the surface delta function. As a result, the accuracy of the surface tension measurement is significantly improved without using the virtual particles. Besides, this improved CSF model also allows an accurate and efficient implementation of the Robin boundary condition which is frequently encountered in heat transfer problems. Several benchmark problems are investigated to validate the performance of the proposed method. The numerical results verify that the improved CSF model in SPH method is well capable of simulating free-surface flows and heat transfer problems.

Highly Accurate Method for Boundary Value Problems with Robin Boundary Conditions

The main aim of the current paper is to construct a numerical algorithm for the numerical solutions of second-order linear and nonlinear differential equations subject to Robin boundary conditions. A basis function in terms of the shifted Chebyshev polynomials of the first kind that satisfy the homogeneous Robin boundary conditions is constructed. It has established operational matrices for derivatives of the constructed polynomials. The obtained solutions are spectral and are consequences of the application of collocation method. This method converts the problem governed by their boundary conditions into systems of linear or nonlinear algebraic equations, which can be solved by any convenient numerical solver. The theoretical convergence and error estimates are discussed. Finally, we support the presented theoretical study by presenting seven examples to ensure the accuracy, efficiency, and applicability of the constructed algorithm. The obtained numerical results are compared with the exact solutions and results from other methods. The method produces highly accurate agreement between the approximate and exact solutions, which are displayed in tables and figures.