Robin Boundary Conditions Research Articles

A Spectral Element Solution of the Poisson Equation with Shifted Boundary Polynomial Corrections: Influence of the Surrogate to True Boundary Mapping and an Asymptotically Preserving Robin Formulation

Abstract We present a new high-order spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order hp-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series, which traditionally is used within the shifted boundary method. Here, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small cut cells. Dirichlet, Neumann, and Robin boundary conditions are enforced weakly through a generalized: (i) Nitsche’s method and (ii) Aubin’s method. A consistent asymptotic preserving formulation of the embedded Robin formulations is presented. Several experiments and analyses of the numerical properties of the various weak forms are showcased. We include convergence studies under polynomial increase of the basis functions, p, mesh refinement, h, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. With this, we assess the influence of errors across variational forms, polynomial order, mesh size, and mappings between the true and surrogate boundaries.

Positive solutions to quasilinear p(y)-Laplacian equation with Robin boundary condition

This article addresses the following Robin boundary value problem: { − div ( b ( y ) | ∇w | p ( y ) − 2 ∇w ) − F ( y ) | w | η ( y ) − 2 w = λG ( y ) | w | γ ( y ) − 2 w in Ω , b ( y ) | ∇w | p ( y ) − 2 ∂ w ∂ ν + H ( y ) | w | p ( y ) − 2 w = 0 on ∂Ω , here Ω ⊂ R N ( N ≥ 2 ) and p ( y ) , η ( y ) , γ ( y ) ∈ C ( Ω ¯ ) . We establish the existence and multiplicity results of the given problem in generalized Sobolev spaces W 1 , p ( y ) ( Ω ) . The main results of this article are derived using the Nehari manifold method under some appropriate assumptions on functions b, F, G and H.

An SPDE with Robin-type boundary for a system of elastically killed diffusions on the positive half-line

We consider a system of particles undergoing correlated diffusion with elastic boundary conditions on the half-line in the limit as the number of particles goes to infinity. We establish existence and uniqueness for the limiting empirical measure valued process for the surviving particles, which is a weak form for an SPDE with a noisy Robin boundary condition satisfied by the particle density. We show that this density process has good L2-regularity properties in the interior of the domain but may exhibit singularities on the boundary at a dense set of times. We make connections to the corresponding absorbing and reflecting SPDEs as the elastic parameter varies.

Reaction-diffusion for fish populations with realistic mobility

Nonlinear reaction-diffusion equations, with Fisher logistic growth and constant diffusion coefficient, have been used in fisheries research to estimate sustainable harvesting rates and critical domain sizes of no-take areas. However, constant diffusivity in a population density corresponds to standard Brownian motion of individuals, with a normal distribution for displacement over a fixed time interval. For available good data sets on mobile fish populations, the distribution is certainly not normal. The data can be fitted with a long-tailed stable Lévy distribution that results from diffusion by fractional Laplacian. Exact multidimensional solutions are developed here for realistic Fisher–Kolmogorov–Petrovski-Piscounov models with diffusion by fractional Laplacian. These can also account for a delay in the reaction term. It is then shown how to modify critical domain sizes of protected areas with Dirichlet and Robin boundary conditions for populations.

Analyzing Torsion Functions under Robin and Dirichlet Boundary Conditions

In this paper, the author introduces the mathematical concepts regarding torsion functions and their derivations under the Robin and Dirichlet boundary conditions which are integral in mechanical engineering, theoretical physics, and other disciplines. Functions that satisfy given or required partial differential equations contain a wealth of information about materials or structures when subjected to stress. We examine these functions under two types of boundary conditions: Robin, which contains Dirichlet and Neumann conditions, Dirichlet that provides conditions on the boundary. Also, in this section, we formally describe the mathematical model used in the study and provide some definitions, notations, and basic propositions that form the foundation for subsequent analysis. In answer to such concerns, the torsion function problems solved herein utilize analytical and numerical computations to reveal the differences and similarities in behavior between the two boundary conditions. Strong empirical evidence further supports our findings, proving the real-life relevance of the revealed theoretical concepts. Lastly, the implications for these findings in the real-world application are also presented and analysed with regards to advantages and disadvantages of various boundary conditions in the engineering of systems. We end this paper with a presentation of the general implications of the results attained from the work done and possible expansions and directions for future studies as a way of continuing the discourse on the theoretical analysis and actual applications. In summary, this paper provides a useful perspective in the analysis of torsion functions solution under mixed boundary conditions which will be of interest for further development of the theory and application into practice.

First‐order asymptotic perturbation theory for extensions of symmetric operators

AbstractThis work offers a new prospective on asymptotic perturbation theory for varying self‐adjoint extensions of symmetric operators. Employing symplectic formulation of self‐adjointness, we use a version of resolvent difference identity for two arbitrary self‐adjoint extensions that facilitates asymptotic analysis of resolvent operators via first‐order expansion for the family of Lagrangian planes associated with perturbed operators. Specifically, we derive a Riccati‐type differential equation and the first‐order asymptotic expansion for resolvents of self‐adjoint extensions determined by smooth one‐parameter families of Lagrangian planes. This asymptotic perturbation theory yields a symplectic version of the abstract Kato selection theorem and Hadamard–Rellich‐type variational formula for slopes of multiple eigenvalue curves bifurcating from an eigenvalue of the unperturbed operator. The latter, in turn, gives a general infinitesimal version of the celebrated formula equating the spectral flow of a path of self‐adjoint extensions and the Maslov index of the corresponding path of Lagrangian planes. Applications are given to quantum graphs, periodic Kronig–Penney model, elliptic second‐order partial differential operators with Robin boundary conditions, and physically relevant heat equations with thermal conductivity.

Qualitative analysis of optimisation problems with respect to non-constant Robin coefficients

Following recent interest in the qualitative analysis of some optimal control and shape optimisation problems, we provide in this article a detailed study of the optimisation of Robin boundary conditions in PDE constrained calculus of variations. Our main model consists of an elliptic PDE of the form −∆u β = f (x, u β) endowed with the Robin boundary conditions ∂ν u β +β(x)u β = 0. The optimisation variable is the function β, which is assumed to take values between 0 and 1 and to have a fixed integral. Two types of criteria are under consideration: the first one is non-energetic criteria. In other words, we aim at optimising functionals of the form J (β) =\int_{Ω or ∂Ω} j(u β). We prove that, depending on the monotonicity of the function j, the optimisers may be of bang-bang type (in other words, the optimisers write 1Γ for some measurable subset Γ of ∂Ω) or, on the contrary, that they may only take values strictly between 0 and 1. This has consequence for a related shape optimisation problem, in which one tries to find where on the boundary Neumann (∂ν u = 0) and constant Robin conditions (∂ν u + u = 0) should be placed in order to optimise criteria. The proofs for this first case rely on new fine oscillatory techniques, used in combination with optimality conditions. We then investigate the case of compliance-type functionals. For such energetic functionals, we give an in-depth analysis and even some explicit characterisation of optimal β * .

Inverse spectral Love problem via Weyl–Titchmarsh function

In this paper, we prove an inverse resonance theorem for the half-solid with vanishing stresses on the surface via Weyl–Titchmarsh function. Using a semi-classical approach, it is possible to simplify this three-dimensional problem of the elastic wave equation for the half-solid as a Schrödinger equation with Robin boundary conditions on the half-line. The goal of the paper is to establish a method to recover the potential from eigenvalues and resonances through the Weyl–Titchmarsh function for non-self-adjoint problems and to establish a one-to-one and onto map between suitable function spaces. Moreover, we produce an algorithm in order to retrieve the shear modulus from the eigenvalues and resonances, via the spectral data.

Surface wave damping by a Robin boundary condition at a permeable seabed

We investigate the damping of inviscid surface gravity waves when they propagate over a permeable seabed. Traditionally, this problem is solved by considering the wave motion in the upper water layer interacting with the Darcy flow in the porous bottom layer through matching of the solutions at the permeable interface. The novel approach in this study is that we describe the interaction between the upper fluid layer and the permeable bottom layer by the application of a Robin boundary condition. By varying the magnitude of the Robin parameter R, we can model the bottom structure of the fluid layer from rigid to completely permeable. In the case when the bottom is a porous medium where Darcy’s law applies, or composed by densely packed vertical Hele-Shaw cells, R is small, and can by determined by comparison with analytical results for a two-layer structure. In this case the wave damping is small. For larger values of R, the damping increases, and for very large R, the wave is almost critically damped. For the nonlinear transport in spatially damped shallow-water waves, increasing bottom permeability (larger R), reduces the magnitude of the horizontal Stokes drift velocity, while the drift profile tends to become parabolic with height. The vertical Stokes drift in the limit of large permeability is linear with height, with a magnitude that is larger than the horizontal drift. It is suggested that the implementation of a permeable bottom bed in some cases could prevent shorelines from damaging erosion by incoming surface waves.

Absolutely stable difference scheme for the delay partial differential equation with involution and Robin boundary condition

This paper examines the initial value problem for a third-order delay partial differential equation with involution and Robin boundary condition. We construct a first-order accurate difference scheme to obtain the numerical solution for this equation. Illustrative numerical results are provided.

An efficient numerical method for the distributed‐order time‐fractional diffusion equation with the error analysis and stability properties

In this manuscript, we study and examine the time‐fractional modified anomalous sub‐diffusion model of distributed‐order. Two numerical approaches are used to study the approximate solutions of the presented model. For the first approach, we use a second‐order difference method based on the L1 formula for the temporal variable. In this case, stability and convergence analysis for time discretization using the L1 formula is displayed. For the second approach, we display and demonstrate the numerical method for the full‐discrete based on the Galerkin weak form based on various kernels and shape functions of reproducing kernel particle method which do not have the ‐Kronecker property. Moreover, in this manuscript, in order to be able to use the necessary boundary conditions, the two straight strategies are used: one is the Lagrange multiplier method, and the other one is the penalty method. By using penalty method, we can the main boundary value model is changed to a new BVP with Robin boundary condition. At the end of the article, some numerical examples are presented and analyzed for the efficiency of the proposed numerical method. Also, the presented numerical method is compared with other numerical approaches, and the results of this comparison are reported in the form of a table.

Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain

AbstractWe consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric.

The Robin Problems in the Coupled System of Wave Equations on a Half-Line

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration.

Research on the Performance of Thermoelectric Self-Powered Systems for Wireless Sensor Based on Industrial Waste Heat.

The issue of energy supply for wireless sensors is becoming increasingly severe with the advancement of the Fourth Industrial Revolution. Thus, this paper proposed a thermoelectric self-powered wireless sensor that can harvest industrial waste heat for self-powered operations. The results show that this self-powered wireless sensor can operate stably under the data transmission cycle of 39.38 s when the heat source temperature is 70 °C. Only 19.57% of electricity generated by a thermoelectric power generation system (TPGS) is available for use. Before this, the power consumption of this wireless sensor had been accurately measured, which is 326 mW in 0.08 s active mode and 5.45 μW in dormant mode. Then, the verified simulation model was established and used to investigate the generation performance of the TPGS under the Dirichlet, Neumann, and Robin boundary conditions. The minimum demand for a heat source is cleared for various data transmission cycles of wireless sensors. Low-temperature industrial waste heat is enough to drive the wireless sensor with a data transmission cycle of 30 s. Subsequently, the economic benefit of the thermoelectric self-powered system was also analyzed. The cost of one thermoelectric self-powered system is EUR 9.1, only 42% of the high-performance battery cost. Finally, the SEPIC converter model was established to conduct MPPT optimization for the TEG module and the output power can increase by up to approximately 47%. This thermoelectric self-powered wireless sensor can accelerate the process of achieving energy independence for wireless sensors and promote the Fourth Industrial Revolution.

Effect of Circadian Rhythm Modulated Blood Flow on Nanoparticle based Targeted Drug Delivery in Virtual In Vivo Arterial Geometries.

A novel integration of nanoparticle-based drug delivery framework with shear-rate dependent blood flow model is presented. The framework is comprised of a unique combination of mechanics-based dispersion model, an asperity model for endothelium surface roughness, and a stochastic nanoparticle-endothelial cell adhesion model. Simulations of MRI based in vivo carotid artery system showcase the effects of vessel geometry on nanoparticle adhesion and retention at the targeted site. Vessel geometry and target site location impact nanoparticle adhesion; curved and bifurcating regions favor local accumulation of drug. It is also shown that aligning drug administration with circadian rhythm and sleep cycle can enhance the efficacy of drug delivery processes. These simulations highlight the potential of the computational modeling for exploring circadian rhythm modulated blood flow for targeted drug delivery while minimizing the in vivo experimentation.

An exponential spectral deferred correction method for multidimensional parabolic problems

We present some efficient algorithms based on an exponential time differencing spectral deferred correction (ETDSDC) method for multidimensional second and fourth-order parabolic problems with non-periodic boundary conditions including Dirichlet, Neumann, Robin boundary conditions. Similar to the Fourier spectral method for periodic problems, the key to the efficiency of our algorithms is to construct diagonal discrete linear operators via Legendre–Galerkin methods with Fourier-like basis functions. In combination with the ETDSDC scheme, the proposed methods are spectrally accurate in space and up to 10th-order accurate in time (as shown in this work). We demonstrate the high-order of convergence and efficiency of our algorithms in solving parabolic equations through a series of two-dimensional and three-dimensional examples including Ginzburg–Landau and Allen–Cahn equations.

Classification of solutions to some elliptic equations with Robin boundary condition and finite Morse index

We deal here with the non-existence of weak finite Morse index solutions to the following nonlinear elliptic equation −Δu=(1+|x|α)|u|p−1uinRN, or in half space with Robin boundary condition, where N≥2, α>−2 and p>1.

Internal/Boundary Control-Based Fixed-Time Synchronization for Spatiotemporal Networks.

This article is concerned about fixed-time (FT) synchronization of spatiotemporal networks (STNs) with the Robin boundary condition. Above all, a switching-type FT stability theorem and an integral inequality are established, which provide a novel theoretical tool for the rigorous analysis of FT control in STNs. Subsequently, three kinds of nontrivial power-law controllers are developed which are separately acted on the interior, the boundary, and the whole of the spatial domain. Based on these control schemes and Lyapunov-like method, several flexible criteria are obtained to achieve FT synchronization of STNs, and the upper bound of the synchronization time is explicitly estimated. Note that, the derived results here are also perfectly applicable to STNs with Neumann or Dirichlet boundary condition. Several illustrate examples are presented at final to confirm the developed controllers and criteria.

Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

Riesz transform and Hardy spaces related to elliptic operators having Robin boundary conditions on Lipschitz domains with their applications to optimal endpoint regularity estimates

The First Regularized Trace of the Sturm-Liouville Operator With Robin Boundary Conditions

The First Regularized Trace of the Sturm-Liouville Operator With Robin Boundary Conditions