Neumann Type Research Articles

Numerical solution of source identification multi-point problem of parabolic partial differential equation with Neumann type boundary condition

We study a source identification boundary value problem for a parabolic partial differential equation with multi-point Neumann type boundary condition. Stability estimates for the solution of the overdetermined mixed BVP for multi-dimensional parabolic equation were established. The first and second order of accuracy difference schemes for the approximate solution of this problem were proposed. Stability estimates for both difference schemes were obtained. The result of numerical illustration in test example was given.

Thin inclusion at the junction of two elastic bodies: non-coercive case.

This article addresses an analysis of the non-coercive boundary value problem describing an equilibrium state of two contacting elastic bodies connected by a thin elastic inclusion. Nonlinear conditions of inequality type are imposed at the joint boundary of the bodies providing a mutual non-penetration. As for conditions at the external boundary, they are Neumann type and imply the non-coercivity of the problem. Assuming that external forces satisfy suitable conditions, a solution existence of the problem analysed is proved. Passages to limits are justified as the rigidity parameters of the inclusion and the elastic body tend to infinity.This article is part of the theme issue 'Non-smooth variational problems with applications in mechanics'.

Well-posedness of a bulk-surface convective Cahn–Hilliard system with dynamic boundary conditions

We consider a general class of bulk-surface convective Cahn–Hilliard systems with dynamic boundary conditions. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn–Hilliard type allow for dynamic changes of the contact angle between the diffuse interface and the boundary, a convection-induced motion of the contact line as well as absorption of material by the boundary. The coupling conditions for bulk and surface quantities involve parameters K,L∈[0,∞]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K,L\\in [0,\\infty ]$$\\end{document}, whose choice declares whether these conditions are of Dirichlet, Robin or Neumann type. We first prove the existence of a weak solution to our model in the case K,L∈(0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K,L\\in (0,\\infty )$$\\end{document} by means of a Faedo–Galerkin approach. For all other cases, the existence of a weak solution is then shown by means of the asymptotic limits, where K and L are sent to zero or to infinity, respectively. Eventually, we establish higher regularity for the phase-fields, and we prove the uniqueness of weak solutions given that the mobility functions are constant.

Boundary value problems associated with Hamiltonian systems coupled with positively-(p, q)-homogeneous systems

We study the multiplicity of solutions for a two-point boundary value problem of Neumann type associated with a Hamiltonian system which couples a system with periodic Hamiltonian in the space variable with a second one with positively-(p, q)-homogeneous Hamiltonian. The periodic problem is also treated.

Structural stability of p(x)-Laplacian kind problems with maximal monotone graphs and Neumann type boundary condition


 
 
 In this work, we study the convergence of a sequence of solutions of degener- ate elliptic problems with variable coercivity and growth exponents. The functional setting involves Lebesgue and Sobolev spaces with variable exponent which varies also with n.
 
 

Numerical Computation of Neumann Controls for the Heat Equation on a Finite Interval

This paper presents a new numerical method which approximates Neumann type null controls for the heat equation and is based on the Fokas method. This is a direct method for solving problems originating from the control theory, which allows the realisation of an efficient numerical algorithm that requires small computational effort for determining the null control with exponentially small error. Furthermore, the unified character of the Fokas method makes the extension of the numerical algorithm to a wide range of other linear PDEs and different type of boundary conditions straightforward.

Spectral norm and von Neumann type trace inequality for dual quaternion matrices

Spectral norm and von Neumann type trace inequality for dual quaternion matrices

Multiple solutions of nonlinear Neumann inclusions

We prove new results on the existence of multiple solutions for elliptic inclusions with nonlinear boundary conditions of Neumann type. Our approach is topological and relies on the fixed point index for multivalued map.

CFL Optimized Forward–Backward Runge–Kutta Schemes for the Shallow-Water Equations

Abstract We present the formulation and optimization of a Runge–Kutta-type time-stepping scheme for solving the shallow-water equations, aimed at substantially increasing the effective allowable time step over that of comparable methods. This scheme, called FB-RK(3,2), uses weighted forward–backward averaging of thickness data to advance the momentum equation. The weights for this averaging are chosen with an optimization process that employs a von Neumann–type analysis, ensuring that the weights maximize the admittable Courant number. Through a simplified local truncation error analysis and numerical experiments, we show that the method is at least second-order in time for any choice of weights and exhibits low dispersion and dissipation errors for well-resolved waves. Further, we show that an optimized FB-RK(3,2) can take time steps up to 2.8 times as large as a popular three-stage, third-order strong stability-preserving Runge–Kutta method in a quasi-linear test case. In fully nonlinear shallow-water test cases relevant to oceanic and atmospheric flows, FB-RK(3,2) outperforms SSPRK3 in admittable time step by factors roughly between 1.6 and 2.2, making the scheme approximately twice as computationally efficient with little to no effect on solution quality. Significance Statement The purpose of this work is to develop and optimize time-stepping schemes for models relevant to oceanic and atmospheric flows. Specifically, for the shallow-water equations we optimize for schemes that can take time steps as large as possible while retaining solution quality. We find that our optimized schemes can take time steps between 1.6 and 2.2 times larger than schemes that cost the same number of floating point operations, translating directly to a corresponding speedup. Our ultimate goal is to use these schemes in climate-scale simulations.

Existence and qualitative analysis of a fully cross-diffusive predator-prey system with nonlinear taxis sensitivity

This paper investigates a fully cross-diffusive predator-prey system with nonlinear taxis sensitivity(⋆){ut=D1uxx−χ1(uk1vx)x+μ1u(λ1−u+a1v),vt=D2vxx+χ2(vk2ux)x+μ2v(λ2−v−a2u), under homogeneous boundary conditions of Neumann type for u and v, in an open bounded interval Ω⊂R, where Di,χi,λi,ai>0 and μi≥0 for i∈{1,2}. In Tao and Winkler (2021) [1]; (2022) [2], Tao and Winkler studied the system (⋆) with k1=k2=1 and obtained the global existence and asymptotic behavior. We study the system (⋆) with k1,k2≠1 and prove that:•If k1∈(1,4031) and k2∈[14,12−9k18−5k1), then the model (⋆) possesses global weak solutions for arbitrarily large positive initial data in H1(Ω).•If k1∈(1,1110) and k2∈[23,12−9k18−5k1), then the global weak solution (u,v) of (⋆) with μ1=μ2=0 stabilizes toward homogeneous equilibria (1|Ω|∫Ωu0,1|Ω|∫Ωv0).•If k1∈(1,98) and k2∈[12,12−9k18−5k1), then the global weak solution (u,v) of (⋆) with μ1,μ2>0 converges to constant stable steady state (λ1,0) provided that λ2≤a2λ1 and the tactic sensitivities χ1 and χ2 are suitably small.

Mathematical Model of Three-Component Suspension Transport

Introduction. Prediction of suspension deposition zones is required to assess and minimize the negative impact on the ecosystem of the reservoir during dredging within the framework of large-scale engineering projects, prediction of suspension deposition zones is required to assess and minimize the negative impact on the ecosystem of the reservoir. It is necessary to build a mathematical model that takes into account many factors that have a significant impact on the accuracy of forecasts. The aim of the work is to construct a mathematical model of transport of multicomponent suspension, taking into account the composition of the soil (different diameter of the suspension particles), the flow velocity of the water flow, the complex geometry of the coastline and bottom, wind stresses and friction on the bottom, turbulent exchange, etc.Materials and Methods. A mathematical model for the transport of a multicomponent suspension and an approximation of the proposed continuous model with the second order of accuracy with respect to the steps of the spatial grid are described, considering the boundary conditions of the Neumann and Robin type. The approximation of the hydrodynamics model is obtained based on splitting schemes by physical processes, which guarantee fulfillment mass conservation for discrete model.Results. The proposed mathematical model formed the basis of the developed software package that allows to simulate the process of sedimentation of a multicomponent suspension. The results of the work of the software package on the model problem of sedimentation of a three-component suspension in the process of soil dumping during dredging are presented.Discussions and Conclusions. The mathematical model of transport of three-component suspension is described and software implemented. The developed software allows to simulate the process of deposition of suspended particles of various diameters on the bottom, and to evaluate its effect on the bottom topography and changes in the bottom composition. The developed software package also allows to analyze the process of sediment movement in the case of resuspension of multicomponent bottom sediments of the reservoir, which causes secondary pollution of the reservoi

Boundary estimates for solutions of the Monge–Ampère equation satisfying Dirichlet–Neumann type conditions in annular domains

We consider smooth solutions of the Monge–Ampère equation on an annular domain, whose boundary consists of two smooth, closed, strictly convex hypersurfaces, subject to mixed boundary conditions. In particular we impose a homogeneous Dirichlet condition on the outer boundary and a Neumann condition on the inner boundary. We demonstrate that in general, global C2 estimates cannot be obtained unless we impose extra restrictions on the principal curvatures of the inner boundary and on the Neumann condition itself. The latter is illustrated by the construction of an explicit counterexample. Under these conditions, we prove a priori C2 estimates and show that our problem admits a smooth solution.

On eigenfunctions and eigenvalues of some boundary value problems for a nonlocal biharmonic operator

In this note, the concept of a nonlocal biharmonic operator is introduced. When introducing this operator, mappings of the type of involution are used. Namely, in the differential expression of this operator, in addition to the variables 12( , ,..., )nxx xx, transformed arguments with mappings of the form 111,...,,,,...,,1jjjjjnS xxxpx xxj n and their multiplication also involved. Spectral problems with Dirichlet and Neumann-type boundary conditions are considered in an n-dimensional parallelepiped for a given nonlocal biharmonic operator. The eigenfunctions and eigenvalues of the problems under consideration are explicitly constructed. When constructing these elements, eigenfunctions and eigenvalues of the classical biharmonic operator with Dirichlet and Neumann type boundary conditions are essentially used. Theorems on the orthonomization and completeness of the systems of eigenfunctions of the problems under consideration are proved. Examples of the corresponding parameters for special cases involved in the problems under consideration are given. In addition, in the two-dimensional case for the corresponding nonlocal biharmonic operator, spectral issues of boundary value problems of the Samarsky-Ionkin type are also investigated. The proper and attached functions of the problem under consideration are found and theorems on the completeness ofthese systems are proved.

Determination of one unknown coefficient in a two-phase free boundary problem in an angular domain with variable thermal conductivity and specific heat

Two different two-phase free boundary Stefan problems in an angular domain with temperature-dependent thermal coefficients are considered. Analytical similarity solutions are obtained imposing a Dirichlet or Neumann type boundary condition, respectively, by solving functional problems. Moreover, formulas are obtained for the determination of one unknown thermal coefficient in the overspecified problem that consists in adding a Neumann condition to the problem with a Dirichlet one if and only if some restrictions on data are verified.

Information loss, mixing and emergent type III1 factors

A manifestation of the black hole information loss problem is that the two-point function of probe operators in a large Anti-de Sitter black hole decays in time, whereas, on the boundary CFT, it is expected to be an almost periodic function of time. We point out that the decay of the two-point function (clustering in time) holds important clues to the nature of observable algebras, states, and dynamics in quantum gravity.We call operators that cluster in time “mixing” and explore the necessary and sufficient conditions for mixing. The information loss problem is a special case of the statement that in type I algebras, there exists no mixing operators. We prove that, in a thermofield double state (KMS state), if mixing operators form an algebra (close under multiplication), the resulting algebra must be a von Neumann type III1 factor. In other words, the physically intuitive requirement that all nonconserved operators should exponentially mix is so strong that it fixes the observable algebra to be an exotic algebra called a type III1 factor. More generally, for an arbitrary out-of-equilibrium state of a general quantum system (von Neumann algebra), we show that if the set of operators that mix under modular flow forms an algebra, it is a type III1 von Neumann factor.In a theory of Generalized Free Fields (GFF), we show that if the two-point function clusters in time, all operators are mixing, and the algebra is a type III1 factor. For example, in \U0001d4a9 = 4 SYM, above the Hawking-Page phase transition, clustering of the single trace operators implies that the algebra is a type III1 factor, settling a recent conjecture of Leutheusser and Liu. We explicitly construct the C∗-algebra and von Neumann subalgebras of GFF associated with time bands and, more generally, open sets of the bulk spacetime using the HKLL reconstruction map.

Improvements of the Brownian walkers method towards the modeling of conduction-radiation coupling

This work directly follows the one presented in a previous article of V. Gonneau, D. Rochais and F. Enguehard (2022) [1]. This contribution describes new results and improvements to existing methods for the modeling of transient thermal conduction within a heterogeneous medium by the movement of Brownian walkers. The material structure is voxelized, and each walker transports an elementary enthalpy during its displacement within the structure. This enthalpy transport associated with the displacement of the walkers represents the conductive flow and makes it possible to simulate transient conduction with a quantitative stochastic approach.This article presents a new method for accounting for Dirichlet type boundary conditions that is quite efficient and that allows to relax quite substantially a constraint of maximum value of the time step of the Brownian walker simulations that had previously been pointed out in [1]. Other boundary condition treatments are also upgraded, in terms of speed (for Neumann type conditions) or generalization (for Robin type conditions). Then, a first non-linear conduction-radiation coupling model in an optically thick and gray semi-transparent medium is solved by Brownian walkers through two approaches: the first one based on a global conductivity function of temperature and allowing to confirm the validity of a stochastic transmission criterion at the interface of two constituents within a heterogeneous medium, and the second one based on the radiative volume power field. This second approach is preceded by the description and the validation of a procedure for the management of a volume power field using Brownian walkers. This procedure introduces the notion of negative Brownian walkers, which prove necessary for representing negative local values of the volume power field as is frequently the case with the radiative volume power.

Commutativity, Majorization, and Reduction in Fan–Theobald–von Neumann Systems

A Fan–Theobald–von Neumann system (Gowda in Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2 ) is a triple $$({{\mathcal {V}}},{{\mathcal {W}}},\lambda )$$ , where $${{\mathcal {V}}}$$ and $${{\mathcal {W}}}$$ are real inner product spaces and $$\lambda :{{\mathcal {V}}}\rightarrow {{\mathcal {W}}}$$ is a norm-preserving map satisfying a Fan–Theobald–von Neumann type inequality together with a condition for equality. Examples include Euclidean Jordan algebras, systems induced by certain hyperbolic polynomials, and normal decompositions systems (Eaton triples). In Gowda (Optimizing certain combinations of linear/distance functions over spectral sets, 2019. arXiv:1902.06640v2 ), we presented some basic properties of such systems and described results on optimization problems dealing with certain combinations of linear/distance and spectral functions. We also introduced the concept of commutativity via the equality in the Fan–Theobald–von Neumann-type inequality. In the present paper, we elaborate on the concept of commutativity and introduce/study automorphisms, majorization, and reduction in Fan–Theobald–von Neumann systems.

Homogenization for Nonlocal Evolution Problems with Three Different Smooth Kernels

In this paper we consider the homogenization of the evolution problem associated with a jump process that involves three different smooth kernels that govern the jumps to/from different parts of the domain. We assume that the spacial domain is divided into a sequence of two subdomains $$A_n \cup B_n$$ and we have three different smooth kernels, one that controls the jumps from $$A_n$$ to $$A_n$$ , a second one that controls the jumps from $$B_n$$ to $$B_n$$ and the third one that governs the interactions between $$A_n$$ and $$B_n$$ . Assuming that $$\chi _{A_n} (x) \rightarrow X(x)$$ weakly in $$L^\infty $$ (and then $$\chi _{B_n} (x) \rightarrow 1-X(x)$$ weakly in $$L^\infty $$ ) as $$n \rightarrow \infty $$ and that the initial condition is given by a density $$u_0$$ in $$L^2$$ we show that there is an homogenized limit system in which the three kernels and the limit function X appear. When the initial condition is a delta at one point, $$\delta _{{\bar{x}}}$$ (this corresponds to the process that starts at $${\bar{x}}$$ ) we show that there is convergence along subsequences such that $${\bar{x}} \in A_{n_j}$$ or $${\bar{x}} \in B_{n_j}$$ for every $$n_j$$ large enough. We also provide a probabilistic interpretation of this evolution equation in terms of a stochastic process that describes the movement of a particle that jumps in $$\Omega $$ according to the three different kernels and show that the underlying process converges in distribution to a limit process associated with the limit equation. We focus our analysis in Neumann type boundary conditions and briefly describe at the end how to deal with Dirichlet boundary conditions.

STABILITY OF FUNCTIONAL INEQUALITIES WITH 3K-VARIABLE BASED ON JORDAN-VON NEUMANN TYPE ADDITIVE FUNCTIONAL EQUATIONS IN BANACH SPACE

In this paper, we study to solve the Cauchy, Jensen and Cauchy-Jensen additive function inequalities with 3k-variables related to Jordan-von Neumann type in Banach space. These are the main results of this paper.

Tomographic imaging of perfectly conducting objects.

In this paper, a new algorithm for tomographic imaging of perfectly conducting scatterers, with boundary conditions of the Dirichlet or Neumann type, is proposed. The boundary value problem is converted into a volume integral equation with a singular double-layer potential. Then, the resulting far-field pattern is expressed in the form of an impact parameter model, i.e.,as a true Fourier transform of the profile function. No approximations are made in the construction of the forward model and derivation of the inversion algorithm. Instead, some elementary facts of the microlocal analysis, in particular the pseudo-locality of the corresponding operator, are used to recover the support of the scattering potential and, therefore, the shape of the obstacle. Mathematically, the problem is reduced to the Radon inversion of a classical computed tomography. It is shown that the algorithm is also capable of classifying the type (Dirichlet or Neumann) of the boundary condition imposed. A relation of the proposed algorithm to a previously known solution based on the physical optics approximation is discussed.