Boundary conditions as symmetry constraints

For n > 1, we consider a reaction-diffusion equationut = Δu + α(y)∇ · G(u) + f(u), (0.2)in an unbounded cylinder Ω := R×D, where D ⊂ Rn−1 is a smooth bounded domain, with a presence of a convection term, under both Neumann and Dirichlet boundary conditions on ∂Ω. For both types of boundary condition, we consider two different forms of convection term, namely : α(y)∇·G(u) and∇ · (α(y)G(u)). The reaction term f is “monostable”. In both Neumann and Dirichlet cases, we prove that there exists a critical speed c⋆ ∈ R such that there exists a travelling front solution of the form u(x, t) = w(x1 −ct, y) with speed c if and only if c ≥ c⋆, where x1 is the coordinate corresponding to theaxis of the cylinder. The critical speed c⋆ often plays an important role for monostable problems by characterizing the long-time behaviour of the initial value problem. The existence of travelling waves for all c ≥ c⋆ is typical of monostable problems such as the prototype Fisher-KPP equation.We give a min-max formula for the speed c⋆. For both types of boundary conditions, we prove that c⋆ is bounded below by a quantity c′ which is related to a certain eigenvalue problem, associated with the linearized problem around 0. Note that under Dirichlet boundary conditions, an extra assumption is needed to ensure that c′ exists, namely, f′(0) has to be greater than the principal eigenvalue of the linearized operator. We discuss two special cases where the equality c⋆ = c′ holds. Under both Neumann and Dirichlet boundary conditions, the first special case is when G = (G1, 0, · · ·, 0), assuming the so-called KPP condition for f and that α(y)G′ 1(u) ≥ α(y)G′ 1(0), for all y ∈ D and all u ∈ (0, 1). The second case is treated only under Neumann boundary conditions : when G′ 1(0) = 0, assuming the KPP condition for f, and that α(y)G′ 1(u) ≥ 0, for all y ∈ D and u ∈ (0, 1). Note that in that case, we give an explicit formula : c⋆ = c′ = 2 p f′(0). Under Dirichlet boundary conditions, we highlight the influence of the domain D, the reaction term f and the convection term α(y)∇ · G(u) on the critical speed c⋆. In the special case where G = (G1, 0, ···, 0), using that c⋆ = c′, we use the eigenvalue problem related to c′ to establish some optimization results for c⋆.

The nonlinear Schrödinger equation in the finite line

Sharp asymptotic behavior of radial solutions of some planar semilinear elliptic problems

We announce a new result which shows that under either Dirichlet, Neumann, or Robin boundary conditions, the corners in a planar domain are a spectral invariant of the Laplacian. For the case of polygonal domains, we show how a locality principle, in the spirit of Kac's "principle of not feeling the boundary" can be used together with calculations of explicit model heat kernels to prove the result. In the process, we prove this locality principle for all three boundary conditions. Albeit previously known for Dirichlet boundary conditions, this appears to be new for Robin and Neumann boundary conditions, in the generality presented here. For the case of curvilinear polygons, we describe how the same arguments using the locality principle fail, but can nonetheless be replaced by powerful microlocal analysis methods.

This work deals with the benchmarking of newly implemented thermal boundary conditions for the incompressible Boussinesq heat transfer solver in OpenFOAM. Internal fluid flows for building physics purposes can be accurately modelled with the incompressible Boussinesq approach. In order to accurately model thermal heat transfer phenomena using CFD more flexible Neumann and Robin thermal boundary condition are required which do not exist in the standard package of OpenFOAM for incompressible Boussinesq solvers. A new boundary condition is derived from other compressible solvers in OpenFOAM and adapted for use with the Boussinesq approximation. This new boundary condition is benchmarked against the commercial software ANSYS CFX by means of an assessment of the mean air temperature and the mean velocities as well as the mean radiant temperature. A simple L-shaped room case is used to compare results between OpenFOAM and ANSYS CFX. The mesh used is for both softwares identical. Six different scenarios are investigated by varying the adjacent wall temperatures (Robin boundary condition). For one scenario, the Neumann boundary condition is also applied. Furthermore, radiation coupling is integrated into the new boundary condition and included in the model. The radiation models view factors (viewFactor) and discrete ordinate method (fvDOM) in OpenFOAM are compared. The comparison the two softwares show very good agreement for air and mean radiant temperatures with a maximum of 0.25 K deviation. However, the CFX results show consistently a more pronounced fluid flow within the room. The results between fvDOM and viewFactor within OpenFOAM are quite similar to each other and furthermore match well with the results obtained in CFX.

This paper investigates the asymptotic stability problem for delayed genetic regulatory networks with reaction---diffusion terms under both Dirichlet boundary conditions and Neumann boundary conditions. First, by constructing a new Lyapunov---Krasovskii functional and using Jensen's inequality, Wirtinger's inequality, Green's second identity and the reciprocally convex approach, we establish delay-dependent asymptotic stability criteria that do not require a restriction of the upper bounds of the delays' derivatives being less than 1. Thus, the stability criteria that we establish are less conservative than the existing criteria and extend the range of applications of the theoretical results. In addition, it is shown that the obtained criterion under Dirichlet boundary conditions retains the information about the reaction---diffusion terms, while these do not exist in the criterion under Neumann boundary conditions. It is then theoretically presented that the stability criteria established in this paper are less conservative than the existing ones. Finally, numerical examples are given to illustrate the effectiveness of the theoretical results.

One-dimensional thermal wave transport in multilayered systems with an interface thermal resistance is studied under the framework of the Cattaneo–Vernotte hyperbolic heat conduction model, considering modulated heat excitation under Dirichlet and Neumann boundary conditions. For a single semi-infinite layer, analytical formulas useful in the measurement of its thermal relaxation time as well as additional thermal properties are presented. For a composite-layered system, in the thermally thin regime, with the Dirichlet boundary condition, the well known effective thermal resistance formula is obtained, while for the Neumann problem, only the heat capacity identity is found. In contrast, in the thermally thick case, an analytical expression for both Dirichlet and Neumann conditions is obtained for the effective thermal diffusivity of the whole system in terms of the thermal properties of the individual layers and their interface thermal resistance. The limits of applicability of this equation, in the thermally thick regime, are shown to provide useful and simple results in the characterization of layered systems and that they can be reduced to the results obtained using the Fourier approach. The role of the thermal relaxation time, the interface thermal resistance, and the implications of these results in the possibility of enhancement in heat transport are discussed.

Background Second-order linear boundary value problems arise in many applications of science and engineering and are commonly treated by reducing them to first-order systems, which increases the computational cost. Direct high-order methods that preserve the second-order structure are therefore of interest. Methods This study proposes fixed-step, high-order diagonally implicit Runge–Kutta–Nyström (DIRKN) methods for the direct numerical solution of special second-order linear boundary value problems. Two schemes, DIRKN(3,4) and DIRKN(4,5), are constructed and combined with a linear shooting technique to handle both Dirichlet and Neumann boundary conditions without transforming the problem into a first-order system. Results The performance of the proposed methods is evaluated using benchmark test problems and application models, including rod heat conduction and reaction–diffusion equations. Numerical results demonstrate that both schemes achieve high accuracy and good efficiency, with DIRKN(4,5) generally producing smaller errors for a comparable number of function evaluations. A linear stability analysis shows absolute stability intervals of (−1.96, 0) for DIRKN(3,4) and (−7.24, 0) for DIRKN(4,5). Conclusions The proposed DIRKN methods provide accurate and efficient solvers for second-order linear boundary value problems with different types of boundary conditions, while preserving the original problem structure and reducing computational cost.

The model problem of a plane angle for a second-order elliptic system subject to Dirichlet, mixed, and Neumann boundary conditions is analyzed. For each boundary condition, the existence of solutions of the form $r^λv$ is reduced to spectral analysis of a particular matrix. Focusing on Dirichlet and mixed boundary conditions, optimal bounds on $|\Re λ|$ are derived, employing tools from numerical range analysis and accretive operator theory. The developed framework is novel and recovers known bounds for Dirichlet boundary conditions. The results for mixed boundary conditions are new and represent the central contribution of this work. Immediate applications of these findings are new regularity results for linear second-order elliptic systems subject to mixed boundary conditions.

This study considers the dynamics of cellular neural network-based inhomogeneous lattice dynamical systems (CNN-based ILDS). The influence of three kinds of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, is elucidated. We reveal that the complete stability of CNN-based ILDS and, under some prescriptions, the topological entropies of CNN-based ILDS with/without the boundary condition are identical.MSC:37B10.

ABSTRACTIn this paper, we study the initial-boundary value problem of the generalized Burgers equation posed on a finite interval with non-homogeneous boundary conditions. The boundary conditions are given in a general form, which covers the usual Dirichlet, Neumann or Robin boundary conditions. For the generalized Burgers equation, we establish the local well-posedness for the weak solution in when the Sobolev index is negative. Besides, for the classical Burgers equation with Dirichlet boundary conditions, we obtain the global well-posedness.

Many disciplines in science and engineering, heat transfer plays a vital role and the problems associated with heat transfer are of great importance. Generally, two-dimensional steady state conduction is governed and transformed into a second order partial differential equation (PDE). Satisfying the differential equation along with four boundary conditions is essential for a solution to be valid. There are analytical solutions available, but only for simple boundary conditions and these are not suitable for complex boundary conditions. A technique for solving partial differential equations is Method of Lines (MOL), in which one dimension is discretized. In this study an analytical approach to a two-dimensional slab with steady state heat conduction under different types of boundary conditions is considered. The complete description of heat flow through slab using MOL is presented and the obtained PDEs are solved. Temperature distribution profiles in the slab by using MOL were plotted and compared with the profiles of analytical solutions by using MATLAB. The steady state analysis of temperature distribution in a slab with specified Dirichlet boundary conditions and Neumann boundary conditions was developed by using Method of Lines. From the profiles it is observed that as the number of lines increases, the error between analytical and MOL (semi analytical) solutions decreases, and the profiles of analytical and MOL converged which indicates the preference of higher number of lines in order to obtain accurate values.

We compute the vacuum polarization for a massless, conformally coupled scalar field on the covering space of global, four-dimensional, anti-de Sitter space-time. Since anti-de Sitter space is not globally hyperbolic, boundary conditions must be applied to the scalar field. We consider general Robin (mixed) boundary conditions for which the classical evolution of the field is well-defined and stable. The vacuum expectation value of the square of the field is not constant unless either Dirichlet or Neumann boundary conditions are applied. We also compute the thermal expectation value of the square of the field. For Dirichlet boundary conditions, both thermal and vacuum expectation values approach the same well-known limit on the space-time boundary. For all other Robin boundary conditions (including Neumann boundary conditions), the vacuum and thermal expectation values have the same limit on the space-time boundary, but this limit does not equal that in the Dirichlet case.

In this paper, we continue to study an initial boundary value problems to a model describing the evolution in time of diffusive phase interfaces in sea ice growth. In a previous paper, the global existence and the large‐time behavior of weak solutions in one space was studied under Dirichlet boundary conditions. Here, we show that the global existence of weak solutions and the large‐time behavior are also studied under Neumann boundary condition. In this paper, we study in space dimension lower than or equal to 3.

We solve the variable coefficient Cauchy---Navier equations of elasticity in the unit square, for Dirichlet and Dirichlet-Neumann boundary conditions, using second order finite difference schemes. The resulting linear systems are solved by the preconditioned conjugate gradient (PCG) method with preconditioners corresponding to to the Laplace operator. The multiplication of a vector by the matrices of the resulting systems and the solution of systems with the preconditioners are performed at optimal and nearly optimal costs, respectively. For the case of Dirichlet boundary conditions, we prove the second order accuracy of the scheme in the discrete $$H^1$$ H 1 norm, symmetry of the resulting matrix and its spectral equivalence to the preconditioner. For the case of Dirichlet---Neumann boundary conditions, we prove symmetry of the resulting matrix. Numerical tests demonstrating the convergence properties of the schemes and PCG are presented.