Neumann-type Boundary Conditions Research Articles

A numerical approach to overcome the very-low Reynolds number limitation of the artificial compressibility for incompressible flows

We propose a numerical approach to solve a long-standing challenge which is the applicability of the artificial compressibility (AC) formulation for solving the incompressible Navier—Stokes equations at very-low Reynolds numbers. A wide range of engineering applications involves very-low Reynolds number flows in Micro-ElectroMechanical Systems (MEMS) and in the fields of chemical-, agricultural- and biomedical engineering. It is known that the already existing numerical methods using the AC approach fail to provide physically correct results at very-low Reynolds numbers (Re ≤ 1). To overcome the limitation of the AC method for these engineering applications, we propose a higher-order Neumann-type pressure outflow boundary condition treatment along with their up to fourth-order numerical approximations. We found that the numerical treatment of the pressure at the outlet boundary plays the main role in overcoming the limitation of the AC method at very-low Reynolds numbers (Re << 1). Therefore, we provide numerical evidence on the accuracy of the AC method beyond its previously reported limitations, e.g., the low Reynolds number Oseen flow (Re << 1) is first presented in this work. A third-order explicit total-variation diminishing (TVD) Runge–Kutta scheme has been employed with standard finite difference spatial discretisation schemes for improving the accuracy of the numerical solution. For modelling strongly viscous flows, the Reynolds number ranges from 10−1 to 10−4. Overall, we found that the accuracy limitation of the AC method below Re < 1 can be overcome with an accurate numerical treatment of the outlet pressure boundary condition instead of using high-order schemes in the governing equations. For the investigated Reynolds number range (10−1 ≤ Re ≤ 10−4), the obtained results show that the relative errors were smaller than 1% for the numerical simulations performed on the configurations of both the two- and three-dimensional, straight microfluidic channels. The imposition of high-order derivative Neumann-type pressure outflow boundary conditions reduced the maximum relative errors of the numerical solutions from 85% and 95% to below than 1% at the outlet section of the two- and three-dimensional, straight microfluidic channel flows, respectively. Taking the advantage of the numerical approach proposed here, two- and three-dimensional benchmark problems employed in the current investigation in comparison with analytical solutions available in the literature, clearly demonstrate that the artificial compressibility can be used beyond its previously known constraints for very-low Reynolds number incompressible flows.

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.

Neumann-type boundary value problem associated with Hamiltonian systems

The aim of this paper is to investigate some multiplicity results for a Hamiltonian system with Neumann-type boundary conditions. A critical point theory is applied in order to show that the the problem has multiple solutions. The crucial part of this paper is that, in contrast to periodic problems where the Poincaré-Birkhoff Theorem has a significant role, no twist condition is required.

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

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Multiplicity results for Hamiltonian systems with Neumann-type boundary conditions

We prove some multiplicity results for Neumann-type boundary value problems associated with a Hamiltonian system. Such a system can be seen as the weak coupling of two systems, the first of which has some periodicity properties in the Hamiltonian function, the second one presenting the existence of a well-ordered pair of lower/upper solutions.

Behavior of solutions to semilinear evolution inequalities in an annulus: The critical cases

In the present paper, we consider the parabolic and hyperbolic inequalities with singular potentials and with critical nonlinearities in the annulus domain. The problems are studied with Neumann-type and Dirichlet-type boundary conditions on the boundary. Moreover, we study the systems of problems too. We have proved that the above problems are globally unsolvable in critical cases, thereby filling the gaps the recent results by Jleli and Samet in [J. Math. Anal. Appl. 514: 2 (2022)] and in [Anal. Math. Phys. 12: 90 (2022)]. Proofs are carried out using the method of test functions with logarithmic arguments, which is being developed for the first time in bounded domains.

Exact Null Controllability of String Equations with Neumann Boundaries

This article focuses on the exact null controllability of a one-dimensional wave equation in noncylindrical domains. Both the fixed endpoint and the moving endpoint are Neumann-type boundary conditions. The control is put on the moving endpoint. When the speed of the moving endpoint is less than the characteristic speed, we can obtain the exact null controllability of this equation by using the Hilbert uniqueness method. In addition, we get a sharper estimate on controllability time that depends on the speed of the moving endpoint.

Hybrid-triggered-based fault-tolerant dissipative control for T-S fuzzy parabolic PDE systems with external disturbances and deception attacks

This paper addresses the fault-tolerant hybrid-triggered control problem for a class of nonlinear parabolic PDE systems with Neumann-type boundary conditions subject to actuator faults, external disturbances and randomly occurring deception attacks. Precisely, in order to provide an accurate representation of the considered nonlinear parabolic PDE system, a Takagi–Sugeno fuzzy PDE model is utilized. Further, in an effort to reduce the amount of communication transmissions, a more general hybrid-triggering method is implemented which includes both time- and event-triggered scheme. For the descriptions of the hybrid-triggered strategy and deception attacks, stochastic variables that follows Bernoulli distribution are presented. At the same time, an actuator fault model is considered in order to manage the impact of faults that occur in the actuators of the system under consideration. By the construction of suitable Lyapunov-Krasovskii functional, a set of adequate requirements is procured that ensure the closed-loop system is asymptotically stable and dissipative. In addition, the necessary controller gain matrices are obtained by making use of linear matrix inequalities. In the end, two numerical examples are exhibited to articulate the efficacy of the proposed control design technique.

Nonlinear stochastic wave equations in 1D with fractional Laplacian, power-law nonlinearity and additive [formula omitted]-regular noise

A qualitative study of nonlinear, 1D stochastic fractional wave equations with dissipative nonlinearities of power-law form is conducted on (t,x)∈[0,+∞)×Dutt+σ2(−uxx)α−a1u+a2‖u‖L2(D)ρu−κut=b0∂W0∂t on (t,x)∈[0,+∞)×D with D=[0,L], where positive fractional α-powers of Laplace operator are allowed, perturbed by additive space–time random noise W0 with fairly general covariance operator Q with finite trace(Q)<+∞ (i.e. Q-regularity). The Q-regular space–time noise W0, which is white in time and generally spatially correlated, is supposed to have a Fourier expansion along the eigenfunctions of the Laplace operator under nonrandom, Dirichlet- and Neumann-type homogeneous boundary conditions. We focus on studying the generalized energy functional V incorporating the fractional diffusive energy part and the energy part which is due to the viscous damping force both in continuous and discrete time. In continuous time, the technique of Fourier expansions and appropriate truncation by finite-dimensional systems leads to the control on the generalized energy functional, and hence we may verify existence, uniqueness and boundedness of moments of Fourier series solutions u. The total mean energy E[E(t)] cannot grow more than linearly in time t (with or without damping). Moreover, under the absence of damping (i.e. κ=0), E[E(t)] is governed by a kind of trace formula in case of additive (state-independent), Q-regular space–time noise W0. For numerical computations and more adequate discretization, we suggest to take nonstandard, partial-implicit midpoint-type methods for the Fourier coefficients. These semi-analytical numerical methods (approximate Fourier series) possess the property of conserving the expected, total energy with random initial data under the absence of nonlinearities. Eventually, we estimate probabilities of large fluctuations of some interesting functionals.

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.

Exact Null Controllability of a Wave Equation with Dirichlet–Neumann Boundary in a Non-Cylindrical Domain

In this paper, by applying the Hilbert Uniqueness Method in a non-cylindrical domain, we prove the exact null controllability of one wave equation with a moving boundary. The moving endpoint of this wave equation has a Neumann-type boundary condition, while the fixed endpoint has a Dirichlet boundary condition. We derived the exact null controllability and obtained an exact controllability time of the wave equation.

Existence and uniqueness of solutions to fractional differential equations with fractional boundary conditions

Boundary conditions involving fractional derivatives of unknown functions are more general and can be used to generalize Dirichlet- or Neumann-type boundary conditions. In this article, we consider a class of nonlinear fractional differential equations having Atangana-Baleanu derivatives equipped with fractional boundary conditions. We employ some standard fixed-point theorems to establish the main results — Leray–Schauder alternative ensures the existence of solutions, whereas Banach contraction principle guarantees uniqueness. Furthermore, we present an implicit numerical scheme based on the trapezoidal method for obtaining a precise numerical estimation of the solution. Some examples are given to illustrate our analytical results and numerical findings. The main distinctive features of this work are as follows. (i) Some important properties of higher-order Atangana-Baleanu operators are introduced. (ii) Fractional boundary conditions are considered for the first time. (iii) A numerical approximation for the solution is expounded.

On the Spectral Properties of a Fourth-Order Self-Adjoint Operator

We consider a spectral problem for a fourth-order differential operator with real periodic coefficients and Neumann-type boundary conditions. For this operator, the eigenvalue asymptotics and a regularized trace formula are obtained.

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.

MIXED BOUNDARY VALUE PROBLEMS FOR THE HELMHOLTZ EQUATION

We consider a new approach to investigating the mixed boundary value problem (BVP) for the Helmholtz equation in the case of a three-dimensional unbounded domain Ω−⊂ℝ3 with compact boundary surface S=∂Ω−, which is divided into two disjoint parts, SD and SN, where the Dirichlet and Neumann type boundary conditions are prescribed respectively. Our approach is based on the classical potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with densities supported on the Dirichlet and Neumann parts of the boundary, respectively. This approach reduces the mixed BVP to a system of integral equations containing neither extensions of the Dirichlet or Neumann data nor a Steklov–Poincaré type operator involving the inverse of the single layer boundary integral operator, which is not available explicitly for arbitrary boundary surfaces. The right-hand sides of the resulting boundary integral equations system are functions coinciding with the given Dirichlet and Neumann data of the problem under consideration. We show that the corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L2-based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the class of functions belonging to the Sobolev space W2, loc1(Ω−) and satisfying the Sommerfeld radiation conditions. We also show that the pseudodifferential matrix operator thus obtained is invertible in the Lp-based Besov spaces and prove that under appropriate boundary data a solution to the mixed BVP possesses the Cα-Hölder continuity property in the closed domain Ω¯− with α=12−𝜀, where 𝜀>0 is an arbitrarily small number.

Study of solute dispersion under linear sorption in a semi-infinite porous formation

Solute dispersion in a porous formation is Mathematically expressed by partial differential equation well known as advection-dispersion equation (ADE). The present study deals with the solute transport governing equation in a semi-infinite homogeneous porous formation under linear sorption. A constant background solute concentration is assumed initially throughout the solute transport domain. Dirichlet and Neumann type boundary conditions are considered to examine the solute concentration distribution profile in the semi-infinite porous medium. The analytical and numerical solutions of the model problem are derived by Laplace transform technique and Crank-Nicolson method, respectively. Solute dispersion behaviour is studied for various form of flow velocities. Solutions obtained by analytical and numerical techniques are illustrated graphically with the help of MATLAB software. Also, the numerical solution is compared with the analytical solution and found great similarity between them.

Space-time formulation, discretization, and computational performance studies for phase-field fracture optimal control problems

The purpose of this work is the development of space-time discretization schemes for phase-field optimal control problems. Specifically in the optimal control minimization problem, a tracking-type cost functional is minimized to steer the crack via the phase-field variable into a desired pattern. To achieve such optimal solutions, Neumann type boundary conditions need to be determined. First, a time discretization of the forward problem is derived using a discontinuous Galerkin formulation. Here, a challenge is to include regularization terms and the crack irreversibility constraint. The optimal control setting is formulated by means of the Lagrangian approach from which the primal part, adjoint, tangent and adjoint Hessian are derived. Herein the overall Newton algorithm is based on a reduced approach by eliminating the state constraint, namely the displacement and phase-field unknowns, but keeping the control variable as the only unknown. From the low-order discontinuous Galerkin discretization, adjoint time-stepping schemes are finally obtained. Both our formulation and algorithmic developments are substantiated and illustrated with six numerical experiments.

Lattice Boltzmann method for antiplane shear deformation: non-lattice-conforming boundary conditions

In this work, two different approaches to treat boundary conditions in a lattice Boltzmann method (LBM) for the wave equation are presented. We interpret the wave equation as the governing equation of the displacement field of a solid under simplified deformation assumptions, but the algorithms are not limited to this interpretation. A feature of both algorithms is that the boundary does not need to conform with the discretization, i.e., the regular lattice. This allows for a larger flexibility regarding the geometries that can be handled by the LBM. The first algorithm aims at determining the missing distribution functions at boundary lattice points in such a way that a desired macroscopic boundary condition is fulfilled. The second algorithm is only available for Neumann-type boundary conditions and considers a balance of momentum for control volumes on the mesoscopic scale, i.e., at the scale of the lattice spacing. Numerical examples demonstrate that the new algorithms indeed improve the accuracy of the LBM compared to previous results and that they are able to model boundary conditions for complex geometries that do not conform with the lattice.

On an alternative approach for mixed boundary value problems for the Laplace equation

Abstract In this paper, we consider a special approach to the investigation of a mixed boundary value problem (BVP) for the Laplace equation in the case of a three-dimensional bounded domain Ω ⊂ ℝ 3 {\Omega\subset\mathbb{R}^{3}} , when the boundary surface S = ∂ ⁡ Ω {S=\partial\Omega} is divided into two disjoint parts S D {S_{D}} and S N {S_{N}} where the Dirichlet—Neumann-type boundary conditions are prescribed, respectively. Our approach is based on the potential method. We look for a solution to the mixed boundary value problem in the form of a linear combination of the single layer and double layer potentials with the densities supported respectively on the Dirichlet and Neumann parts of the boundary. This approach reduces the mixed BVP under consideration to a system of pseudodifferential equations. The corresponding pseudodifferential matrix operator is bounded and coercive in the appropriate L 2 {L_{2}} -based Bessel potential spaces. Consequently, the operator is invertible, which implies the unconditional unique solvability of the mixed BVP in the Sobolev space W 2 1 ⁢ ( Ω ) {W^{1}_{2}(\Omega)} . Using a special structure of the obtained pseudodifferential matrix operator, it is also shown that it is invertible in the L p {L_{p}} -based Besov spaces, which under appropriate boundary data implies C α {C^{\alpha}} -Hölder continuity of the solution to the mixed BVP in the closed domain Ω ¯ {\overline{\Omega}} with α = 1 2 - ε {\alpha=\frac{1}{2}-\varepsilon} , where ε &gt; 0 {\varepsilon&gt;0} is an arbitrarily small number.