Dirichlet-type Boundary Conditions Research Articles

Uniqueness for entropy solution to a class of nonlinear degenerate weighted elliptic p(.)-Laplacian problem with L1-data

In the present paper, we prove the uniqueness of entropy solution to a class of nonlinear degenerate weighted elliptic p(·)-Laplacian problems. These particular problems are delineated by Dirichlet-type boundary conditions and L1 data. We use a priori estimates in the framework of weighted variable exponent Sobolev spaces theory.

Existence of entropy solutions to nonlinear degenerate weighted elliptic p(.)-Laplacian problem and L1-data

Our aim in this paper is to study the existence result of entropy solution for a specific type of nonlinear degenerate weighted elliptic p(.)-Laplacian problems with Dirichlet-type boundary condition and L1 data. In the framework of the theory of weighted Sobolev spaces with variable exponents, we use the regularization approach combined with a priori estimates.

Initial‐boundary value and interface problems on the real half line for the fractional advection–diffusion‐type equation

We use the unified transform method (UTM) to solve initial‐boundary value problems for the fractional advection–diffusion‐type equation (FADE) on the real half line. We generalize this equation using the modified definition of the Atangana–Baleanu fractional derivative of order in order to satisfy the initial condition. A solution methodology is proposed when the UTM is implemented in fractional differential equations with boundary conditions of Dirichlet and Robin type, in particular when using the modified definitions of fractional operators with non‐singular kernel. In addition, an interface problem is stated and solved in the adjacent domains and , where perfect contact continuity conditions are imposed. The exact solutions obtained include as a particular case, the diffusion and advection–diffusion equations with integer‐order derivatives. Finally, representative curves for the solution are shown by varying the fractional order.

Higher order evolution inequalities involving Leray–Hardy potential singular on the boundary

We consider a higher order (in time) evolution inequality posed in the half ball, under Dirichlet type boundary conditions. The involved elliptic operator is the sum of a Laplace differential operator and a Leray–Hardy potential with a singularity located at the boundary. Using a unified approach, we establish a sharp nonexistence result for the evolution inequalities and hence for the corresponding elliptic inequalities. We also investigate the influence of a nonlinear memory term on the existence of solutions to the Dirichlet problem, without imposing any restrictions on the sign of 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.

Existence of non-negative periodic solutions for a degenerate anisotropic parabolic equation with strongly nonlinear source

Abstract The purpose of the present work is investigating a degenerate parabolic equation with anisotropic p-Laplacian operator and strongly nonlinear source under boundary conditions of Dirichlet type; the existence of a periodic non-negative solution is shown. The proof is based on the Leray–Schauder topological degree, which is tricky to work with in this kind of equations.

Well‐posedness and exponential stability for a nonlinear wave equation with acoustic boundary conditions

In this paper, we prove the well‐posedness of a nonlinear wave equation coupled with boundary conditions of Dirichlet and acoustic type imposed on disjoints open boundary subsets. The proposed nonlinear equation models small vertical vibrations of an elastic medium with weak internal damping and a general nonlinear term. We also prove the exponential decay of the energy associated with the problem. Our results extend the ones obtained in previous results to allow weak internal dampings and removing the dimensional restriction . The method we use is based on a finite‐dimensional approach by combining the Faedo‐Galerkin method with suitable energy estimates and multiplier techniques.

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.

Unsteady two-phase flow with dirichlet and Robin-type boundary conditions by employing new weber transform

The axisymmetric unsteady two-phase flow problem is explored. The flow domain is defined by two co-axial circular cylinders and is axial symmetric. The Dirichlet-type boundary condition is used on the inner cylindrical surface, whereas the Robin-type boundary condition is used on the outer cylindrical surface. The velocities are computed analytically using a new form of the Weber transform that is suited for these boundary conditions. The effect of the slip parameter on velocities is investigated using numerical simulations and graphical representations. The studied problem is new in the literature because there do not exist any analytical studies regarding the problems with boundary conditions of Dirichlet type on the inner cylinder (the no-slip on the wall) and boundary conditions of Robin type on the outer cylinder (the mixture slipping on the wall). A new integral transform of Weber type has been employed to determine analytical solutions for such problems, together with the Laplace transform. The studied problem could generate analytical solutions for more two-phase flow problems in annular domains since the translational motions of the inner cylinder and the outer cylinder are given by arbitrary functions of the time t.

Effective data sampling strategies and boundary condition constraints of physics-informed neural networks for identifying material properties in solid mechanics

Material identification is critical for understanding the relationship between mechanical properties and the associated mechanical functions. However, material identification is a challenging task, especially when the characteristic of the material is highly nonlinear in nature, as is common in biological tissue. In this work, we identify unknown material properties in continuum solid mechanics via physics-informed neural networks (PINNs). To improve the accuracy and efficiency of PINNs, we develop efficient strategies to nonuniformly sample observational data. We also investigate different approaches to enforce Dirichlet-type boundary conditions (BCs) as soft or hard constraints. Finally, we apply the proposed methods to a diverse set of time-dependent and time-independent solid mechanic examples that span linear elastic and hyperelastic material space. The estimated material parameters achieve relative errors of less than 1%. As such, this work is relevant to diverse applications, including optimizing structural integrity and developing novel materials.

Non-isothermal non-Newtonian fluids: The stationary case

The stationary Navier–Stokes equations for a non-Newtonian incompressible fluid are coupled with the stationary heat equation and subject to Dirichlet-type boundary conditions. The viscosity is supposed to depend on the temperature and the stress depends on the strain through a suitable power law depending on [Formula: see text] (shear thinning case). For this problem we establish the existence of a weak solution as well as we prove some regularity results both for the Navier–Stokes and the Stokes cases. Then, the latter case with the Carreau power law is approximated through a FEM scheme and some error estimates are obtained. Such estimates are then validated through some two-dimensional numerical experiments.

A system of wave inequalities with inverse-square potentials in an exterior domain

ABSTRACT We study the existence and nonexistence of solutions to a system of wave inequalities with inverse-square potentials in an exterior domain of R N , under inhomogeneous Dirichlet-type boundary conditions. Our study yields naturally existence and nonexistence results for the corresponding stationary wave system.

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.

Higher-order evolution inequalities involving convection and Hardy-Leray potential terms in a bounded domain

AbstractWe consider a class of nonlinear higher-order evolution inequalities posed in $(0,\infty)\times B_1\backslash\{0\}$, subject to inhomogeneous Dirichlet-type boundary conditions, where B1 is the unit ball in $\mathbb{R}^N$. The considered class involves differential operators of the form \begin{equation*} \mathcal{L}_{\mu_1,\mu_2}=-\Delta +\frac{\mu_1}{|x|^2}x\cdot \nabla +\frac{\mu_2}{|x|^2},\qquad x\in \mathbb{R}^N\backslash\{0\}, \end{equation*}where $\mu_1\in \mathbb{R}$ and $\mu_2\geq -\left(\frac{\mu_1-N+2}{2}\right)^2$. Optimal criteria for the nonexistence of weak solutions are established. Our study yields naturally optimal nonexistence results for the corresponding class of elliptic inequalities. Notice that no restriction on the sign of solutions is imposed.

On the critical curve for systems of hyperbolic inequalities in an exterior domain of the half-space

We establish blow-up results for a system of semilinear hyperbolic inequalities in an exterior domain of the half-space. The considered system is investigated under an inhomogeneous Dirichlet-type boundary condition depending on both time and space variables. In certain cases, an optimal criterium of Fujita-type is derived. Our results yield naturally sharp nonexistence criteria for the corresponding stationary wave system and equation.

ASYMPTOTIC ANALYSIS OF STURM-LIOUVILLE PROBLEM WITH DIRICHLET AND NONLOCAL TWO-POINT BOUNDARY CONDITIONS

In this study, we obtain asymptotic expansions for eigenvalues and eigenfunctions of the one–dimensional Sturm–Liouville equation with one classical Dirichlet type boundary condition and two-point nonlocal boundary condition. We analyze the characteristic equation of the boundary value problem for eigenvalues and derive asymptotic expansions of arbitrary order. We apply the obtained results to the problem with two-point nonlocal boundary condition.

Limits and consistency of nonlocal and graph approximations to the Eikonal equation

Abstract In this paper, we study a nonlocal approximation of the time-dependent (local) Eikonal equation with Dirichlet-type boundary conditions, where the kernel in the nonlocal problem is properly scaled. Based on the theory of viscosity solutions, we prove existence and uniqueness of the viscosity solutions of both the local and nonlocal problems, as well as regularity properties of these solutions in time and space. We then derive error bounds between the solution to the nonlocal problem and that of the local one, both in continuous time and forward Euler time discretization. We then turn to studying continuum limits of nonlocal problems defined on random weighted graphs with $n$ vertices. In particular, we establish that if the kernel scale parameter decreases at an appropriate rate as $n$ grows then, almost surely, the solution of the problem on graphs converges uniformly to the viscosity solution of the local problem as the time step vanishes and the number vertices $n$ grows large.

Geometric multigrid method for solving Poisson's equation on octree grids with irregular boundaries

A method is presented to include irregular domain boundaries in a geometric multigrid solver. Dirichlet boundary conditions can be imposed on an irregular boundary defined by a level set function. Our implementation employs quadtree/octree grids with adaptive refinement, a cell-centered discretization and pointwise smoothing. Boundary locations are determined at a subgrid resolution by performing line searches. For grid blocks near the interface, custom operator stencils are stored that take the interface into account. For grid block away from boundaries, a standard second-order accurate discretization is used. The convergence properties, robustness and computational cost of the method are illustrated with several test cases. New version program summaryProgram Title: AfivoCPC Library link to program files:https://doi.org/10.17632/5y43rjdmxd.2Developer's repository link:https://github.com/MD-CWI/afivoLicensing provisions: GPLv3Programming language: FortranJournal reference of previous version: Comput. Phys. Commun. 233 (2018) 156–166. https://doi.org/10.1016/j.cpc.2018.06.018Does the new version supersede the previous version?: Yes.Reasons for the new version: Add support for internal boundaries in the geometric multigrid solver.Summary of revisions: The geometric multigrid solver was generalized in several ways: a coarse grid solver from the Hypre library is used, operator stencils are now stored per grid block, and methods for including boundaries via a level set function were added.Nature of problem: The goal is to solve Poisson's equation in the presence of irregular boundaries that are not aligned with the computational grid. It is assumed these irregular boundaries are defined by a level set function, and that a Dirichlet type boundary condition is applied. The main applications are 2D and 3D simulations with octree-based adaptive mesh refinement, in which the mesh frequently changes but the irregular boundaries do not.Solution method: A geometric multigrid method compatible with octree grids is developed, using a cell-centered discretization and point-wise smoothing. Near irregular boundaries, custom operator stencils are stored. Line searches are performed to locate interfaces with sub-grid resolution. To increase the methods robustness, this line search is modified on coarse grids if boundaries are otherwise not resolved. The multigrid solver uses OpenMP parallelization.

Formula omitted]-regularity for steady flows of electrorheological fluids in 2D

We prove interior C1,α-regularity of every weak solution and the existence of global C1,α-solutions for steady flows of electrorheological fluids with Dirichlet type boundary condition under 32<p(x)<∞ in 2D.