Dirichlet-type Boundary Conditions Research Articles

Neumann boundary control for degenerate wave equations in time-varying domains

Degenerate wave equation with Neumann boundary control in time-varying domains is considered. The fixed endpoint has a Dirichlet-type boundary condition, a Neumann boundary control applied to the moving end. By using the Hilbert Uniqueness Method, we establish the exact boundary controllability for the degenerate wave equation and obtain estimation for the control time.

High-order approximation of Caputo–Prabhakar derivative with applications to linear and nonlinear fractional diffusion models

Abstract In this study, we devise a high-order numerical scheme to approximate the Caputo–Prabhakar derivative of order α ∈ (0, 1), using an rth-order time stepping Lagrange interpolation polynomial, where 3 ≤ r ∈ N $3\le r\in \mathbb{N}$ . The devised scheme is a generalization of the existing schemes developed earlier. Further, we adopt the discussed scheme for solving a linear time fractional advection–diffusion equation and a nonlinear time fractional reaction–diffusion equation with Dirichlet type boundary conditions. We show that the discussed method is unconditionally stable, uniquely solvable and convergent with convergence order O(τ r+1−α , h 2), where τ and h are the temporal and spatial step sizes, respectively. Without loss of generality, applicability of the discussed method is established by illustrative examples for r = 4, 5.

Fine boundary regularity for the singular fractional p-Laplacian

We study the boundary weighted regularity of weak solutions u to a s-fractional p-Laplacian equation in a bounded C1,1 domain Ω with bounded reaction and nonlocal Dirichlet type boundary condition, with s∈(0,1). We prove optimal up-to-the-boundary regularity of u, which is Cs(Ω‾) for any p>1 and, in the singular case p∈(1,2), that u/dΩs has a Hölder continuous extension to the closure of Ω, dΩ(x) meaning the distance of x from the complement of Ω. This last result is the singular counterpart of the one in [30], where the degenerate case p⩾2 is considered.

Applying satellite sea surface temperature as Dirichlet-type surface thermal boundary condition in an ocean model

Following the Dirichlet-type boundary condition that specifies value at boundary of a system in mathematics and physics, this study suggests applying satellite sea surface temperature (SST) as Dirichlet-type surface thermal boundary condition (STBC) in ocean models. Numerical experiments with Dirichlet-type and Combined-type STBC with different configurations of satellite SST, for the period of January to April 2019 in the Northwest Pacific Ocean, were carried out based on the Princeton Ocean Model. The experiments were assessed using satellite and in situ observations of temperature, GOFS3.1 analysis and GLORYS12v1 reanalysis. Results show that applying satellite SST as Dirichlet-type STBC could constrain the modeled SST and near-surface temperature in upper 50 m depth well, which is better than the scheme that uses satellite SST as the relaxation terms of Combined-type STBC. The temperature section along the 137°E resulted from the Dirichlet-type STBC is comparable with GOFS3.1 analysis and better than GLORYS12v1 reanalysis in upper 50 m depth. These results suggest that applying high-accuracy satellite SST as Dirichlet-type STBC in ocean models has a promising prospect in numerical simulation.

Analytical and numerical study of thermo-solutal convection in pulsating flows between two coaxial cylinders

This research reports an analytical and numerical study of thermo-solutal convection in pulsating flows between two coaxial cylinders under a pressure gradient and subject to Dirichlet-type boundary conditions for temperature and concentration along the vertical walls of the annulus. The numerical procedure used in this work is mainly based on the solution of the momentum equations and coupled with the energy and concentration equations. The finite difference method is adopted to solve governing equations using the implicit Crank-Nicolson time marching scheme in the fluid domain and the alternating direction implicit method inside the inner cylinder. The effect of the various parameters such as the fluid-solid conductivity ratio, thermal diffusivity ratio, buoyancy ratio, Lewis number, Richardson number, aspect ratio, and radius ratio on the heat and mass transfer characteristics and the flow structure is presented and discussed. The governing equations are solved analytically after separating into a steady case and an oscillatory case, where an analytical solution for velocity profile, temperature, and concentration distributions is obtained in terms of Bessel’s functions, while the analytical solutions of temperature and concentration profiles are keeping symmetry with the increase of the radius ratio. The results show that the analytical profile can reproduce the three fields obtained numerically and thus for the whole range of parameters.

Existence of Weak Solutions for the Class of Singular Two-Phase Problems with a ψ-Hilfer Fractional Operator and Variable Exponents

In this paper, we prove the existence of at least two weak solutions to a class of singular two-phase problems with variable exponents involving a ψ-Hilfer fractional operator and Dirichlet-type boundary conditions when the term source is dependent on one parameter. Here, we use the fiber method and the Nehari manifold to prove our results.

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.

Numerical Approximation of Spatially Loaded Time-Fractional Diffusion Equation

This article is dedicated to numerically solving the spatially loaded time-fractional diffusion equation with initial and Dirichlet-type boundary conditions. We apply finite difference approximation to the considered problem and use the well-known L1 method to approximate the Caputo fractional derivative. The numerical approximation of the problem yields a loaded implicit difference scheme. To obtain the solution, we employ a special parametric representation of solutions of auxiliary linear systems using the superposition property. In conclusion, we showcase several numerical tests, validate the outcomes, and observe the convergence of errors.

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.