Dirichlet Boundary Research Articles

Homogenization of a quasilinear elliptic problem in domains with small holes

This work aims to provide the asymptotic behavior of some class of quasilinear problems posed in a domain with small holes in for N>2, as . A nonlinear Robin boundary condition is prescribed on some of the holes, while a Dirichlet boundary one is imposed on the other holes as well as on the outer boundary. We apply the Periodic Unfolding Method for the homogenization of this problem.

NUMERICAL SIMULATION OF A SINGLE RING INFILTRATION EXPERIMENT WITH hp-ADAPTIVE SPACE-TIME DISCONTINUOUS GALERKIN METHOD

We present a novel hp-adaptive space-time discontinuous Galerkin (hp-STDG) method for the numerical solution of the nonstationary Richards equation equipped with Dirichlet, Neumann and seepage face boundary conditions. The hp-STDG method presented in this paper is a generalization of a hp-STDG method which was developed for time dependent non-linear convective-diffusive problems. We describe the method and the single ring experiment, and then we present a numerical experiment which clearly demonstrates the superiority of the hp-STDG method over a discontinuous Galerkin method based on a static fine mesh.

On a two-dimensional fractional thermoelastic system with nonlocal constraints describing a fractional Kirchhoff plate

We show herein the existence and uniqueness of solutions for coupled fractional order partial differential equations modeling a thermoelastic fractional Kirchhoff plate model associated with initial, Dirichlet, and nonlocal boundary conditions involving fractional Caputo derivative. Some efficient results of existence and uniqueness are obtained by employing the energy inequality method.

Determining the Optimal Source Points in MFS by Minimizing an Energy Gap Functional for 3D Laplace Operator

In this paper, an extended version of the method of minimizing an energy gap functional for determining the optimal source points in the method of fundamental solutions (MFS) is applied to the 3D Laplace operator subject to the Dirichlet and Neumann boundary conditions. As we know, the MFS is a more popular meshless method for solving boundary or initial-boundary value problems due to its simplicity and high accuracy. However, the accuracy of the MFS depends strongly on the distribution of the source points. Finally, some of the numerical experiments are carried out to express the simplicity and effectiveness of the presented method.

Boundary value problems for singular p- and p(x)- Laplacian equations in a domain with conical point on the boundary

This paper is a survey of our last results about solutions to the Dirichlet and Robin boundary problems, the Robin transmission problem for an elliptic quasilinear second-order equation with the constant p- and variable p(x)-Laplacians, as well as to the degenerate oblique derivative problem for elliptic linear and quasilinear second-order equations in a conical bounded n-dimensional domain.

Electronic structure calculations of compressed Li atom using composite technique under Ritz variational framework

AbstractThe structural properties of Li atom in 1s2nl [n = 2–5, l = 0–4] state inside an impenetrable cavity is studied. The system is assumed to be composed of two parts namely the core (1s2) electrons and the valence electron (nl). A model potential is considered to mimic the interaction between the outer electron with the core. The wavefunction of the core electrons is expanded in explicitly correlated multi‐exponent Hyllerass type basis set while the wavefunction of the valence electron is expanded in pure exponential basis set. Both the wavefunctions are consistent with Dirichlet's boundary conditions. The energy levels are determined by using Ritz variational method and are pushed towards continuum as the effect of confinement increases. The results are in reasonable agreement with the existing numerical estimates. We noticed incidental degeneracy and the subsequent level‐crossing phenomena along with evolution of quasibound states of confined Li atom. The thermodynamic pressures felt by the Li atom, Li+ and Li2+ ions inside the impenetrable cavity.

Novel convergence to steady-state for Nicholson’s blowflies equation with Dirichlet boundary

This short note is concerned with Nicholson’s blowflies equation with Dirichlet boundary. For the challenging non-monotone case, we show the exponential convergence rate by the energy method, where the key observation is that the energy integrations for the perturbations on the inner boundary will be disappeared automatically, even the solutions at the boundary are non-zero.

Thermal stresses in a functionally graded rotating disk: An approximate closed form solution

The purpose of this study is to analyze a circular annulus made of a functionally graded material (FGM) subjected to thermomechanical loading. The governing nonlinear differential equation for heat transfer, comprising of all the modes of heat transfer, such as conduction, convection, radiation, and internal heat generation, was formulated and then solved using the homotopy perturbation method (HPM). The stress field in the circular annulus due to thermomechanical loading was obtained using a HPM based approximate closed-form solution for a steady-state nonhomogeneous temperature field coupled with the solution of the classical theory of elasticity. The present work considered both Dirichlet and Neumann boundary conditions. A rigorous study on the effect of various thermomechanical parameters and grading parameters on temperature as well as the stress field is presented. The present approximate closed-form solution was validated with the finite element method (FEM) based solution. The close agreement between HPM based solutions of this work with the results of the ANSYS based FEM confirms the effectiveness and the viability of the present solution method for a FGM rotating disk with multiple variable nonlinearities. The present closed-form solution is more rational and computationally efficient over FEM and other numerical solutions.

Ambiguities in the local thermal behavior of the scalar radiation in one-dimensional boxes

This paper reports certain ambiguities in the calculation of the ensemble average $\left<T_\mu{}_\nu\right>$ of the stress-energy-momentum tensor of an arbitrarily coupled massless scalar field in one-dimensional boxes in flat spacetime. The study addresses a box with periodic boundary condition (a circle) and boxes with reflecting edges (with Dirichlet's or Neumann's boundary conditions at the endpoints). The expressions for $\left<T^\mu{}^\nu\right>$ are obtained from finite-temperature Green functions. In an appendix, in order to control divergences typical of two dimensions, these Green functions are calculated for related backgrounds with arbitrary number of dimensions and for scalar fields of arbitrary mass, and specialized in the text to two dimensions and for massless fields. The ambiguities arise due to the presence in $\left<T^\mu{}^\nu\right>$ of double series that are not absolutely convergent. The order in which the two associated summations are evaluated matters, leading to two different thermodynamics for each type of box. In the case of a circle, it is shown that the ambiguity corresponds to the classic controversy in the literature whether or not zero mode contributions should be taken into account in computations of partition functions. In the case of boxes with reflecting edges, it results that one of the thermodynamics corresponds to a total energy (obtained by integrating the non homogeneous energy density over space) that does not depend on the curvature coupling parameter $\xi$ as expected; whereas the other thermodynamics curiously corresponds to a total energy that does depend on $\xi$. Thermodynamic requirements (such as local and global stability) and their restrictions to the values of $\xi$ are considered.

An efficient meshless radial point collocation method for nonlinear p-Laplacian equation

This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

Identification of a space- and time-dependent source in a variable coefficient advection-diffusion equation from Dirichlet and Neumann boundary measured outputs

Abstract This paper considers the inverse problem of identifying an unknown space- and time-dependent source function F ⁢ ( x , t ) F(x,t) in the variable coefficient advection-diffusion equation u t = ( D ⁢ ( x ) ⁢ u x ) x - ( V ⁢ ( x ) ⁢ u ) x + F ⁢ ( x , t ) u_{t}=(D(x)u_{x})_{x}-(V(x)u)_{x}+F(x,t) from the Dirichlet ν ⁢ ( t ) := u ⁢ ( ℓ , t ) \nu(t):=u(\ell,t) and Neumann f ⁢ ( t ) := - D ⁢ ( 0 ) ⁢ u x ⁢ ( 0 , t ) f(t):=-D(0)u_{x}(0,t) , t ∈ ( 0 , T ] t\in(0,T] , boundary measured outputs. This problem was motivated by several important real-world applications in the field of contaminant hydrogeology, and the novel analysis presented here is highly relevant to problems of practical interest. The input-output operators corresponding to the Dirichlet and Neumann measured boundary data are introduced. The inverse problem is then formulated as a system of operator equations consisting of these operators and the measured outputs. The compactness and Lipschitz continuity of the input-output operators are proved in the relevant classes of admissible source functions ℱ and F r \mathcal{F}_{r} . These results together with the derived trace estimates allow us to show the existence of a quasi-solution of the inverse source problem as a minimum of the Tikhonov functional, under minimal regularity assumptions with respect to the source function and other inputs. An explicit gradient formula for the Fréchet gradient of the Tikhonov functional is also derived by means of an appropriate adjoint problem.

Application of boundary integral equation method to numerical solution of elliptic boundary-value problems in R3

In this paper we propose numerical methods for solving interior and exterior boundary-value problems for the Helmholtz and Laplace equations in complex three-dimensional domains. The method is based on their reduction to boundary integral equations in R2. Using the potentials of the simple and double layers, we obtain boundary integral equations of the Fredholm type with respect to unknown density for Dirichlet and Neumann boundary value problems. As a result of applying integral equations along the boundary of the domain, the dimension of problems is reduced by one. In order to approximate solutions of the obtained weakly singular Fredholm integral equations we suggest general numerical method based on spline approximation of solutions and on the use of adaptive cubatures that take into account the singularities of the kernels. When constructing cubature formulas, essentially non-uniform graded meshes are constructed with grading exponent that depends on the smoothness of the input data. The effectiveness of the method is illustrated with some numerical experiments.

The numerical study of advection–diffusion equations by the fourth-order cubic B-spline collocation method

A fourth-order numerical method based on cubic B-spline functions has been proposed to solve a class of advection–diffusion equations. The proposed method has several advantageous features such as high accuracy and fast results with very small CPU time. We have applied the Crank–Nicolson method to solve the advection–diffusion equation. The stability analysis is performed, and the method is shown to be unconditionally stable. Error analysis is carried out to show that the proposed method has fourth-order convergence. The efficiency of the proposed B-spline method has been checked by applying on ten important advection–diffusion problems of three types, having Dirichlet, Neumann and periodic boundary conditions. Considered examples prove the mentioned advantages of the method. The computed results are also compared with those available in the literature, and it is found that our method is giving better results.

Stabilization method for the Saint-Venant equations by boundary control

In this paper, we are interested in the stabilization of the flow modeled by the Saint-Venant equations. We have solved two problems in this study. The first, we have proved that the operator associated to the Saint-Venant system has a finite number of unstable eigenvalues. Consequently, the system is not exponentially stable on the space [Formula: see text], but is exponentially stable on a subspace of the space [Formula: see text], ([Formula: see text] is a given domain). The second problem, if the advection is dominant, the natural stabilization is very slow. To solve these problems, we have used an extension method due to Russel (1974) and Fursikov (2002). Thanks to this method, we have determined a boundary Dirichlet control able to accelerate the stabilization of the flow. Also, the boundary Dirichlet control is able to kill all the unstable eigenvalues to get an exponentially stable solution on the space [Formula: see text]. Then, we extend this method to the finite difference equations analog of the continuous Saint-Venant equations. Also, in this case, we obtained similar results of stabilization. A finite difference scheme is used to compute the control and several numerical experiments are performed to illustrate the efficiency of the control.

Blow-Up Phenomena of a Cancer Invasion Model with Nonlinear Diffusion and Haptotaxis Term

In this paper, we consider a nonlinear cancer invasion mathematical model with proliferation, growth and haptotaxis effects. We obtain lower bounds for the finite-time blow-up of solutions of the considered system with nonlinear diffusion operator when blow-up occurs. We have assumed both the Dirichlet and Neumann boundary conditions in $${\mathbb {R}}^n, n\ge 2$$ to attain the desire result.

Steklov approximations of Green’s functions for Laplace equations

Purpose This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator. Design/methodology/approach The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary. Findings The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary. Originality/value This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

Corrosion shape reconstruction of the mixed boundary in electrostatic imaging

We consider the corrosion detection problem in terms of the Laplace equation and study a simply connected bounded domain with Wentzell-type GIBC boundary condition. We derive the systems of integral equations and establish the equivalence to the inverse shape problem in a Sobolev space setting. For the direct problem, we use potential theory to simulate the Neumann data from Dirichlet data on Dirichlet boundary. Then we propose a Newton iterative approach based on the boundary integral equations derived from Green’s representation theorem. After describing the linearization and the iteration scheme for the inverse shape, we compute the Fréchet derivatives with respect to the unknowns. We conclude by presenting several numerical examples for shape reconstructions to show the validity of the proposed method.

Passivity Analysis of Markov Jumping Delayed Reaction-Diffusion Neural Networks under Different Boundary Conditions

This work analyzes the passivity for a set of Markov jumping reaction-diffusion neural networks limited by time-varying delays under Dirichlet and Neumann boundary conditions, respectively, in which Markov jumping is used to describe the variations among system parameters. To overcome some difficulties originated from partial differential terms, the Lyapunov–Krasovskii functional that introduces a new integral term is proposed and some inequality techniques are also adopted to obtain the delay-dependent stability conditions in terms of linear matrix inequalities, which ensures that the designed neural networks satisfy the specified performance of passivity. Finally, the advantages and effectiveness of the obtained results are verified via displaying two illustrated examples.

Stable Solutions of $-\Delta u+\lambda u=|u|^{p-1}u $ in Strips

This paper is devoted to study the following equation $-\Delta u+\lambda u= |u|^{p-1}u \;\;\text{in}\;\Omega $ , with homogeneous Dirichlet or Neumann boundary conditions where $p>1$ , $\lambda >0$ , $\Omega =\mathbb{R}^{n-k}\times \omega $ , $n\geq 2$ , $k\geq 1$ , and $\omega $ is a smoothly bounded domain of $\mathbb{R}^{k}$ . We prove Liouville-type theorems for $C^{2}$ solutions which are stable or stable outside a compact set of $\Omega $ . We first provide an integral estimate using stability argument which combined with Pohozaev-type identity allow to obtain nonexistence results for $p\in [p_{s}(n), p_{s}(n-k)]$ , where $p_{s}(n)$ is the classical Sobolev exponent in dimension $n$ . Also, we establish monotonicity formula to prove the nonexistence of nontrivial solution which is stable or stable outside a compact set of $\Omega $ for all $p>1$ .

Anti-collocated observer-based output feedback control of wave equation with cubic velocity nonlinear boundary and Dirichlet control

This study focuses on the observer and observer-based output feedback control design for a one-dimensional wave equation with van der Pol type nonlinear boundary condition and Dirichlet boundary control. The contribution of this work is divided into two parts. Firstly, the anti-collocated backstepping observer design for the nonlinear wave system is investigated, and the well-posedness and the asymptotical stability analysis of the observer error are studied. Secondly, an output feedback observer-based controller is set up by using boundary displacement measurement only to stabilise the nonlinear wave system; also the asymptotical stability of the closed-loop system is shown. The validity of the theoretical results is demonstrated by numerical simulations.