Dirichlet Boundary Conditions Research Articles

A robust septic hermite collocation technique for dirichlet boundary condition Heat conduction equation

Abstract In the current manuscript, approximate solution for 1D heat conduction equation will be sought with the Septic Hermite Collocation Method (SHCM). To achieve this goal, by means of the roots of both Chebyschev and Legendre polynomials used at the inner collocation points, the pseudo code of the method is found out and applied using Matlab which is one of the widely utilized symbolic programming platforms. The unconditional stability of the scheme is shown by the traditional von-Neumann stability technique. To illustrate the accuracy and effectiveness of this newly current numerical scheme, a comparison among analytical and the computed numerical results is presented in tabular forms. It has been illustrated that the scheme is both accurate and effective one and at the same time can be used in a successful way for finding out numerical solutions of several nonlinear problems as well as linear ones.

Analysis of efficient preconditioner for solving Poisson equation with Dirichlet boundary condition in irregular three-dimensional domains

This paper analyzes modified ILU (MILU)-type preconditioners for efficiently solving the Poisson equation with Dirichlet boundary conditions on irregular domains. In [1], the second-order accuracy of a finite difference scheme developed by Gibou et al. [2] and the effect of the MILU preconditioner were presented for two-dimensional problems. However, the analyses do not directly extend to three-dimensional problems. In this paper, we first demonstrate that the Gibou method attains second-order convergence for three-dimensional irregular domains, yet the discretized Laplacian exhibits a condition number of O(h−2) for a grid size h. We show that the MILU preconditioner reduces the order of the condition number to O(h−1) in three dimensions. Furthermore, we propose a novel sectored-MILU preconditioner, defined by a sectorized lexicographic ordering along each axis of the domain. We demonstrate that this preconditioner reaches a condition number of order O(h−1) as well. Sectored-MILU not only achieves a similar or better condition number than conventional MILU but also improves parallel computing efficiency, enabling very efficient calculation when the dimensionality or problem size increases significantly. Our findings extend the feasibility of solving large-scale problems across a range of scientific and engineering disciplines.

Nonexistence for a system of time‐fractional diffusion equations in an exterior domain

A system of time‐fractional diffusion equations posed in an exterior domain of ( ) under homogeneous Dirichlet boundary conditions is investigated in this paper. The time‐fractional derivatives are considered in the Caputo sense. Using nonlinear capacity estimates specifically adapted to the nonlocal properties of the Caputo fractional derivative, the geometry of the domain, and the boundary conditions, we obtain sufficient conditions for the nonexistence of a weak solution to the considered system.

A two-level semi-hybrid-mixed model for Stokes–Brinkman flows with divergence-compatible velocity–pressure elements

A two-level version for a recent semi-hybrid-mixed finite element approach for modeling Stokes and Brinkman flows is proposed. In the context of a domain decomposition of the flow region Ω, composite divergence-compatible finite elements pairs in H(div,Ω)×L2(Ω) are utilized for discretizing velocity and pressure fields, using the same approach previously adopted for two-level mixed Darcy and stress mixed elasticity models. The two-level finite element pairs of spaces in the subregions may have richer internal resolution than the boundary normal trace. Hybridization occurs by the introduction of an unknown (traction) defined over element boundaries, playing the role of a Lagrange multiplier to weakly enforce tangential velocity continuity and Dirichlet boundary condition. The well-posedness of the method requires a proper choice of the finite element space for the traction multiplier, which can be achieved after a proper velocity FE space enrichment with higher order bubble fields. The method is strongly locally conservative, yielding exact divergence-free velocity fields, demonstrating pressure robustness, and facilitating parallel implementations by limiting the communication of local common data to at most two elements. Easier coupling strategies of finite elements regarding different polynomial degree or mesh widths are permitted, provided that mild mesh and normal trace consistency properties are satisfied. Significant improvement in computational performance is achieved by the application of static condensation, where the global system is solved for coarse primary variables. The coarse primary variables are a piecewise constant pressure variable over the subregions, velocity normal trace and tangential traction over subdomain interfaces, as well as a real number used as a multiplier ensuring global zero-mean pressure. Refined details of the solutions are represented by secondary variables, which are post-processed by local solvers. Numerical results are presented for the verification of convergence histories of the method.

Non-relativistic expansion of open strings and D-branes

We expand the relativistic open bosonic string in powers of 1/c2 where c is the speed of light. We perform this expansion to next-to-leading order in 1/c2 and relate our results to known descriptions of non-relativistic open strings obtained by taking limits. Just as for closed strings the non-relativistic expansion is well-defined if the open string winds a circle in the target space. This direction must satisfy Dirichlet boundary conditions. It is shown that the endpoints of the open string behave as Bargmann particles in the non-relativistic regime. These open strings end on nrDp-branes with p ≤ 24. When these nrDp-branes do not fluctuate they correspond to (p + 1)-dimensional Newton-Cartan submanifolds of the target space. When we include fluctuations and worldvolume gauge fields their dynamics is described by a non-relativistic version of the DBI action whose form we derive from symmetry considerations. The worldvolume gauge field and scalar field of a nrD24-brane make up the field content of Galilean electrodynamics (GED), and the effective theory on the nrD24-brane is precisely a non-linear version of GED. We generalise these results to actions for any nrDp-brane by demanding that they have the same target space gauge symmetries that the non-relativistic open and closed string actions have. Finally, we show that the nrDp-brane action is transverse T-duality covariant. Our results agree with the findings of Gomis, Yan and Yu in [1].

Existence of global weak solutions and simulations to a Dirichlet problem for a generalized Swift–Hohenberg equation

In this paper, we shall investigate an initial–boundary value problem of a generalized Swift–Hohenberg model subject to homogeneous Dirichlet boundary conditions in two spatial dimensions. The model consists of a nonlinear term of the form ψ2Δ2ψ2 in the free energy functional, which is used to model the stability of fronts between hexagons and squares in pinning effect. We first prove the global-in-time existence and uniqueness of weak solutions to this initial–boundary value problem in the case with the parameter β<0, where we employ the energy method and make use of various techniques to derive delicate a priori estimates. At the end, a few numerical experiments of the model are also performed to study the competition between hexagons and squares.

Dynamics of a nonlinear infection viral propagation model with one fixed boundary and one free boundary

In this paper we study a nonlinear infection viral propagation model with diffusion, in which, the left boundary is fixed and with homogeneous Dirichlet boundary conditions, while the right boundary is free. We find that the habitat always expands to the half line [0,∞), and that the virus and infected cells always die out when the Basic Reproduction NumberR0≤1, while the virus and infected cells have persistence properties when R0>1. To obtain the persistence properties of virus and infected cells when R0>1, the most work of this paper focuses on the existence and uniqueness of bounded positive equilibrium solutions for subsystems and the existence of positive equilibrium solutions for the entire system.

Local Exact Controllability of the One-Dimensional Nonlinear Schrödinger Equation in the Case of Dirichlet Boundary Conditions

Local Exact Controllability of the One-Dimensional Nonlinear Schrödinger Equation in the Case of Dirichlet Boundary Conditions

Global well-posedness of strong solutions to the two-dimensional inhomogeneous biaxial nematic liquid crystal flow with vacuum

This paper considers the inhomogeneous biaxial nematic liquid crystal flow in a smooth bounded domain Ω⊂R2, where the velocity u and the orthogonal unit vector fields (m,n) admit the Dirichlet and Neumann boundary condition, respectively. By applying piecewise estimate and continuity method, we get the global existence of strong solutions, provided that the basic energy is suitably small. Our result may be regarded as an extension and improvement of Gong-Lin (2022) and Li-Liu-Zhong (2017) to the Neumann boundary condition, where the initial vacuum is allowed. Some new techniques are developed in order to deal with integral estimates caused by the boundary condition, and more complicated model.

Two Nitsche-based mixed finite element discretizations for the seepage problem in Richards’ equation

This paper proposes two algorithms to impose seepage boundary conditions in the context of Richards’ equation for groundwater flows in unsaturated media. Seepage conditions are non-linear boundary conditions, that can be formulated as a set of unilateral constraints on both the pressure head and the water flux at the ground surface, together with a complementarity condition: these conditions in practice require switching between Neumann and Dirichlet boundary conditions on unknown portions on the boundary. Upon realizing the similarities of these conditions with unilateral contact problems in mechanics, we take inspiration from that literature to propose two approaches: the first method relies on a strongly consistent penalization term, whereas the second one is obtained by an hybridization approach, in which the value of the pressure on the surface is treated as a separate set of unknowns. The flow problem is discretized in mixed form with div-conforming elements so that the water mass is preserved. Numerical experiments show the validity of the proposed strategy in handling the seepage boundary conditions on geometries with increasing complexity.

Inverse initial‐value problems for time fractional diffusion equations in fractional Sobolev spaces

AbstractWe study the time fractional diffusion equation , , in a bounded domain with an elliptic operator and a locally Lipschitz nonlinearity on fractional Sobolev spaces, subjected to the homogeneous Dirichlet boundary condition. Data have not been measured at the initial time , but at a final time , that is, is given instead of . The problem is, therefore, called an inverse initial‐value problem. We first establish the well‐posedness of this problem on fractional Sobolev spaces and the regularity of the solution by assuming only the local Lipschitz continuity of . Second, an susceptible‐infected (shortly, SI) model with heterogeneity and a Navier–Stokes equation have been exemplified. Finally, a spatial ‐estimate for the solution and its gradient has been provided. The essential tools are asymptotic behaviours of Mittag–Leffler functions, fractional power spaces, fractional Sobolev spaces and embedding, weighted functional spaces, and estimates for heat semigroup.

Virtual Body-fitted Grid-based Immersed Boundary Method for Simulation of Thermal Flows with Dirichlet and Neumann Boundary Conditions

In this work, a virtual body-fitted grid-based immersed boundary method (IBM) combined with the thermal lattice Boltzmann flux solver (TLBFS) is proposed to efficiently simulate thermal flows with Dirichlet and Neumann boundary conditions. The method involves generating a virtual body-fitted grid along the immersed boundary and implementing the boundary conditions using a fractional step technique. In the prediction step, the finite-volume TLBFS is used to obtain the predicted flow field on the Eulerian mesh without considering the immersed boundary, while in the correction step, the correction process of the flow field is conducted on the virtual grid. Thus, the effect of the boundary is transferred to the fluid domain through the virtual grid indirectly. For Dirichlet boundary conditions, such as the no-slip condition and the specified temperature condition, the velocity and temperature corrections are considered unknowns and solved through a matrix system formed from enforcing the boundary conditions. As for the Neumann boundary condition, such as the specified heat flux condition, the temperatures at virtual layers inside the immersed body are regarded as unknowns and computed from those at virtual layers outside the wall according to the quadratic distribution. Several benchmark cases including natural, forced, and mixed convection problems are well simulated to validate the method. The numerical results are in good agreement with reference data, demonstrating the capacity of the present method to simulate thermal flows with Dirichlet and Neumann boundary conditions while requiring fewer Lagrangian points and achieving higher computational efficiency.

Some useful tools in the study of nonlinear elliptic problems

This paper gives an overview of some basic aspects concerning the qualitative analysis of nonlinear, nonhomogeneous elliptic problems. We are concerned with two classes of elliptic equations with Dirichlet boundary condition. The first problem is driven by a general nonhomogeneous differential operator, which includes several usual operators (such as the (p,q)-Laplace operator introduced by P. Marcellini). Next, we focus on differential operators with unbalanced growth in the nonautonoumous case. Our analysis will point out some relevant differences between balanced and unbalanced growth problems. The presentation is done in the context of Dirichlet problems but a similar analysis can be developed for other boundary conditions, such as Neumann or Robin.

Fractional stable random fields on the Sierpiński gasket

We define and study fractional stable random fields on the Sierpiński gasket. Such fields are formally defined as (−Δ)−sWK,α, where Δ is the Laplace operator on the gasket and WK,α is a stable random measure. Both Neumann and Dirichlet boundary conditions for Δ are considered. Sample paths regularity and scaling properties are obtained. The techniques we develop are general and extend to the more general setting of the Barlow fractional spaces.

Uniform convergence of finite element method on Bakhvalov-type mesh for a 2-D singularly perturbed convection–diffusion problem with exponential layers

Analyzing uniform convergence of finite element method for a 2-D singularly perturbed convection–diffusion problem with exponential layers on Bakhvalov-type mesh remains a complex, unsolved problem. Previous attempts to address this issue have encountered significant obstacles, largely due to the constraints imposed by a specific mesh. These difficulties stem from three primary factors: the width of the mesh subdomain adjacent to the transition point, constraints imposed by the Dirichlet boundary condition, and the structural characteristics of exponential layers. In response to these challenges, this paper introduces a novel analysis technique that leverages the properties of interpolation and the relationship between the smooth function and the layer function on the boundary. By combining this technique with a simplified interpolation, we establish the uniform convergence of optimal order k under an energy norm for finite element method of any order k. Numerical experiments validate our theoretical findings.

On a parabolic equation in microelectromechanical systems with an external pressure

The parabolic problem on a bounded domain of with Dirichlet boundary condition models the microelectromechanical systems (MEMS) device with an external pressure term. In this paper, we classify the behavior of the solutions to this equation. We first show that under certain initial conditions, there exist critical constants and such that when , there exists a global solution, while for or , the solution quenches in finite time. The estimates of voltage , quenching time , and pressure term are investigated. The quenching set is proved to be a compact subset of with an additional condition on , provided is a convex bounded set. In particular, if is radially symmetric, then the origin is the only quenching point. Furthermore, we not only derive the two‐sided bound estimate for the quenching solution but also obtain its asymptotic behavior near the quenching time.

Numerical analysis for nonlinear wave equations with boundary conditions: Dirichlet, Acoustics and Impenetrability

In this article, we present an error estimation in the L2 norm referring to three wave models with variable coefficients, supplemented with initial and boundary conditions. The first two models are nonlinear wave equations with Dirichlet, Acoustics, and nonlinear dissipative impenetrability boundary conditions, while the third model is a linear wave equation with Dirichlet, Acoustics, and linear dissipative impenetrability boundary conditions. In the field of numerical analysis, we establish two key theorems for estimating errors to the semi-discrete and totally discrete problems associated with each model. Such theorems provide theoretical results on the convergence rate in both space and time. For conducting numerical simulations, we employ linear, quadratic, and cubic polynomial basis functions for the finite element spaces in the Galerkin method, in conjunction with the Crank-Nicolson method for time discretization. For each time step, we apply Newton's method to the resulting nonlinear problem. The numerical results are presented for all three models in order to corroborate with the theoretical convergence order obtained.

Gradient estimates for Yamabe type equations under different curvature conditions and applications

The aim of this paper is to give various aspects of gradient estimates for positive solutions to the Yamabe type equation ut=Δfu+au+buα. We study the problem in many different settings. As result, we obtain several types of gradient estimates for the solution. When the manifold is complete without boundary, we study gradient estimates of Shi type, and of Hamilton type. Moreover, we also derive gradient estimates if the manifold is evolved under the (k,m)-super Perelman-Ricci flows. Besides, we prove gradient estimates under the integral Ricci curvature bounds. Finally, when the manifold has compact boundary, we investigate gradient estimates with Dirichlet boundary conditions. Our theorems generalize and improve many previous works.

Two point boundary value problems for ordinary differential systems with generalized variable exponents operators

In recent years an increasing interest in more general operators generated by Musielak–Orlicz functions is under development since they provided, in principle, a unified treatment to deal with ordinary and partial differential equations with operators containing the p-Laplace operator, the ϕ-Laplace operator, operators with variable exponents and the double phase operators. These kind of consideration lead us in García-Huidobro et al. (2024), to consider problems containing the operator (S(t,u′))′, where ′=ddt and look for period solutions of systems of nonlinear systems of differential equations. In this paper we extend our approach to deal with systems of differential equations containing the operator (S(t,u′))′ this time under Dirichlet, mixed and Neumann boundary conditions. As in García-Huidobro et al. (2024) our approach is to work in C1 spaces to obtain suitable abstract fixed points theorems from which several applications are obtained, including problems of Liénard and Hartman type.

Retarded and advanced Green functions for first-order differential equations

Retarded and advanced Green functions are examined through the lens of linear first-order ordinary differential equations. They are constructed for a finite interval with emphasis on the imposition of homogeneous Dirichlet boundary conditions at the interval ends. It is shown that both types of Green functions exhibit the property of facilitating solutions.