Inhomogeneous Boundary Value Problem Research Articles

Impulsive Boundary Value Problems for Planar Hamiltonian Systems

We give an existence and uniqueness theorem for solutions of inhomogeneous impulsive boundary value problem (BVP) for planar Hamiltonian systems. Green's function that is needed for representing the solutions is obtained and its properties are listed. The uniqueness of solutions is connected to a Lyapunov type inequality for the corresponding homogeneous BVP.

An inhomogeneous boundary value problem for the stationary motion equations of Jeffreys viscoelastic medium

We consider the inhomogeneous Dirichlet problem for the stationary motion equations of an incompressible viscoelastic medium of Jeffreys type. We prove the existence of weak solutions of this problem. Moreover, we show that the weak solution set is sequentially weakly closed.

An electromechanically coupled micro-sphere framework: application to the finite element analysis of electrostrictive polymers

The number of industrial applications of electroactive polymers (EAPs) is increasing and, consequently, the need for reliable modelling frameworks for such materials as well as related robust simulation techniques continuously increases. In this context, we combine the modelling of non-linear electroelasticity with a computational micro-sphere formulation in order to simulate the behaviour of EAPs. The micro-sphere approach in general enables the use of physics-based constitutive models like, for instance, the so-called worm-like chain model. By means of the micro-sphere formulation, scalar-valued micromechanical constitutive relations can conveniently be extended to a three-dimensional continuum setting. We discuss several electromechanically coupled numerical examples and make use of the finite element method to solve inhomogeneous boundary value problems. The incorporated material parameters are referred to experimental data for an electrostrictive polymer. The numerical examples show that the coupled micro-sphere formulation combined with the finite element method results in physically sound simulations that mimic the behaviour of an electrostrictive polymer.

Asymptotic analysis of boundary value and spectral problems in thin perforated regions with rapidly changing thickness and different limiting dimensions

Boundary value and spectral problems for an elliptic differential equation with rapidly oscillating coefficients in a thin perforated region with rapidly changing thickness are investigated. Descriptions of asymptotic algorithms for solutions of such problems in thin regions with different limiting dimensions are combined. For a mixed inhomogeneous boundary value problem a corrector is constructed and an asymptotic estimate in the corresponding Sobolev space is established. Asymptotic bounds for eigenvalues and eigenfunctions of the Neumann spectral problems are also found. Full asymptotic expansions for the eigenvalues and eigenfunctions are constructed under certain symmetry assumptions about the structure of the thin perforated region and the coefficients of the equations.Bibliography: 21 titles.

Spectral Method for Mixed Inhomogeneous Boundary Value Problems in Three Dimensions

In this paper, we investigate spectral method for mixed inhomogeneous boundary value problems in three dimensions. Some results on the three-dimensional Legendre approximation in Jacobi weighted Sobolev space are established, which improve and generalize the existing results, and play an important role in numerical solutions of partial differential equations. We also develop a lifting technique, with which we could handle mixed inhomogeneous boundary conditions easily. As examples of applications, spectral schemes are provided for three model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms is proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy, and confirm the theoretical analysis well.

Modified Parseval and Carleman equalities for Sturm-Liouville problems with interior singularities

We develop a spectral method to solve inhomogeneous boundary-value problems with singular interior point. Particularly, we derive an eigenfunction expansion formula for the Green's function for which the modified Parseval and Carleman equalities are found.

Domain Decomposition Spectral Method for Mixed Inhomogeneous Boundary Value Problems of High Order Differential Equations on Unbounded Domains

In this paper, we develop domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations defined on unbounded domains. We introduce an orthogonal family of new generalized Laguerre functions, with the weight function x ? , ? being any real number. The corresponding quasi-orthogonal approximation and Gauss-Radau type interpolation are investigated, which play important roles in the related spectral and collocation methods. As examples of applications, we propose the domain decomposition spectral methods for two fourth order problems, and the spectral method with essential imposition of boundary conditions. The spectral accuracy is proved. Numerical results demonstrate the effectiveness of suggested algorithms.

Electrostriction in electro-viscoelastic polymers

Electrostrictive polyurethane (PU) elastomers belong to the class of materials commonly referred to as electroactive polymers (EAP). These materials have their use in a variety of applications, including biomimetics and microrobotics where traditional engineering components may fall short. PU is one of the materials considered for so called dielectric actuators, where the electromechanical response is generally due to Coulomb forces on the actuator electrodes, giving rise to a compressive pressure and thereby large deformations of the polymer. On the other hand, for more moderate electric fields, which may be more attractive in certain applications, a large part of the electroactive response for PU elastomers is due to inherent electrostriction. The latter phenomena is the focus of this work. As is common in elastomers, PU elastomers are viscoelastic. A coupled electro-viscoelastic phenomenological constitutive model for electrostrictive PU is proposed and fitted to experimental data available in the literature. The possibility of performing simulations of EAP is of interest as the number of applications grow. Considering this, the computational model is embedded in a coupled finite element formulation and, based on this, representative simulations of inhomogeneous boundary value problems are presented.

Weak and entropy solutions for a class of nonlinear inhomogeneous Neumann boundary value problem with variable exponent

(N≥ 3) is a bounded open domain with smooth boundary and ηis the unitoutward normal on ∂Ω.The study of various mathematical problems with variable exponent has recieved considerableattention in recent years (see [4,7,8-15,17,19,24-27,29,30,33,34]). These problems concern applica-tions (see [21,22,31,32,35]) and raise many difficult mathematical problems.The operator −div a(x,∇u) is called p(x)-Laplace, which becomes p-Laplace when p(x) ≡ p(aconstant). It possesses more complicated nonlinearities than the p-Laplace. For related resultsinvolving the p-Laplace, see [2,3]. In [2], the authors studied the problem−div a(x,∇u)+γ(u)∋ φin Ω,a(x,∇u).η+β(u)∋ ψ on ∂Ω,(1.2)where ηis the unit outward normal on ∂Ω, ψ∈ L

Solvability of inhomogeneous boundary-value problems for fourth-order differential equations

We consider a Cauchy-type boundary-value problem, a problem with three boundary conditions, and the Dirichlet problem for a general typeless fourth-order differential equation with constant complex coefficients and nonzero right-hand side in a bounded domain Ω ⊂ R 2 with smooth boundary. By the method of the Green formula, the theory of extensions of differential operators, and the theory of L-traces (i.e., traces associated with the differential operation L), we establish necessary and sufficient (for elliptic operators) conditions of the solvability of each of these problems in the space H m (Ω), m ≥ 4.

Asymptotic analysis of boundary-value problems in thin perforated domains with rapidly varying thickness

For a second-order symmetric uniformly elliptic differential operator with rapidly oscillating coefficients, we study the asymptotic behavior of solutions of a mixed inhomogeneous boundary-value problem and a spectral Neumann problem in a thin perforated domain with rapidly varying thickness. We obtain asymptotic estimates for the differences between solutions of the original problems and the corresponding homogenized problems. These results were announced in Dopovidi Akademii Nauk Ukrainy, No. 10, 15‐19 (1991). The new results obtained in the present paper are related to the construction of an asymptotic expansion of a solution of a mixed homogeneous boundary-value problem under additional assumptions of symmetry for the coefficients of the operator and for the thin perforated domain.

Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems on polygons

The authors investigate Petrov-Galerkin spectral element method. Some results on Legendre irrational quasi-orthogonal approximations are established, which play important roles in Petrov-Galerkin spectral element method for mixed inhomogeneous boundary value problems of partial differential equations defined on polygons. As examples of applications, spectral element methods for two model problems, with the spectral accuracy in certain Jacobi weighted Sobolev spaces, are proposed. The techniques developed in this paper are also applicable to other higher order methods.

Inhomogeneous boundary value problem for nonstationary compressible Navier–Stokes equations

The existence of global weak renormalized solutions to the evolution flow problems for compressible Navier-Stokes equations is established. The in/out flow problem in a bounded domain in three spatial dimensions is considered. A general mathematical theory for the flow problem is developed. Bibliography: 15 titles.

Elastic lateral-torsional buckling of restrained web-tapered I-beams

This paper presents an investigation on the elastic lateral-torsional buckling behaviour of discretely restrained tapered beams. Through a model problem consisting of a doubly symmetric web-tapered I-section cantilever acted by a tip load, it is shown how the effects of linearly elastic or rigid restraints can be included in the one-dimensional model previously developed by the authors to characterize the elastic buckling behaviour of thin-walled tapered beams. The restraints may have a translational, torsional, minor axis bending and/or warping character. The resulting self-adjoint eigenproblem is cast in non-dimensional form over a fixed reference domain, a process that leads to the identification of a complete set of independent non-dimensional parameters. It is then reformulated as a standard inhomogeneous boundary value problem and solved numerically with the collocation package colnew. A parametric study is carried out in order to examine in some detail (i) the effectiveness of different types of restraint, (ii) the influence of the restraint stiffness and (iii) the interplay between these two aspects and the degree of web tapering. This study provides insight into the peculiarities of the lateral-torsional buckling behaviour of tapered beams and some seemingly paradoxical results are given a physical explanation.

An Approximate Optimal Damping Control Method for Nonlinear Time-delay Systems with Disturbances

The optimal damping control for nonlinear time-delay systems with persistent disturbances is considered. Based on successive approximation approach (SAA), the optimal damping control (ODC) law is achieved by solving a decoupled sequence of inhomogeneous linear two-point boundary value (TPBV) problems without time-delay and time-advance terms. The ODC law of the original problem consists of accurate state feedback term, disturbance rejection term and a nonlinear time-delay compensation term, which is the limit of the adjoint vector sequence. By using the finite-time iteration of the compensation sequence, we can obtain an approximate optimal disturbance rejection control law. The proposed algorithms not only solve optimal control problems in the nonlinear time-delay system but also reduce the computation time and improve the precision. Numerical examples are included to illustrate the procedures.

Inhomogeneous boundary value problem for Hartree-type equation

In this paper, we settle the problem for time-dependent Hartree equation with inhomogeneous boundary condition in a bounded smooth domain in RN. A global existence and uniqueness result for H1 solution is derived.

On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition on the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Ω = Ω 0 × ( 0 , L ) ∈ R 3 . We show existence of a solution ( v , ρ ) ∈ W p 2 ( Ω ) × W p 1 ( Ω ) , p > 3 , where v is the velocity of the fluid and ρ is the density, that is a small perturbation of a constant flow ( v ¯ ≡ [ 1 , 0 , 0 ] , ρ ¯ ≡ 1 ). We also show that this solution is unique in a class of small perturbations of ( v ¯ , ρ ¯ ) . The term u ⋅ ∇ w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence ( v n , ρ n ) that is bounded in W p 2 ( Ω ) × W p 1 ( Ω ) and satisfies the Cauchy condition in a larger space L ∞ ( 0 , L ; L 2 ( Ω 0 ) ) what enables us to deduce that the weak limit of a subsequence of ( v n , ρ n ) is in fact a strong solution to our problem.

Matrix decomposition RBF algorithm for solving 3D elliptic problems

In this study, we propose an efficient algorithm for the evaluation of the particular solutions of three-dimensional inhomogeneous elliptic partial differential equations using radial basis functions. The collocation points are placed on concentric spheres and thus the resulting global matrix possesses a block circulant structure. This structure is exploited to develop an efficient matrix decomposition algorithm for the solution of the resulting system. Further savings in the matrix decomposition algorithm are obtained by the use of fast Fourier transforms. The proposed algorithm is used, in conjunction with the method of fundamental solutions for the solution of three-dimensional inhomogeneous elliptic boundary value problems.

Inhomogeneous boundary value problems for compressible Navier–Stokes and transport equations

In the paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. The well-posedness for inhomogeneous boundary value problems of elliptic–hyperbolic type is shown.

Boundary-value problems for differential equations in a Banach space

We establish a criterion for the existence of solutions of linear inhomogeneous boundary-value problems in a Banach space. We obtain conditions for the normal solvability of such problems and consider their special cases, namely, countable-dimensional boundary-value problems.