Nonhomogeneous Boundary Value Problems Research Articles

A nonhomogeneous boundary value problem for the Kuramoto–Sivashinsky equation in a quarter plane

We study the initial boundary value problem for the one‐dimensional Kuramoto–Sivashinsky equation posed in a half line with nonhomogeneous boundary conditions. Through the analysis of the boundary integral operator, and applying the known results of the Cauchy problem of the Kuramoto–Sivashinsky equation posed on the whole line , the initial boundary value problem of the Kuramoto–Sivashinsky equation is shown to be globally well‐posed in Sobolev space for any s>−2. Copyright © 2017 John Wiley & Sons, Ltd.

Mathematical modelling of the axially moving panels subjected to thermomechanical actions

ABSTRACTThe paper is devoted to a stability and out-of-plane deformation analysis of an axially moving elastic web modelled as a panel (a plate undergoing cylindrical deformation). The panel is under homogeneous pure mechanical in-plane tension and thermal strains corresponding to the thermal tension and bending. In accordance with the static approach of stability analysis the problem of out-of-plane thermomechanical divergence (buckling) is reduced to an eigenvalue problem which is analytically solved. This problem corresponds to the case of in-plane thermomechanical tension and zero thermal bending. The general case of deformations induced by combined thermomechanical bending and tension is reduced to nonhomogeneous boundary-value problem and analyzed with the help of Fourier series.

Layer Potentials and Boundary-Value Problems for Second Order Elliptic Operators with Data in Besov Spaces

This monograph presents a comprehensive treatment of second order divergence form elliptic operators with bounded measurable t-independent coefficients in spaces of fractional smoothness, in Besov and weighted L classes. We establish: (1) Mapping properties for the double and single layer potentials, as well as the Newton potential; (2) Extrapolation-type solvability results: the fact that solvability of the Dirichlet or Neumann boundary value problem at any given L space automatically assures their solvability in an extended range of Besov spaces; (3) Well-posedness for the non-homogeneous boundary value problems. In particular, we prove well-posedness of the non-homogeneous Dirichlet problem with data in Besov spaces for operators with real, not necessarily symmetric coefficients. 2010 Mathematics Subject Classification. Primary 35J25, Secondary 31B20, 35C15, 46E35.

Nonhomogeneous boundary value problem for Navier–Stokes equations in 2D symmetric unbounded domains

In this paper, we study the nonhomogeneous boundary value problem for the stationary Navier–Stokes equations in two-dimensional symmetric domains with finitely many outlets to infinity. The domains may have no self-symmetric outlet (-type domain), one self-symmetric outlet (-type domain) or two self-symmetric outlets (-type domain). We construct a symmetric solenoidal extension of the boundary value satisfying the Leray–Hopf inequality. After having such an extension, the nonhomogeneous boundary value problem is reduced to homogeneous one and the existence of at least one weak solution follows. Notice that we do not impose any restrictions on the size of the fluxes over the inner and outer boundaries. Moreover, the Dirichlet integral of the solution can be either finite or infinite depending on the geometry of the domains.

On nonlinear thermo-electro-elasticity.

Electro-active polymers (EAPs) for large actuations are nowadays well-known and promising candidates for producing sensors, actuators and generators. In general, polymeric materials are sensitive to differential temperature histories. During experimental characterizations of EAPs under electro-mechanically coupled loads, it is difficult to maintain constant temperature not only because of an external differential temperature history but also because of the changes in internal temperature caused by the application of high electric loads. In this contribution, a thermo-electro-mechanically coupled constitutive framework is proposed based on the total energy approach. Departing from relevant laws of thermodynamics, thermodynamically consistent constitutive equations are formulated. To demonstrate the performance of the proposed thermo-electro-mechanically coupled framework, a frequently used non-homogeneous boundary-value problem, i.e. the extension and inflation of a cylindrical tube, is solved analytically. The results illustrate the influence of various thermo-electro-mechanical couplings.

A Nonhomogeneous Boundary Value Problem for the Boussinesq Equation on a Bounded Domain

In this paper, we study the well-posedness of an initial-boundary-value problem (IBVP) for the Boussinesq equation on a bounded domain, utt−uxx+(u 2 )xx+uxxxx=0, x∈(0,1), t>0, u(x,0)= ϕ(x), ut(x,0)= ψ(x), u(0,t)=h1(t), u(1,t)=h2(t), uxx(0,t)=h3(t), uxx(1,t)=h4(t). It is shown that the IBVP is locally well-posed in the space H s (0,1) for any s≥0 with the initial data ϕ, ψ lie in H s (0,1) and H s−2 (0,1), respectively, and the naturally com- patible boundary data h1, h2 in the space H (s+1)/2 loc (R + ), and h3, h4 in the the space of H (s−1)/2 loc (R + ) with optimal regularity.

A new approach to non-homogeneous hyperbolic boundary value problems using hybrid-Trefftz stress finite elements

A new approach to the solution of non-homogeneous hyperbolic boundary value problems is casted here using the hybrid-Trefftz stress/flux elements. Similarly to the Dual Reciprocity Method, the technique adopted in this paper uses a Trefftz-compliant set of functions to approximate the complementary solution of the problem and an additional trial basis to approximate its particular solution. However, the particular and complementary solutions are obtained here in a single step, and not sequentially, as typical of the Dual Reciprocity Method. The trial functions used for both particular and complementary solutions are merged together in the same basis and offered full flexibility to combine so as to recover the enforced equations in the best possible way. This option enables Trefftz-compliant functions to compensate for deficiencies of the particular solution basis, meaning that accurate total solutions can be obtained with relatively poor particular solution approximations. The formulation preserves the Hermitian, sparse and localized structure that typifies the matrix of coefficients of hybrid-Trefftz finite elements and avoids the drawbacks of the collocation procedures that arise in the Dual Reciprocity Method. Moreover, all terms of the matrix of coefficients are reduced to boundary integral expressions provided the particular solution trial functions satisfy the static problem obtained after discarding both non-homogeneous and time derivative terms from the governing equation.

A finite element method for computing accurate solutions for Poisson equations with corner singularities using the stress intensity factor

In this article, we consider the Poisson equation with homogeneous Dirichlet boundary conditions, on a polygonal domain with one reentrant corner. The solution of the Poisson equation with a concave corner yields a singular decomposition, u=w+ληs, where w is regular, s is a singular function, and the coefficient λ is the so called stress intensity factor. This stress intensity factor can be computed using the extraction formula. We introduce a new non-homogeneous boundary value problem, which has ‘zero’ stress intensity factor. Using the solution of this new partial differential equation, we can compute an accurate solution of the original problem, simply by adding singular part. We obtain an optimal convergence rate with smaller errors when compared with others.

Bending Deflection Analysis of a Semi-Trailer Chassis by Using Symmetric Smoothed Particle Hydrodynamics

In this paper, a simple approach is presented for the calculation of bending deflection of a semi trailer chassis. The 3D model of the chassis is used to determine the moment of inertia of the cross section function and then the mathematical model of the chassis is presented as a Euler Bernoulli Beam which has the variable cross section. Different loading conditions coming from the semi trailer test procedures are applied. The bending deflections of the semi trailer chassis are numerically calculated by using the Symmetric Smoothed Particle Hydrodynamics (SSPH) method. The first time the performance of the SSPH method for the fourth order non-homogeneous variable coefficents linear boundary value problems is evaluated. For the calculations different number of terms in the TSEs are employed when the number of nodes in the problem domain increases. The comparisons are made with the results of experiments. It is observed that the SSPH method has the conventional convergence properties yields smaller L 2 error. Finally, the approach presented here may be used and found reliable for the calculation of deflection of the semi trailer chassis before the release of detail design.

Fourth order elliptic operator-differential equations with unbounded operator boundary conditions in the Sobolev-type spaces

Conditions for well-posed and unique solvability of a non-homogeneous boundary value problem for a class of fourth order elliptic operator-differential equations with an unbounded operator in boundary conditions are found in this work. Note that these solvability conditions are sufficient, and they are expressed only in terms of the properties of operator coefficients of the boundary value problem. Besides, the estimates for the norms of intermediate derivative operators in a Sobolev-type space are obtained, and their close relationship with the solvability conditions is established.

On Nonhomogeneous Boundary Value Problem for the Steady Navier-StokesSystem in Domain with Paraboloidal and Layer Type Outlets to Infinity

The nonhomogeneous boundary value problem for the steady Navier-Stokes system is studied in a domain $\Omega$ with two layer type and one paraboloidaloutlets to infinity. The boundary$\partial\Omega$ is multiply connected and consists ofthe outer boundary $S$ and the inner boundary $\Gamma$. The boundary value ${a}$ is assumed to have a compact support. The flux of ${a}$ over the inner boundary $\Gamma$is supposed to be sufficiently small. We do not impose any restrictions on fluxesof ${a}$ over the unbounded components of the outer boundary $S$. Theexistence of at least one weak solution is proved.

Congestion Control for Nonlinear TD-SCDMA Discrete Networks Based on TCP/IP

A successive approximation approach (SAA) is developed to obtain a new congestion controller for the nonlinear TD-SCDMA network control systems based on TCP/IP. By using the successive approximation approach, the original optimal control problem is transformed into a sequence of nonhomogeneous linear two-point boundary value (TPBV) problems. The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term that is the limit of the solution sequence of the adjoint vector differential equations. By using the finite-time iteration of nonlinear compensation term of optimal solution sequence, we can obtain a suboptimal control law for TD-SCDMA network control systems based on TCP/IP.

On nonhomogeneous boundary value problems for the stationary Navier–Stokes equations in two-dimensional symmetric semi-infinite outlets

We study the stationary nonhomogeneous Navier–Stokes problem in a two-dimensional symmetric domain with a semi-infinite outlet (for instance, either paraboloidal or channel-like). Under the symmetry assumptions on the domain, boundary value and external force, we prove the existence of at least one weak symmetric solution without any restriction on the size of the fluxes, i.e. the fluxes of the boundary value [Formula: see text] over the inner and the outer boundaries may be arbitrarily large. Only the necessary compatibility condition (the total flux is equal to zero) has to be satisfied. Moreover, the Dirichlet integral of the solution can be finite or infinite depending on the geometry of the domain.

Generalized HPC method for the Poisson equation

An efficient and innovative numerical algorithm based on the use of Harmonic Polynomials on each Cell of the computational domain (HPC method) has been recently proposed by Shao and Faltinsen (2014) [1], to solve Boundary Value Problem governed by the Laplace equation. Here, we extend the HPC method for the solution of non-homogeneous elliptic boundary value problems. The homogeneous solution, i.e. the Laplace equation, is represented through a polynomial function with harmonic polynomials while the particular solution of the Poisson equation is provided by a bi-quadratic function. This scheme has been called generalized HPC method. The present algorithm, accurate up to the 4th order, proved to be efficient, i.e. easy to be implemented and with a low computational effort, for the solution of two-dimensional elliptic boundary value problems. Furthermore, it provides an analytical representation of the solution within each computational stencil, which allows its coupling with existing numerical algorithms within an efficient domain-decomposition strategy or within an adaptive mesh refinement algorithm.

Multiple Solutions of Neumann Problems: An Orlicz–Sobolev Space Setting

In the present paper, we establish the range of two parameters for which a non-homogeneous boundary value problem admits at least three weak solutions. The proof of the main results relies on recent variational principles due to Ricceri.

An iterative method for suboptimal control of linear time-delayed systems

This article presents a new approach for solving the Optimal Control Problem (OCP) of linear time-delay systems with a quadratic cost functional. The proposed method can also be used for designing optimal control time-delay systems with disturbance. In this study, the Variational Iteration Method (VIM) is employed to convert the original Time-Delay Optimal Control Problem (TDOCP) into a sequence of nonhomogeneous linear two-point boundary value problems (TPBVPs). The optimal control law obtained consists of an accurate linear feedback term and a nonlinear compensation term which is the limit of an adjoint vector sequence. The feedback term is determined by solving Riccati matrix differential equation. By using the finite-step iteration of a nonlinear compensation sequence, we can obtain a suboptimal control law. Finally, Illustrative examples are included to demonstrate the validity and applicability of the technique.

Two constant sign solutions for a nonhomogeneous Neumann boundary value problem

Abstract We consider a nonlinear Neumann problem with a nonhomogeneous elliptic differential operator. With some natural conditions for its structure and some general assumptions on the growth of the reaction term we prove that the problem has two nontrivial solutions of constant sign. In the proof we use variational methods with truncation and minimization techniques.

Thermal Stresses Induced by a Variable Heat Source in a Rectangle and Variable Pressure at Its Boundary by Finite Fourier Transform

This paper deals with the two-dimensional, non-homogeneous boundary value problem for static, isotropic and thermoelastic material occupying an infinitely long cylinder with a rectangular cross-section. The cylinder is surrounded by a given temperature and subjected to variable pressures at its boundaries. We deal with static, uncoupled, linear thermoelasticity. The equations of heat conduction and mechanical problem are considered separately. The technique of the finite Fourier transform is used for the solution. The thermoelastic behavior, due to an internal heat generation within the domain, is discussed. The results for displacement and stresses have been computed from the Airy stress function and are illustrated graphically.

Solution method for a boundary value problem with fuzzy forcing function

In this paper, we present a new approach to a non-homogeneous fuzzy boundary value problem. We consider a linear differential equation with real coefficients but with a fuzzy forcing function and fuzzy boundary values. We assume that the forcing function is a triangular fuzzy function. Unlike previous studies, we look for a solution that is a fuzzy set of real functions (not a fuzzy-valued function). Each of these real functions satisfies the boundary value problem with some membership degree. We have developed a method that finds this solution, and demonstrated its effectiveness using a test example.To show that the approach can be extended to other types of fuzzy numbers, we extended it to the trapezoidal case. For a particular example, we used the product t-norm to demonstrate how a new solution type can be obtained.

The existence theorem for steady Navier-Stokes equations in the axially symmetric case

We study the nonhomogeneous boundary value problem for the Navier-Stokes equations of steady motion of a viscous incompressible fluid in a bounded three-dimensional domain with multiply connected boundary. We prove that this problem has a solution in some axially symmetric cases, in particular, when all components of the boundary intersect the axis of symmetry.