Shooting methods for a PT-symmetric periodic eigenvalue problem

We present a rigorous analysis of the performance of some one-step discretization schemes for a class of PT-symmetric singular boundary eigenvalue problem which encompasses a number of different problems whose investigation has been inspired by the 2003 article of Benilov et al. (J Fluid Mech 497:201–224, 2003). These discretization schemes are analyzed as initial value problems rather than as discrete boundary problems, since this is the setting which ties in most naturally with the formulation of the problem which one is forced to adopt due to the presence of an interior singularity. We also devise and analyze a variable step scheme for dealing with the singular points. Numerical results show better agreement between our results and those obtained from small-ϵ asymptotics than has been shown in results presented hitherto.     

Construction of Solutions and L^1-error Estimates of Viscous Methods for Scalar Conservation Laws with Boundary

This paper is concerned with an initial boundary value problem for strictly convex conservation laws whose weak entropy solution is in the piecewise smooth solution class consisting of finitely many discontinuities. By the structure of the weak entropy solution of the corresponding initial value problem and the boundary entropy condition developed by Bardos-Leroux Nedelec, we give a construction method to the weak entropy solution of the initial boundary value problem. Compared with the initial value problem, the weak entropy solution of the initial boundary value problem includes the following new interaction type： an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary. According to the structure and some global estimates of the weak entropy solution, we derive the global L^1-error estimate for viscous methods to this initial boundary value problem by using the matching travelling wave solutions method. If the inviscid solution includes the interaction that an expansion wave collides with the boundary and the boundary reflects a new shock wave which is tangent to the boundary, or the inviscid solution includes some shock wave which is tangent to the boundary, then the error of the viscosity solution to the inviscid solution is bounded by O（ε^1/2） in L^1-norm; otherwise, as in the initial value problem, the L^1-error bound is O（ε｜ In ε｜）.     

State observability of elastic string vibrations under the boundary conditions of the first kind

We solve state observation problems for string vibrations, i.e., problems in which the initial conditions generating the observed string vibrations should be reconstructed from a given string state at two distinct time instants. The observed vibrations are described by the boundary value problem for the wave equation with homogeneous boundary conditions of the first kind. The observation problem is considered for classical and L ₂-generalized solutions of this boundary value problem.     

Soliton solutions of an integrable boundary problem on the half-line for the discrete toda chain

We write formulas for soliton solutions of the discrete Toda chain and pose the integrable boundary value problem for this chain. We find conditions for the parameters (discrete spectrum points, transmission coefficients, and the corresponding factors) whereby solutions of the integrable boundary value problem are selected from all soliton solutions. As a result, we construct two hierarchies of soliton solutions of the specified problem with even and odd soliton numbers and find an explicit form of the conditions for the parameters. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 148, No. 3, pp. 387–397, September, 2006.     

Control-theoretic properties of structural acoustic models with thermal effects,I. Singular estimates

We consider a class of structural acoustics models with thermoelastic flexible wall. More precisely, the PDE system consists of a wave equation (within an acoustic chamber) which is coupled to a system of thermoelastic plate equations with rotational inertia; the coupling is strong as it is accomplished via boundary terms. Moreover, the system is subject to boundary thermal control. We show that—under three different sets of coupled (mechanical/thermal) boundary conditions—the overall coupled system inherits some specific regularity properties of its thermoelastic component, as it satisfies the same singular estimates recently established for the thermoelastic system alone. These regularity estimates are of central importance for (i) well-posedness of Differential and Algebraic Riccati equations arising in the associated optimal control problems, and (ii) existence of solutions to the semilinear initial/boundary value problem under nonlinear boundary conditions. The proof given uses as a critical ingredient a sharp trace theorem pertaining to second-order hyperbolic equations with Neumann boundary data.     

Explicit solutions of an integrable boundary value problem for the two-dimensional toda lattice

We obtain an explicit solution of the integrable boundary value problem for the two-dimensional Toda lattice using the inverse scattering method. We interpret the integrability property in terms of the corresponding linear problem, the Gel’fand-Levitan-Marchenko equation, and the dressing procedure. The simplest initial solutions of the boundary value problem become new nontrivial solutions after the dressing procedure is applied.     

The 3D compressible Euler equations with damping in a bounded domain

We proved global existence and uniqueness of classical solutions to the initial boundary value problem for the 3D damped compressible Euler equations on bounded domain with slip boundary condition when the initial data is near its equilibrium. Time asymptotically, the density is conjectured to satisfy the porous medium equation and the momentum obeys to the classical Darcy's law. Based on energy estimate, we showed that the classical solution converges to steady state exponentially fast in time. We also proved that the same is true for the related initial boundary value problem of porous medium equation and thus justified the validity of Darcy's law in large time.     

Solutions of the Navier–Stokes equations with various types of boundary conditions

In this paper we consider the initial boundary value problem of the Navier–Stokes system with various types of boundary conditions. We study the global-in-time existence and uniqueness of a solution of this system. In particular, suppose that the problem is solvable with some given data (the initial velocity and the external body force). We prove that there exists a unique solution for data which are small perturbations of the previous ones.     

Well-posedness of the IBVP for 2-D Euler equations with damping

In this paper we focus on the initial-boundary value problem of the 2-D isentropic Euler equations with damping. We prove the global-in-time existence of classical solution to the initial-boundary value problem for small smooth initial data by the method of local existence of solution combined with a priori energy estimates, where the appropriate boundary condition plays an important role.     

环肋加劲圆柱壳在静水压力作用下的初始后屈曲分析

本文用Koiter理论分析环肋加劲圆柱壳在静水压力作用下的后屈曲性能.前屈曲状态采用与边界条件一致的非线性有矩方程,本征值问题的解用伽辽金方法求出,得到的临界载荷与经典线性解作了比较.具体计算了三种不同环肋参数的外肋加劲圆柱壳.结果表明,肋的强弱不仅显著影响临界载荷值,同时也改变了柱壳的缺陷敏感度.     

On a family of initial-boundary value problems for the heat equation

We consider the heat problem with nonlocal boundary conditions containing a real parameter. For the zero value of the parameter, this problem is well known as the Samarskii-Ionkin problem and has been comprehensively studied. We analyze the spectral problem for the operator of second derivative subjected to the boundary conditions of the original problem. By separation of variables, we prove the existence and uniqueness of a classical solution for any nonzero value of the parameter. The obtained a priori estimates for a solution imply the stability of the problem with respect to the initial data.     

Continuité et différentiabilité d'Éléments propres: Application à l'optimisation de structures

The buckling load of a structure may usually be computed with an eigenvalue problem: it is the eigenvalue of smallest absolute value. In optimizing structures with a constraint on the buckling load, repeated eigenvalues are likely to occur. We prove continuity and differentiability results of eigenelements with respect to design variables using the variational characterization of eigenvalues. We illustrate these results with a classical problem: buckling of a beam. Application to arch buckling is presented in another article.     

Simple layer potential and the first boundary value problem for a parabolic system on the plane

By the method of boundary integral equations, we construct a classical solution of the first initial–boundary value problem for a one-dimensional (with respect to x) parabolic system in a domain with nonsmooth lateral boundary for the case in which the right-hand sides of the boundary conditions only have continuous derivatives of order 1/2. We study the smoothness of the solution.     

Specific features of application of perturbation techniques in problems of nonlinear oscillations of a liquid with free surface in cavities of noncylindrical shape

We consider the problem of nonlinear oscillations of an ideal incompressible liquid in a tank of a body-of-revolution shape. It is shown that the ordinary way of application of perturbation techniques results in the violation of solvability conditions of the problem. To avoid this contradiction we introduce some additional conditions and revise previously used approaches. We construct a discrete nonlinear model of the investigated problem on the basis of the Hamilton-Ostrogradskii variational formulation of the mechanical problem, preliminarily satisfying the kinematic boundary conditions and solvability conditions of the problem. Numerical examples testify to the efficiency of the constructed model. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 1, pp. 44–70, January, 2007.     

Stability of some boundary value methods for the solution of initial value problems

The stability properties of three particular boundary value methods (BVMs) for the solution of initial value problems are considered. Our attention is focused on the BVMs based on the midpoint rule, on the Simpson method and on an Adams method of order 3. We investigate their BV-stability regions by considering the scalar test problem and constant stepsize. The study of the conditioning of the coefficient matrix of the discrete problem is extended to the case of variable stepsize and block ODE problems. We also analyse an appropriate choice for the stepsize for stiff problems. Numerical tests are reported to evidentiate the effectiveness of the BVMs and the differences among the BVMs considered.Work supported by the Ministero della Ricerca Scientifica, 40% project, and C.N.R. (contract of research # 92.00535.01).     

Convergence analysis of a conforming adaptive finite element method for an obstacle problem

The adaptive algorithm for the obstacle problem presented in this paper relies on the jump residual contributions of a standard explicit residual-based a posteriori error estimator. Each cycle of the adaptive loop consists of the steps ‘SOLVE’, ‘ESTIMATE’, ‘MARK’, and ‘REFINE’. The techniques from the unrestricted variational problem are modified for the convergence analysis to overcome the lack of Galerkin orthogonality. We establish R-linear convergence of the part of the energy above its minimal value, if there is appropriate control of the data oscillations. Surprisingly, the adaptive mesh-refinement algorithm is the same as in the unconstrained case of a linear PDE—in fact, there is no modification near the discrete free boundary necessary for R-linear convergence. The arguments are presented for a model obstacle problem with an affine obstacle χ and homogeneous Dirichlet boundary conditions. The proof of the discrete local efficiency is more involved than in the unconstrained case. Numerical results are given to illustrate the performance of the error estimator.     

Numerical homogenization of foam-like structures based on the FE2-approach

The computation of foam–like structures is still a topic of research. There are two basic approaches: the microscopic model where the foam–like structure is entirely resolved by a discretization (e.g. with Timoshenko beams) on a micro level, and the macroscopic approach which is based on a higher order continuum theory. A combination of both of them is the FE²-approach where the mechanical parameters of the macroscopic scale are obtained by solving a Dirichlet boundary value problem for a representative microstructure at each integration point. In this contribution, we present a two–dimensional geometrically nonlinear FE²-framework of first order (classical continuum theories on both scales) where the microstructures are discretized by continuum finite elements based on the p-version. The p-version elements have turned out to be highly efficient for many problems in structural mechanics. Further, a continuum–based approach affords two additional advantages: the formulation of geometrical and material nonlinearities is easier, and there is no problem when dealing with thicker beam–like structures. In our numerical example we will investigate a simple macroscopic shear test. Both the macroscopic load displacement behavior and the evolving anisotropy of the microstructures will be discussed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Runge-Kutta methods without order reduction for linear initial boundary value problems

Summary. It is well-known the loss of accuracy when a Runge–Kutta method is used together with the method of lines for the full discretization of an initial boundary value problem. We show that this phenomenon, called order reduction, is caused by wrong boundary values in intermediate stages. With a right choice, the order reduction can be avoided and the optimal order of convergence in time is achieved. We prove this fact for time discretizations of abstract initial boundary value problems based on implicit Runge–Kutta methods. Moreover, we apply these results to the full discretization of parabolic problems by means of Galerkin finite element techniques. We present some numerical examples in order to confirm that the optimal order is actually achieved. Received July 10, 2000 / Revised version received March 13, 2001 / Published online October 17, 2001     

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier-Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x|^−μ, μ ∈ (1/2,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1, n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles. Alla memoria di Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 147–202.