Acoustic scattering by a thin cylindrical screen with Dirichlet and impedance boundary conditions on opposite sides of the screen

A problem on scattering acoustic waves by a thin cylindrical screen is studied. In doing so, the Dirichlet condition is specified on one side of the screen, while the impedance boundary condition is specified on the other side of the screen. The solution of the problem is subject to the radiating condition at infinity and to the propagative Helmholtz equation. By using the potential theory the scattering problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original scattering problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Reissner厚板弹性弯曲的一般解析解

针对大型工程建设中的Reisner厚板弹性弯曲问题,本文采用复级数方法求解相应的常系数偏微分方程组的边值问题,并首次得到了任意边界条件下的一般解析解.该解形式简单,计算方便、可靠.以四边简支和三边固支一边自由两种支撑条件下厚板承受均布载荷为例进行了分析验算,与已有的计算结果相比,计算结果相当满意.同时本文还着重对解的收敛速度、正确性(合理性)及边界满足情况进行了考察.     

The harmonic dirichlet problem for a cracked domain with jump conditions on cracks

The harmonic problem in a cracked domain is studied in R ^m , m?>?2. The boundary of the domain is assumed to be nonsmooth, while cracks are smooth. The Dirichlet condition is specified on the boundary of the domain. Jumps of the unknown function and its normal derivative are specified on the cracks. Uniqueness and solvability results are obtained. The problem is reduced to the uniquely solvable integral equation, its solution is given explicitely in the form of a series. The estimates of the solution of the problem depending on the boundary data are obtained.     

An inverse problem for a family of two parameters time fractional diffusion equations with nonlocal boundary conditions

The determination of a space‐dependent source term along with the solution for a 1‐dimensional time fractional diffusion equation with nonlocal boundary conditions involving a parameter β>0 is considered. The fractional derivative is generalization of the Riemann‐Liouville and Caputo fractional derivatives usually known as Hilfer fractional derivative. We proved existence and uniqueness results for the solution of the inverse problem while over‐specified datum at 2 different time is given. The over‐specified datum at 2 time allows us to avoid initial condition in terms of fractional integral associated with Hilfer fractional derivative.     

Simulation of Rayleigh waves in cracked plates

The aim of this paper is to develop new numerical procedures to detect micro cracks, or superficial imperfections, in thin plates using excitation by Rayleigh waves. We shall consider a unilateral contact problem between the two sides of the crack in an elastic plate subjected to suitable boundary conditions in order to reproduce a single Rayleigh wave cycle. An approximate solution of this problem will be calculated by using one of the Newmark methods for time discretization and a finite element method for space discretization. To deal with the nonlinearity due to the contact condition, an iterative algorithm involving one multiplier will be used; this multiplier will be approximated by using Newton's techniques. Finally, we will show numerical simulations for both cracked and non‐cracked plates. Copyright © 2006 John Wiley & Sons, Ltd.     

Mindlin板条中弹性波传播问题的分析

基于Mindlin弹性厚板理论，采用Hamilton状态空间求解方法，研究了板条横向两侧为自由边界条件下的弹性：波传播问题，给出了问题的分析解，以及各种振动模态和波传播模的存在性．采用本征函数展开法给出了条形板中导波模的频散方程，并与直接根据Mindlin提出的解获得的频散关系进行了比较，通过比较发现：在短波时，频散曲线相差较大，即前几阶低频模式有一定差别。根据Hamilton体系得到的截止频率高一些，长波时频散曲线几乎相同，即高频模式无差别。     

The scattering of plane elastic waves by a one‐dimensional periodic surface

The two‐dimensional scattering problem for time‐harmonic plane waves in an isotropic elastic medium and an effectively infinite periodic surface is considered. A radiation condition for quasi‐periodic solutions similar to the condition utilized in the scattering of acoustic waves by one‐dimensional diffraction gratings is proposed. Under this condition, uniqueness of solution to the first and third boundary‐value problems is established. We then proceed by introducing a quasi‐periodic free field matrix of fundamental solutions for the Navier equation. The solution to the first boundary‐value problem is sought as a superposition of single‐ and double‐layer potentials defined utilizing this quasi‐periodic matrix. Existence of solution is established by showing the equivalence of the problem to a uniquely solvable second kind Fredholm integral equation. Copyright © 1999 John Wiley & Sons, Ltd.     

Combined Method for Calculating the Field Generated by a Point Source in a Waveguide. II

A two-dimensional problem on the propagation of waves that are excited by a point source in a waveguide is studied. The Dirichlet (Neumann) boundary condition is given on the upper (respectively, lower) boundary of the waveguide. The representation of the exact solution as a sum of geometric-optical waves, normal waves, and a remainder is given. Sufficient conditions on the total amount of detached normal and geometric-optical waves are obtained. The remainder is expressed by a simple formula. Bibliography: 19 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 297, 2003, pp. 116–135.     

Spectral problem with Steklov condition on a thin perforated interface

A two-dimensional Steklov-type spectral problem for the Laplacian in a domain divided into two parts by a perforated interface with a periodic microstructure is considered. The Steklov boundary condition is set on the lateral sides of the channels, a Neumann condition is specified on the rest of the interface, and a Dirichlet and Neumann condition is set on the outer boundary of the domain. Two-term asymptotic expansions of the eigenvalues and the corresponding eigenfunctions of this spectral problem are constructed.     

Stress state of freely supported multilayered elliptical plates of anisotropic materials

The problem of a stressed state in elliptic plates has been considered in general for a rigid contour fixation. It is much more difficult to obtain a solution for the freely supported plates, even for isotropic materials. In this paper we suggest an approach for defining the stressed state of thin elliptic plates with layered structure under the condition of a freely supported contour. The solution is obtained in a rectangular cartesian coordinate system. The displacements, which are the fundamental unknowns, are given in the form of polynomials with unknown coefficients defined by a system of algebraic equations. The resolving equations and three out of the four boundary conditions are satisfied precisely. One boundary condition, is satisfied by means of collocation method of separate points of the contour. Estimation of the accuracy of the suggested approach is carried out by comparing the obtained results with the known ones. The problem of deformation of a twolayered plate has been discussed, in which the principal direction of elasticity does not coincide with the coordinate directions.S. P. Timoshenko Institute of Mechanics, National Academy of Science of the Ukraine, Kiev. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 496–504, July–August, 1997. Original article submitted March 19     

Asymptotics of the solution of Laplace’s equation with third-type boundary conditions on the boundaries of two small holes

A boundary value problem for Laplace’s equation in a bounded domain with two small holes is considered. Third-type boundary conditions are set on the boundaries of the holes. A Neumann condition is specified on the outer boundary of the domain. A uniform asymptotic approximation of the solution is constructed and justified up to an arbitrary power of a small parameter.     

The Neumann problem for the Laplace equation on general domains

The solution of the weak Neumann problem for the Laplace equation with a distribution as a boundary condition is studied on a general open set G in the Euclidean space. It is shown that the solution of the problem is the sum of a constant and the Newtonian potential corresponding to a distribution with finite energy supported on ∂G. If we look for a solution of the problem in this form we get a bounded linear operator. Under mild assumptions on G a necessary and sufficient condition for the solvability of the problem is given and the solution is constructed. The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.     

Study of generalized eigenfunctions of a perturbed isotropic elastic half‐space

We consider the self‐adjoint operator governing the propagation of elastic waves in a perturbed isotropic half‐space (perturbation with compact support of a homogeneous isotropic half‐space) with a free boundary condition. We propose a method to obtain, numerical values included, a complete set of generalized eigenfunctions that diagonalize this operator. The first step gives an explicit representation of these functions using a perturbative method. The unbounded boundary is a new difficulty compared with the method used by Wilcox [25], who set the problem in the complement of bounded open set. The second step is based on a boundary integral equations method which allows us to compute these functions. For this, we need to determine explicitly the Green's function of (A₀ – ω²), where A₀ is the self‐adjoint operator describing elastic waves in a homogeneous isotropic half‐space. Copyright © 2000 John Wiley & Sons, Ltd.     

Scattering of the plane wave by a transparent wedge

In the paper it is proved that the problem of scattering of the plane wave by a transparent wedge has a unique solution, provided that the radiation condition should be meant in the following form: if one subtracts from the solution the incident wave and all reflected and refracted waves, then the remainder satisfies the radiation condition in integral form. The problem is scalar, the velocities of the wave inside and outside the wedge are not equal, the wave process is described by the classical Helmholtz equations, and the conjugation boundary condition is satisfied on the sides of the wedge. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 354, 2008, pp. 5–18.     

对称迭层矩形板的平面应力分析

常用的对称迭层板为各向异性板．根据平面应力问题的基本方程精确地用应力函数解法求得了各向异性板的一般解析解．推导出平面内应力和位移的一般公式,其中积分常数由边界条件来决定．一般解包括三角函数和双曲函数组成的解,它能满足4个边为任意边界条件的问题．还有代数多项式解,它能满足4个角的边界条件．因此一般解可用以求解任意边界条件下的平面应力问题．以4边承受均匀法向和切向载荷以及非均匀法向载荷的对称迭层方板为例,进行了计算和分析．     

The Third Problem for the Laplace Equation with a Boundary Condition from L p

The third problem for the Laplace equation is studied on an open set with Lipschitz boundary. The boundary condition is in L_p and it is fulfilled in the sense of the nontangential limit. The existence and the uniqueness of a solution is proved and the solution is expressed in the form of a single layer potential. For domains with C¹ boundary the explicit solution of the problem is calculated.     

Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition

An initial boundary-value problem for a quasilinear system of partial differential equations with a nonlocal boundary condition involving a delayed argument is considered. The existence of a unique solution to this problem is proved by reducing it to a system of nonlinear integral-functional equations. The inverse problem of finding a solution-dependent coefficient of the system from additional information on a solution component specified at a fixed point of space as a function of time is formulated. The uniqueness of the solution of the inverse problem is proved. The proof is based on the derivation and analysis of an integral-functional equation for the difference between two solutions of the inverse problem.     

Initial-boundary value problems for incomplete singular perturbations of hyperbolic systems

The general problem studied has as a prototype the full non-linear Navier-Stokes equations for a slightly viscous compressible fluid including the heat transfer. The boundaries are of inflow-outflow type, i.e. non-characteristic, and the boundary conditions are the most general ones with any order of derivatives. It is assumed that the uniform Lopatinsky condition is satisfied. The goal is to prove uniform existence and boundedness of solution as the viscosity tends to zero and to justify the boundary layer asymptotics. The paper consists of two parts. In Part I the linear problem is studied. Here, uniform lower and higher order tangential estimates are derived and the existence of a solution is proved. The higher order estimates depend on the smoothness of coefficients; however this smoothness does not exceed the smoothness of the solution. In Part II the quasilinear problem is studied. It is assumed that for zero viscosity the overall initial-boundary value problem has a smooth solutionu ₀ in a time interval 0≦t≦T ₀. As a result the boundary laye, is weak and is uniformlyC ¹ bounded. This makes the linear theory applicable. an iteration scheme is set and proved to converge to the viscous solution. The convergence takes place for small viscosity and over the original time interval 0≦t≦T ₀.     

Time Evolution of the Scattering Data for the Forced Toda Lattice

Determination of the time evolution of the scattering data for an inverse scattering transform solution of the forced Toda lattice appears to require an overspecification of the boundary condition at the end of the lattice. This appears in the form of an apparent need to specify the values of two functions at the boundary rather than one. We present three different approaches to the resolution of this problem. One approach gives the Maclaurin series (in time) for the scattering data. The second approach gives the scattering data in terms of the solution to a nonlinear, nonlocal partial differential equation. The third approach gives the scattering data in terms of the solution to a linear integral equation. All three approaches reduce to one the number of functions which must be specified to determine a solution. The advantages and limitations of each approach are discussed.     

Helmholtz equation outside an open arc in a plane with a mixed boundary condition

A boundary value problem for the Helmholtz equation outside an open arc in a plane is studied with mixed boundary conditions. In doing so, the Dirichlet condition is specified on one side of the open arc and the boundary condition of the third kind is specified on the other side of the open arc. We consider non-propagative Helmholtz equation, real-valued solutions of which satisfy maximum principle. By using the potential theory the boundary value problem is reduced to a system of singular integral equations with additional conditions. By regularization and subsequent transformations, this system is reduced to a vector Fredholm equation of the second kind and index zero. It is proved that the obtained vector Fredholm equation is uniquely solvable. Therefore the integral representation for a solution of the original boundary value problem is obtained. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)