Transfinite Thin Plate Spline Interpolation

Duchon’s method of thin plate splines defines a polyharmonic interpolant to scattered data values as the minimizer of a certain integral functional. For transfinite interpolation, i.e., interpolation of continuous data prescribed on curves or hypersurfaces, Kounchev has developed the method of polysplines, which are piecewise polyharmonic functions of fixed smoothness across the given hypersurfaces and satisfy some boundary conditions. Recently, Bejancu has introduced boundary conditions of Beppo–Levi type to construct a semicardinal model for polyspline interpolation to data on an infinite set of parallel hyperplanes. The present paper proves that, for periodic data on a finite set of parallel hyperplanes, the polyspline interpolant satisfying Beppo–Levi boundary conditions is in fact a thin plate spline, i.e., it minimizes a Duchon type functional. The construction and variational characterization of the Beppo–Levi polysplines are based on the analysis of a new class of univariate exponential ℒ-splines satisfying adjoint natural end conditions.     

A.V. Bitsadze's observation on bianalytic functions and the Schwarz problem

According to an observation of A.V. Bitsadze from 1948 the Dirichlet problem for bianalytic functions is ill-posed. A natural boundary condition for the polyanalytic operator, however, is the Schwarz condition. An integral representation for the solutions in the unit disc to the inhomogeneous polyanalytic equation satisfying Schwarz boundary conditions is known. This representation is extended here to any simply connected plane domain having a harmonic Green function. Some other boundary value problems are investigated with some Dirichlet and Neumann conditions illuminating that just the Schwarz problem is a natural boundary condition for the Bitsadze operator.     

一个与硅氧化有关的粘性可压缩流体的自由边界问题

本文研究一个描述硅的氧化过程的自由边界问题．它的数学模型是一个可压缩的Ｎａｖｉｏｒ－Ｓｔｏｋｅｓ方程与一个抛物方程以及一个双曲方程的耦合，其中在自由边界上存在表面张力并且密度方程是非齐次的．本文将证明只要已知数据满足相容性条件，则上述问题有唯一局部强解．     

Free boundary problem for a viscous compressible flow associated with oxidation of silicon

This paper is concerned with a free boundary problem describing the oxidation process of silicon. Its mathematical model is a compressible Navier-Stokes equations coupling a parabolic equation and a hyperbolic one. Surface tension is involved at the free boundary and density equation is non-homogeneous. It is proved that for arbitrary data satisfying only natural consistency conditions the problem is uniquely solvable on some finite time interval. Supported by National Natural Science Foundation of China     

The problem of the bending of thin orthotropic bars in threedimensional formulation

For the three-dimensional problem of the theory of elasticity of an orthotropic body we obtain solutions satisfying the equilibrium equations inside a bar and boundary conditions on its planar faces. These solutions make it possible to satisfy a given set of boundary conditions on lateral surface of the bar. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 27–34.     

Effect of hydrostatic pressure on vibrating functionally graded circular plate coupled with bounded fluid

This study investigates free vibration of a thick FG circular plate in contact with an inviscid, and incompressible fluid. Analysis of plate is based on First-order Shear Deformation Reissner–Mindlin Theory (FSDT) with consideration of rotational inertial effects and transverse shear stresses. Potential theory together Bernouli's equation are utilized to obtain the fluid pressure on the free surface of the plate. The governing equation of the oscillatory behavior of the fluid is obtained by solving Laplace equation and satisfying its boundary conditions. The natural frequencies and mode shapes of the plate are determined using Chebyshev polynomials. The effects of the geometrical parameters such as plate thickness to its radius ratio, boundary conditions, fluid density, volume fraction index, and height of the fluid on natural frequencies and mode shapes are investigated. Comparison of analytically outcome of this study is made with similar publications in the literature.     

Discrete kiguradze type inequalities

Solutions of nth order difference inequalities satisfying n boundary conditions of predetermined sign are compared to solutions of difference equalities satisfying n?1 homogenous boundary conditions. The inequalities obtained here apply to Green's functions of corresponding boundary value problems     

The method of homogeneous solutions in the axisymmetric static problem for a finite cylinder with homogeneous conditions on the lateral surface

The stresses and displacements in a finite cylinder with homogeneous conditions on the lateral surface can be represented in terms of a single function, the radial displacement. Using the generalized orthogonality property of P. A. Shiff for eigenfunctions, we develop a method of satisfying the boundary conditions at the ends of the cylinder and determine a class of boundary conditions that admit a solution without reduction to an infinite system of algebraic equations.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 73–77.     

Solution of a mixed boundary-value problem of thermodiffusion for layered bodies of canonical shape

We propose a method of solving coupled thermodiffusion problems for layered bodies of canonical shape. The method is based on separating the coupled system of equations and boundary conditions into independent boundary-value problems. The solution contains arbitrary functions of time determined from a system of second-order integral equations of convolution type obtained as a result of satisfying the boundary conditions.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 87–91.     

Buckling and free vibration analysis of high speed rotating carbon nanotube reinforced cylindrical piezoelectric shell

In this paper, buckling and free vibration behavior of a piezoelectric rotating cylindrical carbon nanotube-reinforced (CNTRC) shell is investigated. Both cases of uniform distribution (UD) and FG distribution patterns of reinforcements are studied. The accuracy of the presented model is verified with previous studies and also with those obtained by Navier analytical method. The novelty of this study is investigating the effects of critical voltage and CNT reinforcement as well as satisfying various boundary conditions implemented on the piezoelectric rotating cylindrical CNTRC shell. The governing equations and boundary conditions have been developed using Hamilton's principle and are solved with the aid of Navier and generalized differential quadrature (GDQ) methods. In this research, the buckling phenomena in the piezoelectric rotating cylindrical CNTRC shell occur as the natural frequency is equal to zero. The results show that, various types of CNT reinforcement, length to radius ratio, external voltage, angular velocity, initial hoop tension and boundary conditions play important roles on critical voltage and natural frequency of piezoelectric rotating cylindrical CNTRC shell.     

Extension of higher-order infinitesimal bendings of a simply connected surface of positive curvature with boundary conditions

We study continuous and infinitesimal bendings of orderk (k-bendings) of a simply connected surface of positive curvature with boundary conditions. We study the question of the extension ofk-bendings satisfying some boundary condition to continuous bendings satisfying an analogous boundary condition.Translated from Ukrainskií Geometricheskií Sbornik, No. 30, 1987, pp. 41–49.     

Boundary value problems with operator right-hand side

For a second-order boundary value problem with operator right-hand side and with functional boundary conditions, we prove solvability theorems with mixed and Dirichlet boundary conditions assuming the existence of a lower and an upper function. These theorems are analogs of theorems for the corresponding boundary value problems for an ordinary second-order differential equation with right-hand side satisfying the Carathéodory conditions.     

On boundary value problems for an <Emphasis Type="Italic">n</Emphasis>th-order equation

For an nth-order differential equation with right-hand side satisfying the Carathéodory conditions, we consider a two-point boundary value problem with boundary conditions of which the first two have a general form and the remaining conditions are simplest. We show that the existence of a lower and an upper function and the compactness condition imply the solvability of this boundary value problem.     

Functions with presribed singular values of the gradient

We prove the existence of infinitely many vector-valued Lipschitz-continuous functions u on an open set satisfying suitable Dirichlet boundary conditions such that the singular values of the gradient matrix u, i.e. the square roots of the eigenvalues of the symmetrized matrix , agree a.e. on with N given positive, bounded and lower semicontinuous functions. Received May 28, 1997     

可换四元数空间中某些双曲型方程的Riemann-Hilbert边值问题

考察了在可交换四元数空间(基元为1,i,j,k满足条件i~2=j~2=-1,ij=ji= k)中的某些双曲型方程的Riemann-Hilbert边值问题,分别在不同的情况下获得了问题的可解条件和通解.     

Application of reproducing kernel method to third order three-point boundary value problems

This paper investigates the analytical approximate solutions of third order three-point boundary value problems using reproducing kernel method. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel method can not be used directly to solve third order three-point boundary value problems, since there is no method of obtaining reproducing kernel satisfying three-point boundary conditions. This paper presents a method for solving reproducing kernel satisfying three-point boundary conditions so that reproducing kernel method can be used to solve third order three-point boundary value problems. Results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

Perfect splines with boundary conditions and their application to certain extremal problems

In this paper we consider two problems. The first is connected with the optimal recovery of functions satis fyiog boundary conditions. The second is the characterization of the unique function whose r-th derivative has minimum L_∞-norm, taking given values of alternating signs and satisfying boundary conditions. Partially supported by Ministry of Science under Project MM-414.     

The coupling of bem and fem for the two-dimensional viscous flow problem

In this paper we propose a numerical scheme for treating the problem of sJow viscous flow past an obstacle in the plane. This scheme is a combination of boundary element and finite element methods. By introducing an auxiliary boundary curve, we divide the region under consideration into two subregions, an inner and an outer region. In the inner region, we employ a finite element method (FEM) for solving a system of simplified field equations with proper natural boundary conditions. In the outer region, the solution is expressed in the form of a simple-layer potential with density function satisfying a system of modified integral equations of the first kind. The latter are solved by a boundary element method (BEM). Both solutions are matched on the common auxiliary boundary curve. Error estimates in suitable function spaces are derived in terms of the mesh widths as well as the small parameters, the Reynolds numbers     

Integrable Boundary Value Problem for the Boussinesq Equation

We discuss a method for seeking integrable boundary conditions for nonlinear equations. For the Boussinesq equation, we find a new boundary condition that is compatible with the Lax pair and has an infinite set of higher symmetries and a Bäcklund transformation. We construct a class of explicit partial solutions of an equation satisfying this boundary condition.     

Asymptotic solutions of non-classical boundary-value problems of the natural vibrations of orthotropic shells

The natural vibrations of orthotropic shells are considered in a three-dimensional formulation for different versions of the boundary conditions on the faces: rigid clamping rigid clamping, rigid clamping free surface, and mixed conditions. Asymptotic solutions of the corresponding dynamic equations of the three-dimensional problem of the theory of elasticity are obtained. The principal values of the frequencies of natural vibrations are determined. It is shown that three types of natural vibrations occur in the shell: two shear vibrations and a longitudinal vibration, which are due solely to the boundary conditions on the faces. It is proved that each boundary layer has its own natural frequency. The boundary-layer functions are determined and the rates at which they decrease with distance from the faces inside the shell are established.