Riemann boundary problem with infinite index of logarithmic order on a spiral-form contour. I

We study the Riemann boundary problem with infinite index of logarithmic order on an open rectifiable Jordan spiral-form contour, here the influence of the contour on the solvability of the problem is comparable with the influence of the argument of its coefficient. An explicit solution of the problem is constructured in the class of functions admitting weak-power singularities at the ends of a line of conjugacy.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1509–1517, November, 1990.     

Steady-state vibrations of a thin two-layer plate with delamination

We consider the problem of harmonic vibrations of a thin two-layer plate with horizontal crack. The problem is solved with the help of the null-field approach. The influence of the shape of the crack contour on the amplitude-frequency characteristics of plate vibrations is investigated. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 2, pp. 83–89, April–June, 1998.     

Determination of the underground contour of a submerged apron with a region of constant velocity when there is a salt backwater

Within the framework of two-dimensional seepage theory, the underground contour of a submerged apron with a region of constant velocity in the case where there is a layer of stagnant salt water under the apron is constructed. The solution of the corresponding boundary-value problem is found by Polubarinova-Kochina's method [1] using the results obtained in [2]. The results of numerical calculations are given and the influence of the fundamental defining parameters of the model on the shape and size of the underground contour of the apron is analysed. Mention is made of special and limiting cases: a scheme with a water-confining stratum [3], an unsubmerged apron [2] and flow around a tongue [4,5].     

Application in Aerohydrodynamics of the Solution of an Inverse Boundary Value Problem for Analytic Functions

We consider a modified inverse boundary value problem of aerohydrodynamics in which it is required to find the shape of an airfoil streamlined by a potential flow of an incompressible nonviscous fluid, when the distribution of the velocity potential on one section of the airfoil is given as a function of the abscissa, while, on other sections of the airfoil, as a function of the ordinate of the point. The velocity of the undisturbed flow streamlining the sought-for airfoil is determined in the process of solving the problem. It is shown that, under rather general conditions on the initially set functions, the sought-for contour is closed unlike the inverse problem in the case when, on the unknown contour, the velocity distribution is given as a function of the arc abscissa of the contour point. We also consider the case when, on the entire desired contour, the distribution of the velocity potential is given as a function of one and the same Cartesian coordinate of the contour point.     

Interaction between two neighboring circular holes in a prestretched simply supported orthotropic strip under bending

In this work, the influence of initial stretching of a simply supported plate-strip containing two circular holes on the stress concentration around the holes caused by bending of the strip is examined using the finite-element method. The mathematical formulation of the corresponding boundary-value problem is presented within the frame work of the three-dimensional linearized theory of elasticity (TDLTE) under a plane strain state. The material of the plate-strip is linearly elastic, homogeneous, and orthotropic. The numerical results obtained in investigating the influence of the initial stretching and the location of holes on the stress concentration are presented. In particular, it is established that the initial stretching significantly decreases the stress concentration at some characteristic points on the contour of the holes. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 6, pp. 827–838, November–December, 2008.     

Variational statement of deformation problems for a composite latticed plate with various types of lattices

The variational statement of various boundary value problems for tangential displacements and forces in a latticed plate with an arbitrary piecewise smooth contour is investigated. The lattice consists of several families of bars made of a homogeneous composite material with a matrix of relatively low shear stiffness. The energy method reduces the problem to the variational problem of minimizing the energy functional. The conditions on the plate contour are established under which the functional is minimal and positive definite, which ensures that the problem is well posed.     

Dynamic Shear Model for Layered Plates

A variant of a geometrically nonlinear dynamic shear model for layered plates is offered, where the inertial forces are taken into account at the stage of formation of hypotheses. A closed-form solution to the problem of forced vibrations is obtained for layered plates clamped or hinge-supported on their contour. The amplitude-frequency characteristics for various kinds of external loads are constructed. The influence of layer distribution across the package thickness is analyzed.     

Differaction of a pulsed longitudinal shear wave in an orthrotropic massif with a cavita

We propose a method for numerical-analytic solution of a boundary-value problem for diffraction of a longitudinal shear wave in the form of a triangular pulse on a cylindrical tunnel cavity of circular crosssection in an unbounded rectilinearly orthotropic massif. The basis of the proposed approach consits of methods of spectral decomposition of periodically continued pulses with the introduction of small correcting perturbations of the elastic constants of the massif. The characteristics of the stress-strain state in the basic stationary problems are expressed in the terms of the generalized wave potentials which become the classical potentials of longitudinal and transverse waves for an isotropic medium. We study the influence of te space-time structure of the pulse on the concentration of the contour dynamical stresses under different phases of mutual potision of the leading edge of the pulse and the contour of the section of the cavity in a massif of isotropic and anisotropic basalt. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 67–74.     

Transverse flexure of a physically nonlinear rectangular grid plate elastically fixed at its contour

The transverse flexure of a rectangular grid plate consisting of two families of physically nonlinear rods is considered. On the basis of the calculation scheme, the hypothesis of a continuum calculation model is adopted. A numerical algorithm based on the elastic-solution method is proposed for the investigation of the initial nonlinear boundary problem. The calculation results for a plate with a rhombic grid formed by H-beam rods with a tension-compression diagram in the form of a cubic binomial are subjected to detailed analysis. The influence of physical nonlinearity of the material, as well as the parameters of the grid and elastic fixing of the contour, on the stress-strain state and the limiting load is shown.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 18, pp. 49–55, 1987.     

Robust Contour Tracking Model Using a Variational Level-Set Algorithm

In this work, we provide a novel variational level-set based object contour tracking approach. Thus, a mathematically rigorous variant of the Chan-Vese algorithm for image segmentation via geometric active contour model is proposed here. With respect to the original contour detection algorithm, the level set function ?(t) defining the evolving contour S _t  = {x; ?(t, x) = 0} is iteratively computed from a nonlinear parabolic boundary value problem that is well posed in the space of functions with bounded variations. We provide a robust mathematical justification of the proposed level-set model.     

An integral method for solving nonlinear eigenvalue problems

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour.     

Convolution operators on a finite interval with periodic kernel-Fredholm property and invertibility

Finite interval convolution operators with periodic kernel-functions are studied from the point of view of Fredholm properties and invertibility. These operators are associated with Wiener-Hopf operators with matrix-valued symbols defined on a space of functions whose domain is a contour consisting of two parallel straight-lines. For the Fredholm study a Wiener-Hopf operator is considered on a space of functions defined on a contour composed of two closed curves having a common multiple point. Invertibility of the finite interval operator is studied for a subclass of symbols related to the problem of wave diffraction by a strip grating.The present work was sponsored by JNICT (Portugal) under grant n. 87422/MATM and Programa Ciência.     

Parabolic and hyperbolic contours for computing the Bromwich integral

Some of the most effective methods for the numerical inversion of the Laplace transform are based on the approximation of the Bromwich contour integral. The accuracy of these methods often hinges on a good choice of contour, and several such contours have been proposed in the literature. Here we analyze two recently proposed contours, namely a parabola and a hyperbola. Using a representative model problem, we determine estimates for the optimal parameters that define these contours. An application to a fractional diffusion equation is presented.

     

Simultaneous stabilization: Construction of a universal stabilizer for linear plants with delay with the use of spectral reducibility

We consider the problem of the construction of a common stabilizing controller for a family of k linear stationary nth-order plants with commensurable delays in the controls and the state variables. Our approach to stabilization is based on the construction of a common two-loop stabilizer. The controller of the first (internal) loop solves the problem of reducing each plant of the family to a finite spectrum, and the controller of the second (external) contour solves the stabilization problem.     

Evaluation of generalized Mittag–Leffler functions on the real line

This paper addresses the problem of the numerical computation of generalized Mittag–Leffler functions with two parameters, with applications in fractional calculus. The inversion of their Laplace transform is an effective tool in this direction; however, the choice of the integration contour is crucial. Here parabolic contours are investigated and combined with quadrature rules for the numerical integration. An in-depth error analysis is carried out to select suitable contour’s parameters, depending on the parameters of the Mittag–Leffler function, in order to achieve any fixed accuracy. We present numerical experiments to validate theoretical results and some computational issues are discussed.     

A CLASS OF COMPOUND VECTOR-VALUED PROBLEM AND FACTORIZATION OF MATRIX FUNCTION

In this article,we consider a class of compound vector-valued problem on upper-half plane C+,which consists of vector Riemann problem along a closed contour in C+ with matrix coefficient in Hlder class and vector Hilbert problem on the real axis with essential bounded measurable matrix coefficient.Under appropriate assumption we obtain its solution by use of Corona theorem and factorization of matrix functions in decomposed Banach algebras.     

Fock asymptotics in the problem of diffraction by a moving smooth contour

In the paper the asymptotics are obtained of the solution of the diffraction problem in the Fock region for the incidence of a nonstationary modulated wave on a smooth moving contour. The formulas obtained generalize the solution of V. A. Fock to the stationary problem in a neighborhood of a point of tangency of an ray to the case of a nonstationary wave and a moving scatterer. A new characteristic of the problem is introduced the effective radius of curvature of the contour, depending on the geometric and kinematic properties of the scatterer. The boundaries are established of the applicability of the asymptotic formulas.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Institute im. V. A. Steklova AN SSSR, Vol. 195, pp. 69–81, 1991.In conclusion I want to thank the participants of the seminars of LGU and LOMI on the diffraction and propagation of waves for useful discussions.     

Exceptional case of the linear conjugation problem in weighted Hölder classes

We study the linear conjugation problem for the case in which the coefficient of the problem may have finitely many zeros and/or pole singularities on the contour. All studies are carried out in weighted H¨older classes with complex weight. We obtain a closed-form expression for the solution and the solvability conditions.     

A factorization method for products of holomorphic matrix functions

A class of matrix functions defined on a contour which bounds a finitely connected domain in the complex plane is considered. It is assumed that each matrix function in this class can be explicitly represented as a product of two matrix functions holomorphic in the outer and the inner part of the contour, respectively. The problem of factoring matrix functions in the class under consideration is studied. A constructive method reducing the factorization problem to finitely many explicitly written systems of linear algebraic equations is proposed. In particular, explicit formulas for partial indices are obtained.     

A result on large surfaces of prescribed mean curvature in a Riemannian manifold

Plateau's problem (PP) is studied for surfaces of prescribed mean curvature spanned by a given contour in a 3-d Riemannian manifold. We consider the local situation where a neighborhood of a given point on the manifold is described by a single normal chart. Under certain conditions on and the contour, existence of a small -surface to (PP) is guaranteed by [HK]. The purpose of this paper is the investigation of large -surfaces. Our result states: For sufficiently large (constant) mean curvature and a sufficiently small contour depending on the local geometry of the manifold, (PP) has at least two solutions, a small one and a large one. The proof is based on mountain pass arguments and uses – in contrast to results in the 3-d Euclidean space and in order to derive conformality directly – also a deformation constructed by variations of the independent variable. Received November 8, 1995 / Accepted April 29, 1996