Green functions of three-dimensional static thermoelasticity problems for a piecewise homogeneous space

The Green functions of thermoelasticity problems for a piecewise homogeneous body, composed of two perfectly contacting semiinfinite isotropic bodies, have been constructed in closed form. We have used here generalized functions and Green functions for the corresponding systems of ordinary differential equations. As the limiting cases, we have obtained the Green functions of thermoelasticity problems for a semiinfinite body, when its surface, thermally insulated or maintained at zero temperature, is load-free or rigidly restrained. We also present some results of numerical studies.     

Solution of two-dimensional problems of elasticity and thermoelasticity for a rectangular region

We propose a separation-of-variables method for the biharmonic equation and construct a complete system of orthogonal functions for constructing exact solutions in the form of non-periodic trigonometric series for two-dimensional problems of the theory of elasticity and thermoelasticity for a rectangular region. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 19–25.     

A direct method of integrating the equations of two-dimensional problems of elasticity and thermoelasticity for orthotropic materials

For a plane, a half-plane, and a strip, we propose a direct method of integrating the differential equations of equilibrium and continuity with respect to the stresses in the case of two-dimensional problems of elasticity and thermoelasticity for orthotropic materials. We find the relations between the components of the stress tensor, the key integro-differential equation and the equation of continuity equivalent to it for determining one of the components of the normal stresses. Translated fromMatematichni Metodi i Fiziko-Mekhanichni Polya, Vol. 40, No. 1, pp. 24–29.     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

Minimizing Sequences for a Family of Functional Optimal Estimation Problems

Rates of convergence are derived for approximate solutions to optimization problems associated with the design of state estimators for nonlinear dynamic systems. Such problems consist in minimizing the functional given by the worst-case ratio between the ℒ _p -norm of the estimation error and the sum of the ℒ _p -norms of the disturbances acting on the dynamic system. The state estimator depends on an innovation function, which is searched for as a minimizer of the functional over a subset of a suitably-defined functional space. In general, no closed-form solutions are available for these optimization problems. Following the approach proposed in (Optim. Theory Appl. 134:445–466, 2007), suboptimal solutions are searched for over linear combinations of basis functions containing some parameters to be optimized. The accuracies of such suboptimal solutions are estimated in terms of the number of basis functions. The estimates hold for families of approximators used in applications, such as splines of suitable orders.     

Transition functions of the Il'yushin approximation method

Transition functions are constructed whose direct substitution for combinations of elastic constants in solutions of problems of elasticity theory permits writing the solution of the corresponding viscoelastic problems. Examples of using the transition functions for obtaining solutions of viscoelastic problems are given: circularly symmetric solid and anular plates, plate with a small circular hole, and viscoelastic half-plane loaded by a concentrated force.M. M. Gubkin Moscow Institute of the Petrochemical and Gas Industry. Moscow Institute of Electronic Engineering. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 405–416, May–June, 1973.     

Notes on Rankin–Cohen brackets

The Rankin–Cohen product of two modular forms is known to be a modular form. The same formula can be used to define the Rankin–Cohen product of two holomorphic functions f and g on the upper half-plane. Assuming that this product is a modular form, we prove that both f and g are modular forms if one of them is. We interpret this result in terms of solutions of linear ordinary differential equations.     

Estimates on the Green function and existence of positive solutions for some polyharmonic nonlinear equations in the half space

Let G_m,n be the Green function of with Dirichlet boundary conditions We establish some estimates on G_m,n, including a 3G-Theorem. Next, we introduce a Kato class of functions and we exploit properties of these functions to study the existence of positive solutions of some m-polyharmonic nonlinear elliptic problems.Mathematics Subject Classification (2000): 34B27, 35J40     

Mathieu functions and coulomb spheroidal functions in the electrostatic probe theory

A spherical probe placed in a slowly moving collisional plasma with a large Debye length λ_D → ∞ is considered. The partial differential equation describing the electron concentration around the probe is reduced to two ordinary differential equations, namely, to the equation for Coulomb spheroidal functions and Mathieu’s modified equation with the parameter a of the latter related to the eigenvalue λ of the former by the relation a = λ + 1/4. It is shown that the solutions of Mathieu’s equation are Mathieu functions of half-integer order, which are expressed as series in terms of spherical Bessel functions and series of products of Bessel functions. These Mathieu functions are numerically constructed for Mathieu’s modified and usual equations.     

Fast decreasing rational functions

Necessary and sufficient conditions are obtained for the existence of sequences of rational functions of the formr _n(x) =p _n(x)/p_n(−x), withp _n a polynomial of degreen, that decrease geometrically on (0, 1] in accordance with a specified rate function. The technique of proof involves minimum energy problems for Green potentials in the presence of an external field. Applications are given for the construction of rational approximations of |x| and sgn(x) on [−1, 1] having geometric rates of convergence forx ≠ 0. The research of this author was supported, in part, by National Science Foundation grant DMS-9501130.     

Solution of two-dimensional quasistatic problems of thermoelasticity for multilayered bodies

We propose a new method of solving two-dimensional quasistatic problems of thermoelasticity for isotropic multilayered bodies. The method is based on the applicaiton of generalized functions. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 49–52.     

Non-Euclidean visibility problems

We consider the analog of visibility problems in hyperbolic plane (represented by Poincaré half-plane model ℍ), replacing the standard lattice ℤ × ℤ by the orbitz = i under the full modular group SL₂(ℤ). We prove a visibility criterion and study orchard problem and the cardinality of visible points in large circles.     

Density of Sums of Shifts of a Single Function in Hardy Spaces on the Half-Plane

It is proved that there exists a function defined in the closed upper half-plane for which the sums of its real shifts are dense in all Hardy spaces H_p for 2 ≤ p < ∞, as well as in the space of functions analytic in the upper half-plane, continuous on its closure, and tending to zero at infinity.

     

The axisymmetric problem of thermoelasticity of a multilayer thermosensitive tube

We propose a method of solving stationary heat-conduction problems of contacting bodies with coefficient of thermal conductivity that are linear functions of the temperature and the corresponding problems of thermoelasticity based on the method of perturbations. We give a numerical analysis of the thermal stresses in a two-layer tube. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 97–103.     

Sommerfeld Half-plane Problems with Higher Order Boundary Conditions

Sommerfeld-type diffraction problems for a half-plane with arbitrary n-th order generalized impedance boundary conditions arc examined in a Sobolev space setting. The corresponding boundary-transmission problems for the two dimensional Helmholtz equation are shown to be well-posed in a family of Sobolev spaces with finite energy norms, through a reduction to equivalent systems of boundary integral equations of Wiener-Hopf type in [L₂⁺ (IR)]². Formulas for the solutions as well as the so-called edge conditions arc obtained for any n, by explicit canonical generalized factorization of the presymbols of the associated Wiener-Hopf operators.     

The thermoelastic problem for steady-state vibrations of a plante in three-dimensional formulation

The perturbation method is used to reduce a coupled three-dimensional problem of stationary vibrations of thermoelasticity to a number of three-dimensional vibration problems of the theory of thermal stresses. We develop an algorthm for constructing the solution in the zeroth approximation on the basis of the method of homogeneous solutions. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 23, 1992, pp. 119–127.     

On the definition of Bourgain points

This paper is devoted to study of the so-called Bourgain points (B-points) for functions from L^∞(ℝ). In 1993, Bourgain showed that for a real-valued bounded function f, the set E_f of B-points is everywhere dense and has the maximal Hausdorff dimension, dim _H(E_f ) = 1; in addition, the vertical variation of the harmonic extension of f to the upper half-plane is finite at B-points. An essentially simpler definition of B-points is given compared to that in the original works by Bourgain. A geometric characterization of B-points for Cantor-like sets is obtained. Bibliography: 7 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 355, 2008, pp. 219–236.     

On the invertibility of the operator <Emphasis Type="Italic">d/dt</Emphasis> + <Emphasis Type="Italic">A</Emphasis> in certain functional spaces

We prove that the operator d/dt + A constructed on the basis of a sectorial operator A with spectrum in the right half-plane of ℂ is continuously invertible in the Sobolev spaces W _p ¹ (ℝ, D _α), α ≥ 0. Here, D _α is the domain of definition of the operator A ^α and the norm in D _α is the norm of the graph of A ^α. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 8, pp. 1020–1025, August, 2007.     

Applications of the Schwarz lemma to inequalities for entire functions with constraints on zeros

It is shown that new inequalities for certain classes of entire functions can be obtained by applying the Schwarz lemma and its generalizations to specially constructed Blaschke products. In particular, for entire functions of exponential type whose zeros lie in the closed lower half-plane, distortion theorems, including the two-point distortion theorem on the real axis, are proved. Similar results are established for polynomials with zeros in the closed unit disk. The classical theorems by Turan and Ankeny-Rivlin are refined. In addition, a theorem on the mutual disposition of the zeros and critical points of a polynomial is proved. Bibliography: 16 titles. Dedicated to the 100th anniversary of G. M. Goluzin’s birthday __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 101–112.