Green functions of a quasi-static problem

In this paper Green functions are constructed in analytic form for a deformable half-plane of a quasi-static problem of thermoelasticity when the heat flow on the boundary x₂=0 of the half-plane is zero. To construct the Green functions, certain integral representations are used whose kernels are known Green functions of the corresponding problems of elasticity theory. The functions constructed make it possible to obtain a wide class of new solutions of boundary-value problems of thermoelasticity, in particular solutions for a piecewise homogeneous half-plane. Bibliography: 6 titles. Translated fromObchyslyuwval’na ta Pryklandna Matematyka, No. 77, 1993, pp. 97–104.     

Solution of the fundamental problems of elasticity theory for a multiconnected anisotropic half-plane

By analytic continuation of generalized complex potentials to upper half-planes we reduce the boundary conditions on a rectilinear boundary to problems of linear coupling for cuts of a multiconnected extended plane. By solving the latter problems we obtain general representations of the complex potentials in the case of a multiconnected anisotropic half-plane for different types of boundary conditions on intervals of the rectilinear boundary. As particular cases we give expressions for the complex potentials in the cases of action of external forces on the rectilinear boundary and dies both with and without friction. Two figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 44–63.     

Stress concentration in an anisotropic half-plane with holes and cracks

On the basis of general representations of the generalized complex potentials for a multiconnected half-plane, which the authors have obtained, we solve problems for a multiconnected half-plane with holes and cracks when external forces or dies act on the boundary of the half-plane. Using conformal mapping for an ellipse and the method of least squares, we reduce these problems to solving a system of linear algebraic equations. For different anisotropic materials we give the results of studies of the stress distributions and the variation of the stress intensity factors for a half-plane with a crack in the case of tension at infinity, internal pressure on the edges of the crack, and the action of normal forces on the rectilinear boundary. Two figures, 2 tables. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 63–72.     

A Class of Quadratic Integral-Differential Equations

A nonlinear integro-ordinary differential equation built up by a linear ordinary differential operator of n th order with constant coefficients and a quadratic integral term is dealt with. The integral term represents the so-called autocorrelation of the unknown function. Applying the Fourier cosine transformation, the integral-differential equation is reduced to a quadratic boundary value problem for the complex Fourier transform of the solution in the upper half-plane. This problem in turn is reduced to a linear boundary value problem which can be solved in closed form. There are infinitely many solutions of the integral-differential equation depending on the prescribed zeros of a function related to the complex Fourier transform.     

Solution of the fundamental problems of the theory of elasticity for a half-plane with holes and cracks

We obtain the general solution of the fundamental problems of the theory of elasticity for an isotropic half-plane with a finite number of arbitrarily situated elliptic holes whose boundaries may intersect or form rectilinear cuts or boundaries of curvilinear holes. On the rectilinear boundary the first problem and the second or mixed problem of the theory of elasticity are defined. We use general expressions obtained previously by the author for the complex potentials generated by solving the problem of linear coupling for cuts in a multiconnected region, conformal mappings, and the method of least squares. The problem is reduced to solving a system of linear algebraic equations. The results of numerical experiments are given for a half-plane with a crack in the case of the first fundamental problem and the action of various loads. Two figures, two tables. Bibliography: 4 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 157–171.     

Scattering of waves by periodically moving bodies

We consider solutions of the scalar wave equation vanishing on the boundary of an obstacle which undergoes periodic motion. In analogy with the Lax-Phillips theory, we show that the scattering matrix, as a function of frequency, is holomorphic in a lower half-plane, and meromorphic in an upper half-plane, provided rays are not trapped. The poles of the scattering matrix correspond to certain outgoing eigenfunctions, and there is a near-field expansion of finite energy solutions in terms of these eigenfunctions.     

The stress distribution in an anisotropic half-plane with two elliptic holes or cracks

By use of the method of complex potentials, conformal mappings and least squares this problem is reduced to solving a system of linear algebraic equations with respect to the unknown constants that occur in the required functions. We describe the results of numerical studies of the variation of the stress intensity factors for cracks in an anisotropic half-plane under tension of the half-plane and force on its boundary. Two figures, two tables. Bibliography: 7 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 57–61.     

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.     

On the decay rate of solutions of the wave equation in domains with star-shaped boundaries

One studies the large-time decay rate of the weighted energy of solutions of the first mixed problem for the wave equation in domains with smooth boundaries which are star-shaped with respect to the origin. Estimates are established for solutions of the first boundary-value problem for the Helmholtz equation in the upper half-plane. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 26, pp. 390–406, 2007.     

Kadomtsev–Petviashvili Equation on the Half-Plane

We study the boundary value problem for the Kadomtsev–Petviashvili equation on the half-plane y > 0 with a homogeneous condition along the boundary. We show that the problem can be efficiently solved using the dressing method. We present explicit solutions for particular cases of the boundary value problem.     

Interaction of elastic waves with a curvilinear crack in a half-plane

We consider the problem of the interaction of monochromatic displacement waves with a curvilinear crack-cut in a half-plane. We find integral representations of the solution. The boundary-value problem is reduced to a system of singular integral equations. A parametric investigation is carried out for the effect of the form of the load, the fastening conditions on the boundary of the half-plane, and the curvature of the crack on the dynamic coefficients of the stress intensity.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 77–82, 1988.     

On the reducibility of a nonnegatively hamiltonian real periodic matrix to a block-diagonal form

In the paper, under certain conditions, it is shown that a Hamiltonian real periodic matrix can be block-diagonalized by a real matrix of the same period and, in the resulting block-diagonal form, the spectrum of one of the matrices on the diagonal belongs to the open left half-plane and that of the other to the open right half-plane. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 688–695, November, 1999.     

On Dirichlet and Neumann problems for harmonic functions

The aim of the paper is to examine some aspects of the boundary value problems for harmonic functions in half-spaces related to approximation theory. M. V. Keldyshmentioned curious fact on richness in some sense of the solutions of Dirichlet problem in upper half-plane for a fixed continuous boundary data on the real axis. This can be considered as a model version for the Dirichlet problem with continuous boundary data, defined except a single boundary point, with no restrictions imposed on solutions near that point.Some extensions and multi-dimensional versions of Keldysh’s richness are obtained and related questions on existence, representation and richness of solutions for the Dirichlet and Neumann problems discussed.     

Behavior of surface waves at transition through a junction line on the boundary of an elastic homogenous isotropic body

A homogeneous elastic body with stress-free boundary is considered. The boundary of the body, which consists of a smooth cylindrical surface and a half-plane, has a continuous tangent plane, but the curvature of the normal section of the boundary has a discontinuity of the first kind at each point of the junction line. The behavior of two kinds of “whispering gallery” transversal surface waves at transition through the junction line is studied. For waves of the first kind (corresponding to Dirichlet boundary conditions), the displacement vector is normal to the boundary, whereas for waves of the second kind (corresponding to Neumann boundary conditions) the displacement vector is tangent to the boundary and normal to the ray, similarly to the case of Love waves. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 230, 1995, pp. 86–102. Translated by N. Ya. Kirpichnikova.     

On the scattering frequencies of the laplace operator for exterior domains

In this paper we show that the so-called scattering frequencies of the Laplace operator over an exterior domain, subject to Robin or Dirichlet boundary condition, cannot lie in certain portions of the upper half-plane. The excluded sets depend only on the type of boundary condition and the radius of the smallest sphere containing the scattering obstacle.     

Inequalities for Entire Functions of Finite Degree and Polynomials

The extremal properties of polynomials and entire functions of finite degree not vanishing in the upper half-plane are studied. The exact inequalities obtained complement and strengthen the results by Genchev, Gardner and Govil, Turan, and Lax. Proofs are based on a univalence condition established by Dubinin. Bibliography: 15 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 314, 2004, pp. 174–195.     

The difference method in a diffraction problem: The technique of interior boundary condition

The diffraction problem for a plane wave on a half-plane covered by thin layer with an interface is solved by the difference method. The system of difference equations is derived from the variational principle. A boundary solution at infinity must be imposed; this is a radiation condition, which is used in the form of the limit absorption principle. The arising infinite system of difference equations is reduced to a finite part of the boundary (the interface) by using the technique of so-called interior boundary conditions in the sense of Ryaben’kii. The real conditions are found by the Fourier method with respect to one spatial variable in the form of Fourier or Laurent series in the corresponding variable, which converge either inside, outside, or on the unit circle. Above the upper boundary of the layer, all unknowns are eliminated by using the so-called grid Green function, that is, the resolving function for the half-plane satisfying the radiation condition at infinity. For the unknowns on the upper boundary of the layer, an equation in terms of a function of a complex variable of Wiener-Hopf type is obtained, which is solved by factorization. Factorization is performed numerically: the logarithm of the function is expanded in a bi-infinite series, which is replaced by a discrete Fourier series. The closing system in a neighborhood of the interface has order proportional to the number of points on the interface. Solving this system yields all of the required characteristics of the solution.     

上半平面中含参变未知函数的Hilbert边值问题

给出了上半平面中的含参变未知函数的Hilbert边值问题的提法,利用函数的对称扩张,将其转化为无穷直线上含参变未知函数的Riemann边值问题,得到了该问题的一般解和可解性定理.     

Application of the coupling problem to the study of the stressed state of a multiply connected isotropic half-plane

We consider a problem of the theory of elasticity for an isotropic half-plane with holes. By continuing analytically across an unstressed portion of the boundary and solving the resulting Riemann-Hilbert boundary-value problem we find a general representation of the complex potential. The unknown functions occurring in the potential and the coefficients of a polynomial are determined from the boundary conditions on the edges of holes and the equilibrium conditions under punches. As special cases we consider prescribed stresses and displacements on portions of the boundary and the action of punches with and without friction. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 22, pp. 20–26, 1991.     

Periodic and boundary value problems for second order differential equations

In this paper we study second order scalar differential equations with Sturm-Liouville and periodic boundary conditions. The vector fieldf(t,x,y) is Caratheodory and in some instances the continuity condition onx ory is replaced by a monotonicity type hypothesis. Using the method of upper and lower solutions as well as truncation and penalization techniques, we show the existence of solutions and extremal solutions in the order interval determined by the upper and lower solutions. Also we establish some properties of the solutions and of the set they form.