On elastic waves in a wedge

The existence of waves propagating along the edge of an elastic wedge has been established by many authors by physically rigorous arguments on the base of numerical computations. A mathematically rigorous proof for a wedge with aperture angle less than π/2 was presented by I. Kamotskii. We supplement the I. Kamotskii result and prove the existence of fundamental modes for some range of aperture angles greater than π/2. Bibliography: 7 titles.     

Diffraction of impulsive elastic waves by a fluid cylinder

We consider the diffraction of impulsive SV waves by a fluid circular cylinder. The cylinder is embedded in an unbounded isotropic homogeneous elastic medium and it is filled with some acoustic fluid. The line source, generating the incident pulse, is situated outside the cylinder parallel to its axis. We investigate the problem by the method of dual integral transformation as developed by Friedlander. The resulting integrals are evaluated approximately to obtain the short-time estimate of the motion near the wave front in the shadow zone of the elastic medium. We also interpret the approximate solution in terms of Keller’s geometrical theory of diffraction.     

Solution of a logarithmic singular integral equation

A quick and efficient method of solution of a singular integral equation of the first kind involving a logarithmic singularity is explained.     

Diffraction of a plane wave on a slippery elastic wedge

Diffraction of a plane elastic wave on a slippery wedge is considered; by a slippery wedge we mean a wedge in which the tangent tension and the normal component of the displacement vector are equal to zero on its surface. It is known that one can construct an explicit solution of this problem. The Sommerfeld representation of this solution is found in construct the paper. Bibliography: 6 titles.     

Diffraction of elastic waves by an inclined crack in a layer

For the problem of the diffraction of normal modes by an inclined crack in an elastic layer, an integral equation with explicit representation of the Fourier symbol kernel is derived in the form of the product of matrices. The algorithm for calculating the wave fields, based on the analytical representations obtained, enables a rapid parametric analysis to be carried out of the influence of the size and orientation of the crack on the transmission of travelling waves. The influence of the inclination of the crack on the effects of resonance trapping and localization of wave energy, which were established previously for the case of a horizontal crack, is analysed.     

A Liapunov functional for a singular integral equation

We consider a scalar integral equation where a∈L²[0,∞), while C(t,s) has a significant singularity, but is convex when t−s>0. We construct a Liapunov functional and show that g(t,x(t))−a(t)∈L²[0,∞) and that x(t)−a(t)→0 pointwise as t→∞. Small perturbations are also added to the kernel. In addition, we consider both infinite and finite delay problems. This paper offers a first step toward treating discontinuous kernels with Liapunov functionals.     

Solutions of some singular integral equation

A singular integral equation with a Holderian second member function on [a,b] is considered and solved for four different type of kernels in the class of functions that are unbounded at the end points of the interval.     

Interaction of surface water waves with a vertical elastic plate: a hypersingular integral equation approach

In this paper, we present an alternative method to investigate scattering of water waves by a submerged thin vertical elastic plate in the context of linear theory. The plate is submerged either in deep water or in the water of uniform finite depth. Using the condition on the plate, together with the end conditions, the derivative of the velocity potential in the direction of normal to the plate is expressed in terms of a Green’s function. This expression is compared with that obtained by employing Green’s integral theorem to the scattered velocity potential and the Green’s function for the fluid region. This produces a hypersingular integral equation of the first kind in the difference in potential across the plate. The reflection coefficients are computed using the solution of the hypersingular integral equation. We find good agreement when the results for these quantities are compared with those for a vertical elastic plate and submerged and partially immersed rigid plates. New results for the hydrodynamic force on the plate, the shear stress and the shear strain of the vertical elastic plate are also evaluated and represented graphically.     

Nondecreasing solutions of a quadratic singular Volterra integral equation

We study the existence of nondecreasing solutions of a quadratic singular Volterra integral equation in the space of continuous functions on bounded interval. The main tool utilized in our considerations is the technique associated with certain measure of noncompactness related to monotonicity. The results obtained in the paper may be applied to a wide class of singular Volterra integral equations.     

Diffraction of a plane elastic wave by a wedge with a special form of boundary conditions

An exact solution of the antiplane problem of the diffraction of a plane elastic SH-wave with a step profile by a wedge is obtained. The stresses on the wedge sides are assumed to be proportional to a linear combination of the displacements, velocities and higher derivatives with respect to time of the displacements along the wedge axis. A solution of the problem is obtained using integral transformations with subsequent transformation using Cagniard's method. Solutions of the corresponding problems with boundary conditions of the Winkler and inertial types are considered. When a wave with a linear profile is incident on the wedge the stresses suffer a discontinuity of the second kind on the diffraction wave front; the same type of feature is observed in the problem with the inertial condition.     

Positive versus free boundary solutions to a singular elliptic equation

The equation-δu = χ uo(-1/u^Β + λf (x, u)) in Ω with Dirichlet boundary condition on ∂Ω has a maximal solution uλ ≥0 for every λ 0. For λ less than a constant λ*, the solution vanishes inside the domain; and for λ λ*, the solution is positive. We obtain optimal regularity ofuλ even in the presence of the free boundary. Supported in part by H. J. Sussmann’s NSF Grant DMS01-03901. Supported by FAPESP. He also thanks Rutgers University.     

Newton-Kantorovich approximations to nonlinear singular integral equation with shift

The paper is concerned with the applicability of some new conditions for the convergence of Newton-kantorovich approximations to solution of nonlinear singular integral equation with shift of Uryson type. The results are illustrated in generalized Holder space.     

Diffraction of a weak shock by a wedge

We match formal asymptotic expansions with differently scaled variables to obtain a uniform approximation to the similarity solution of the shock-wedge diffraction problem.     

Galerkin solution of a singular integral equation with constant coefficients

Galerkin methods are used to approximate the singular integral equation

with solution φ having weak singularity at the endpoint −1, where a, b≠0 are constants. In this case φ is decomposed as φ(x)=(1−x)^α(1+x)^βu(x), where β=−α, 0<α<1. Jacobi polynomials are used in the discussions. Under the conditions fH^μ[−1,1] and k(t,x)H^μ,μ[−1,1]×[−1,1], 0<μ<1, the error estimate under a weighted L² norm is O(n^−μ). Under the strengthened conditions f^″H^μ[−1,1] and , 2α−<μ<1, the error estimate under maximum norm is proved to be O(n^2α−−μ+), where , >0 is a small enough constant.     

Solution of a singular integral equation on a system of intervals

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 44, pp. 84–88, 1985.     

Solution of a nonlinear singular integral equation with quadratic nonlinearity

Using methods of the theory of boundary-value problems for analytic functions, we prove a theorem on the existence of solutions of the equation

Numerical solution of a singular integral equation appearing in magnetohydrodynamics

The numerical solution for the velocity and induced magnetic field has been obtained for the MHD flow through a rectangular pipe with perfectly conducting electrodes. The problem reduces to the solution of a singular integral equation which has been solved numerically. It is found that as the Hartmann number is increased the velocity profile shows a flattening tendency and the flux through a section is reduced. Also as compared with the case of nonconducting walls the flux is found to be smaller. Graphs and tables are given for the solution of the integral equation and the velocity and induced magnetic field.

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Zusammenfassung Für den MHD Fluß durch ein rechteckiges Rohr mit gut leitenden Elektroden wurde die numerische Lösung für die Geschwindigkeit und das induzierte Feld ermittelt. Das Problem ließ sich auf eine singuläre Integralgleichung zurückführen, die numerisch gelöst wurde. Es hat sich herausgestellt, daß wenn die Hartmann-Zahl größer wird, das Geschwindigkeitsprofil eine Tendenz zur Abflachung zeigt und der Fluß durch den Querschnitt zurückgeht. Im Vergleich mit dem Einsatz von nicht leitenden Wänden wurde ebenfalls ein geringerer Fluß festgestellt. Für die Lösung der Integralgleichung, die Geschwindigkeit und das magnetische induzierte Feld sind graphische Darstellungen und Tabellen angegeben.

     

On the existence of a solution to a singular integral equation in electromagnetic reflection

For a singular integral equation arising in a modified approach to boundary integral equations for exterior boundary-value problems from the theory of electromagnetic reflection an existence proof is given.     

Diffraction by a wedge at skew incidence: integral representations of Cauchy-Carleman for the electromagnetic fields

An electromagnetic diffraction problem in a wedge shaped region is reduced to a system of coupled functional difference equations by means of Sommerfeld integrals and Malyuzhinets theorem. By introducing an integral operator it is shown that the solutions of this system of functional equations can be defined in terms of integral representations whose kernels are solutions of a singular integral equation of Cauchy-Carleman type for which an explicit solution is given.     

On certain additions to the theory of one singular integral equation

Certain additions are given to the theory of a singular integral equation /1/ encountered in some problems of potential theory and in a correspondingly complicated form in two-dimensional elasticity theory.