Numerical solution of the inverse problem for the polarized-radiation transfer equation

An inverse problem for the steady vector transfer equation for polarized radiation in an isotropic medium is studied. For this problem, an attenuation factor is found from a given solution of the equation at a medium boundary. An approach is propounded to solve the inverse problem by using special external radiative sources. A formula is derived which relates the Radon transform of an attenuation factor to a radiation-flux density at the boundary. Numerical experiments show that the algorithm for the polarized-radiation transfer equation has an advantage over the method used in the scalar case.     

Asymptotics of eigenvalues of the Dirichlet boundary value problem for the Lame operator in a three-dimensional domain with a small cavity

A boundary value problem for the Lame operator in a bounded three-dimensional domain with a small cavity is studied. The domain is filled with an elastic homogeneous isotropic medium that is clamped at the boundary, which corresponds to the Dirichlet boundary condition. The leading term of an asymptotic expansion for the eigenvalue is constructed in the case of the Dirichlet limit problem. The asymptotic expansion is constructed in powers of a small parameter ? that is the diameter of the cavity.     

Singular boundary integral equations of boundary value problems of the elasticity theory under supersonic transport loads

We consider a transport boundary value problem for an isotropic elastic medium bounded by a cylindrical surface of arbitrary cross-section and subjected to supersonic transport loads. We pose the corresponding hyperbolic boundary value problem and prove the uniqueness of the solution with regard to shock waves. To solve the problem, we use the method of generalized functions. In the space of generalized functions, we obtain the solution, perform its regularization, and construct a dynamic analog of the Somigliana formula and singular boundary equations solving the boundary value problem.     

Estimations for the characteristic values of a Sturm-Liouville problem

We consider a particular Sturm-Liouville problem, with boundary conditions containing the characteristic values. This problem governs the stability of the interfaces in oil recovery, by using the Hele-Shaw model of a homogeneous porous medium. We obtain estimations of the approximative characteristic values, by using a finite-difference approximation and the Gerschgorin's localization theorem. We obtain estimations of the exact characteristic values by using the Rayleigh's quotient of the initial problem. The upper limit of the approximative characteristic values is (in general) greater than the lower limit of the exact characteristic values. The same upper limit is given by the two methods, for a particular value of a parameter of the problem. This value gives us a large improvement of the stability. A previous convergence result is confirmed.     

Regularity of the free boundary for the one phase Hele-Shaw problem

In this paper we prove that, in a local neighborhood, Lipschitz continuous free boundary of a solution of the one-phase Hele-Shaw problem is indeed smooth if the solution is Lipschitz continuous and non-degenerate in the neighborhood.     

Uniqueness Theorems for the Inverse Problem of Elastodynamic Boundary Scattering

A uniqueness theorem is established for the elastodynamic problemof direct scattering of harmonic small-amplitude waves by anobstacle embedded in an isotropic medium with spatially varyingparameters. Uniqueness theorems are then developed for the inverseproblem of boundary determination of the obstacle associatedwith this direct scattering problem. In particular, it is shownfor nondegenerate elastodynamic problems that knowledge of boththe lamellar and the solenoidal far-field patterns is necessaryto reconstruct the obstacle's boundary.     

Selection in the Saffman-Taylor finger problem and the Taylor-Saffman bubble problem without surface tension

We consider the Saffman-Taylor problem describing the displacement of one fluid by another having a smaller viscosity, in a porous medium or in a Hele-Shaw configuration, and the Taylor-Saffman problem of a bubble moving in a channel containing moving fluid. Each problem is known to possess a family of solutions, the former corresponding to propagating fingers and the latter to propagating bubbles, with each member characterized by its own velocity and each occupying a different fraction of the porous channel through which it propagates. To select the correct member of the family of solutions, the conventional approach has been to add surface tension σ and then take the limit σ → 0. We propose a selection criterion that does not rely on surface tension arguments.     

Analysis and optimization in problems of cloaking of material bodies for the Maxwell equations

We consider control problems for the 3D Maxwell equations describing electromagnetic wave scattering in an unbounded inhomogeneous medium that contains a permeable isotropic obstacle with cloaking boundary. Such problems arise when studying cloaking problems by the optimization method. The boundary coefficient occurring in the impedance boundary condition plays the role of a control. We study the solvability of the control problem and derive optimality systems that describe necessary conditions for the extremum. By analyzing the constructed optimality systems, we justify sufficient conditions imposed on the input data providing the uniqueness and stability of optimal solutions.     

The stressed state of a transversally isotropic medium with a spheroidal inclusion under nonideal mechanical contact

We study the problem of the stressed state of a transversally isotropic medium containing a foreign inclusion in the form of a prolate spheroid under an arbitrary homogeneous stress field at infinity. On the “medium-inclusion” interface there is slipping without flaking. The stressed state is constructed in the medium and in the inclusion using the exterior and interior problems for a prolate spheroid on the basis of potential functions. The solution of the problem is reduced to studying infinite systems of linear algebraic equations. The results of numerical studies are shown as graphs that describe the stress distribution in both the transversally isotropic medium and in the inclusion under various boundary conditions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 15–26.     

Asymptotic convergence of the Stefan problem to Hele-Shaw

We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as , and give the precise asymptotic growth rate for the radius.

     

ON THE DISCRETE-ORDINATES METHOD FOR THE MONOENERGETIC NEUTRON TRANSPORT EQUATION IN A SLAB WITH GENERALIZED BOUNDARY CONDITIONS

This paper is devoted to discussing the discrete-ordinates method for the monoenergetic neutron transport equation in a slab with generalized boundary conditions. For homogeneous medium with isotropic scattering and fission, the convergence theorems for discrete-ordinates approximations are given respectively for critical eigenvalue problem and dominant eigenvalue problems: for inhomogeneous medium with anisotropic scattering and fission, a similar discussion and an estimation for the convergence rate are given for critical eigenvalue problems. Finally, some numerical results are given by use of this method.     

On one proof of the classical solvability of the Hele-Shaw problem with free boundary

We consider the Stefan problem for a parabolic equation with a small parameter as the coefficient of the derivative with respect to time. We justify the limit transition as the small parameter tends to zero, which enables us to prove the classical solvability of the Hele-Shaw problem with free boundary in the small with respect to time. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50; No. 11, pp. 1452–1462, November, 1998.     

Global well-posedness for the one-phase Muskat problem

The free boundary problem for a two-dimensional fluid permeating a porous medium is studied. This is known as the one-phase Muskat problem and is mathematically equivalent to the vertical Hele-Shaw problem driven by gravity force. We prove that if the initial free boundary is the graph of a periodic Lipschitz function, then there exists a global-in-time Lipschitz solution in the strong $L_t^{\infty }{L}_x^2$ sense and it is the unique viscosity solution. The proof requires quantitative estimates for layer potentials and pointwise elliptic regularity in Lipschitz domains. This is the first construction of unique global strong solutions for the Muskat problem with initial data of arbitrary size.     

On complex variable method in finite elasticity

We highlight the alternative presentation of the Cauchy-Riemann conditions for the analyticity of a complex variable function and consider plane equilibrium problem for an elastic transversely isotropic layer, in finite deformation. We state the fundamental problems and consider traction boundary value problem, as an example of fundamental problem-one. A simple solution of “Lame's problem” for an infinite layer is obtained. The profile of the deformed contour is given; and this depends on the order of the term used in the power series specification for the complex potential and on the material constants of the medium.     

The inverse scattering problem by an elastic inclusion

In this work we consider the inverse elastic scattering problem by an inclusion in two dimensions. The elastic inclusion is placed in an isotropic homogeneous elastic medium. The inverse problem, using the third Betti’s formula (direct method), is equivalent to a system of four integral equations that are non linear with respect to the unknown boundary. Two equations are on the boundary and two on the unit circle where the far-field patterns of the scattered waves lie. We solve iteratively the system of integral equations by linearising only the far-field equations. Numerical results are presented that illustrate the feasibility of the proposed method.     

On the solvability of a variational problem about phase transitions in continuum mechanics

We study a variational problem about phase transitions in continuum mechanics under the condition that the surface tension coefficient vanishes. A homogeneous isotropic two-phase elastic medium occupies a ball-shaped domain, the zero displacement field is fixed on the boundary of this domain, and a spherically symmetric force field acts on the medium. The solvability of this problem is established. As is shown, if a force field is nonzero almost everywhere, then the problem has only spherically symmetric solutions. Bibliography: 9 titles. Translated from Problemy Matematicheskogo Analiza, No. 38, December 2008, pp. 61–71.     

The scattering of plane elastic waves by a one‐dimensional periodic surface

The two‐dimensional scattering problem for time‐harmonic plane waves in an isotropic elastic medium and an effectively infinite periodic surface is considered. A radiation condition for quasi‐periodic solutions similar to the condition utilized in the scattering of acoustic waves by one‐dimensional diffraction gratings is proposed. Under this condition, uniqueness of solution to the first and third boundary‐value problems is established. We then proceed by introducing a quasi‐periodic free field matrix of fundamental solutions for the Navier equation. The solution to the first boundary‐value problem is sought as a superposition of single‐ and double‐layer potentials defined utilizing this quasi‐periodic matrix. Existence of solution is established by showing the equivalence of the problem to a uniquely solvable second kind Fredholm integral equation. Copyright © 1999 John Wiley & Sons, Ltd.     

Stability Properties of Non-Radial Steady Ferrofluid Patterns

We consider the dynamic of a fixed volume of ferrofluid in a Hele-Shaw cell under the influence of centrifugal and magnetic forces. The steady-state solutions of the associated moving boundary problem are the periodic solutions of a generalized Laplace-Young equation. We use bifurcation theory to find analytic curves consisting of non-radial steady-state solutions of the problem. The stability of these solutions is discussed by using the exchange of stability theorem.     

The Three-Dimensional Problem of Statics of the Elastic Mixture Theory with Displacements Given on the Boundary

The first three-dimensional boundary value problem is considered for the basic equations of statics of the elastic mixture theory in the finite and infinite domains bounded by the closed surfaces. It is proved that this problem splits into two problems whose investigation is reduced to the first boundary value problem for an elliptic equation which structurally coincides with an equation of statics of an isotropic elastic body. Using the potential method and the theory of Fredholm integral equations of second kind, the existence and uniqueness of the solution of the first boundary value problem is proved for the split equation.     

Periodic solutions of elasticity and electroelasticity theories for a cylinder in R3

General representations are constructed for periodic solutions of the theories of elasticity and electroelasticity for a cylinder in R³. These solutions are utilized to reduce the first boundary value problem in an unbounded isotropic medium weakened by tunnel slit cavities to a system of singular integral equations. The solvability of the characteristic part of the obtained system is proved.Sumy. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 13–20, 1990.