Study of Three-Dimensional Boundary Value Problems of Fluid Filtration in an Anisotropic Porous Medium

Three-dimensional boundary value problems (the first and second boundary value problems and the conjugation problem) of stationary filtration of fluids in anisotropic (orthotropic) and inhomogeneous porous media are posed and studied. A medium is characterized by a symmetric permeability tensor whose components generally depend on the coordinates of points of the space. A nonsingular affine transformation of coordinates is used and the problems are stated in canonical form, which dramatically simplifies their study. In the case of orthotropic and piecewise orthotropic homogeneous medium, the solution of the problem with canonical boundaries (plane and ellipsoid surfaces) can be obtained in finite form. In the general case, where the orthotropic medium is inhomogeneous and the boundary surfaces are arbitrary and smooth, the problem can be reduced to singular and hypersingular integral equations. The problems are topical, for example, in the practice of fluid (water, oil) recovery from natural anisotropic and inhomogeneous soil strata.     

On the solution of boundary value problems on an inhomogeneous plane with a crack and a barrier connected in series

We consider boundary value problems for an equation in divergence form on a plane divided into two inhomogeneous half-planes by a film inclusion in the form of a strongly permeable crack and a weakly permeable barrier connected in series; this models a contact of heterogeneous media under inhomogeneous external conditions. The desired potentials have prescribed singular points (sources, drains, etc.). The coefficients of the equation are nonconstant and may increase or decrease when moving away from the film inclusion along a family of parabolas. We obtain representations of solutions of the considered problems via harmonic functions with the corresponding singular points on the plane.     

On soliton and periodic solutions of Maxwell–Bloch system for two-level medium with degeneracy

Maxwell–Bloch (MB) system describing the resonant propagation of electromagnetic pulses in either two-level media with degeneracy in angle moment projection or a three-level media with equal oscillator forces is considered. An inhomogeneous broadening of energy levels and a polarization of the wave are accounted. The equations are integrated by the binary Darboux transformations technique. Pulses corresponding to a transition between levels with the largest population difference are shown to be stable. The solution describing the propagation of pulses in the medium excited by a periodic wave is obtained. The hierarchy of infinitesimal symmetries is obtained by means of Darboux transformation.     

Numerical solution of steady-state porous flow free boundary problems

Summary A new numerical method is used to solve stationary free boundary problems for fluid flow through porous media. The method also applies to inhomogeneous media, and to cases with a partial unsaturated flow.     

Steady-state oscillations in continuously inhomogeneous medium described by the generalized darboux equation

We refine a result due to L. V. Ovsyannikov on the general formof the second order linear differential equations with a nonzero generalized Laplace which are invariant admitting a Lie group of transformations of the maximal order with n > 2 independent variables for which the associated Riemannian spaces have nonzero curvature. We show that the set of these equations is exhausted by the generalized Darboux equation and the Ovsyannikov equation. We find the operators acting on the set of solutions in every one-parameter family of generalized Darboux equations. For the elliptic generalized Darboux equation possessing the maximal symmetry and describing steadystate oscillations in continuously inhomogeneous medium with a degeneration hyperplane, the group analysis methods yield the exact solutions to boundary value problems for some regions (a generalized Poisson formula) which in particular can be the test solutions in simulating steadystate oscillations in continuously inhomogeneous media.     

An Efficient Cartesian Grid-Based Method for Scattering Problems with Inhomogeneous Media

Boundary integral equations provide a powerful tool for the solution of scattering problems. However, often a singular kernel arises, in which case the standard quadratures will give rise to unavoidable deteriorations in numerical precision, thus special treatment is needed to handle the singular behavior. Especially, for inhomogeneous media, it is difficult if not impossible to find out an analytical expression for Green’s function. In this paper, an efficient fourth-order accurate Cartesian grid-based method is proposed for the two-dimensional Helmholtz scattering and transmission problems with inhomogeneous media. This method provides an alternative approach to indirect integral evaluation by solving equivalent interface problems on Cartesian grid with a modified fourth-order accurate compact finite difference scheme and a fast Fourier transform preconditioned conjugate gradient (FFT-PCG) solver. A remarkable point of this method is that there is no need to know analytical expressions for Green’s function. Numerical experiments are provided to demonstrate the advantage of the current approach, including its simplicity in implementation, its high accuracy and efficiency.     

Steady-state oscillations in the continuously inhomogeneous medium described by the Ovsyannikov equation

Using the group analysis methods, for the Ovsyannikov equation with maximal symmetry which describes steady-state oscillations in a continuous inhomogeneous medium, we obtain exact solutions to boundary-value problems for some regions (generalized Poisson formulas), which in particular can serve as test solutions for simulating steady-state oscillations in continuous inhomogeneous media. We find operators acting on the set of solutions in a one-parameter family of these equations.     

Contact problems for functionally graded materials of complicated structure

An approximate analytical method allowing one to efficiently solve, to a preassigned accuracy, contact problems for materials with properties arbitrarily varying in depth is developed. Its possibilities are illustrated with the example of torsion of an elastic half-space, having a coating inhomogeneous across its thickness, by a circular stamp. All the results obtained are rigorously substantiated. For the approximate solutions constructed, their error is analyzed. The asymptotic properties of the solutions are investigated. The cases of a nonmonotonic change in the elastic properties are considered. In particular, the analytical solutions are examined in the case where the variation gradient of the elastic properties changes its sign many times. The results derived allow one to solve the inverse problems of elasticity theory of inhomogeneous media (e.g., the problem on controlling the variation in the elastic properties of a covering across its thickness).     

The scattering of classical waves from inhomogeneous media

By extending Kato's theory of two Hilbert space scattering, we are able to formulate both optical and accoustical scattering from inhomogeneous media as strictly elliptic problems. We use this formulation to present simple proofs of the existence and completeness of scattering states.     

Legendre spectral method for solving mixed boundary value problems on rectangle

In this paper, we develop a spectral method for mixed inhomogeneous Dirichlet/Neumann/Robin boundary value problems defined on rectangle. Some results on two‐dimensional Legendre approximation in Jacobi‐weighted Sobolev space are established. As examples of applications, spectral schemes are provided for two model problems with mixed inhomogeneous boundary conditions. The spectral accuracy in space of proposed algorithms are proved. Efficient implementations are presented. Numerical results demonstrate their high accuracy and confirm the theoretical analysis well. Copyright © 2016 John Wiley & Sons, Ltd.     

Superconvergence results for Galerkin methods for wave propagation in various porous media

Finite-element methods are considered for numerically solving the equations describing wave propagation in various porous media such as inhomogeneous elastic media, fluid saturated media, composite isotropic inhomogeneous elastic media, composite anisotropic media, etc. Quasi-projection analyses based on an asymptotic expansion to high order of finite-element solutions are given to obtain error estimates in Sobolev spaces of nonpositive index for the approximate solution. Superconvergence phenomena for the finite-element methods under consideration are also investigated. © 1996 John Wiley & Sons, Inc.     

The principles of correspondence between static boundary value problems of thermoviscoelasticity and thermoelasticity

Principles of correspondence between static boundary value problems of thermoviscoelasticity and thermoelasticity are proposed. This class of problems of the inhomogeneous non-linear anisotropic theory of thermoviscoelasticity is reduced by integral transformations to the corresponding class of thermoelasticity problems.     

Trace theorems for three-dimensional, time-dependent solenoidal vector fields and their applications

We study trace theorems for three-dimensional, time-dependent solenoidal vector fields. The interior function spaces we consider are natural for solving unsteady boundary value problems for the Navier-Stokes system and other systems of partial differential equations. We describe the space of restrictions of such vector fields to the boundary of the space-time cylinder and construct extension operators from this space of restrictions defined on the boundary into the interior. Only for two exceptional, but useful, values of the spatial smoothness index, the spaces for which we construct extension operators is narrower than the spaces in which we seek restrictions. The trace spaces are characterized by vector fields having different smoothnesses in directions tangential and normal to the boundary; this is a consequence of the solenoidal nature of the fields. These results are fundamental in the study of inhomogeneous boundary value problems for systems involving solenoidal vector fields. In particular, we use the trace theorems in a study of inhomogeneous boundary value problems for the Navier-Stokes system of viscous incompressible flows.

     

Effect of polarization on the trajectory of dissipative solitons

We show that the optical Magnus effect for dissipative solitons is determined not only by the helicity but also by the topological index, i.e., by the magnetic quantum number or by the projection of the soliton orbital moment on its trajectory. In the case of inhomogeneous media, we find a relation between the optical Magnus effect and the nonholonomy of the field of unit vectors tangent to the trajectory. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 1, pp. 126–134, January, 2009.     

A Hopf bifurcation in a free boundary problem with inhomogeneous media

We shall consider an interfacial problem arising reaction–diffusion models with inhomogeneous media. The purpose of this paper is to analyze the occurrence of Hopf bifurcation in the interfacial problem and to examine the effects of an inhomogeneous media. Conditions for existence of stationary solutions and Hopf bifurcation for a certain class of inhomogeneity are obtained analytically and numerically.     

Kansa-RBF algorithms for elliptic problems in regular polygonal domains

We propose matrix decomposition algorithms for the efficient solution of the linear systems arising from Kansa radial basis function discretizations of elliptic boundary value problems in regular polygonal domains. These algorithms exploit the symmetry of the domains of the problems under consideration which lead to coefficient matrices possessing block circulant structures. In particular, we consider the Poisson equation, the inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. Numerical examples demonstrating the applicability of the proposed algorithms are presented.     

Complete second order differential equations in Banach spaces with dynamic boundary conditions

In this paper, we deal with the mixed initial boundary value problem for complete second order (in time) linear differential equations in Banach spaces, in which time-derivatives occur in the boundary conditions. General wellposedness theorems are obtained (for the first time), which are used to solve the corresponding inhomogeneous problems. Examples of applications to initial boundary value problems for partial differential equations are also presented.     

Connection between holomorphic vector bundles and cohomology on a Riemann surface and conjugation boundary value problems

This paper studies interconnections between holomorphic vector bundles on compact Riemann surfaces and the solution of the homogeneous conjugation boundary value problem for analytic functions on the one hand, and cohomology and the solution of the inhomogeneous problem on the other. We establish that constructing the general solution to the homogeneous problem with arbitrary coefficients in the boundary conditions is equivalent to classifying holomorphic vector bundles. Solving the inhomogeneous problem is equivalent to checking the solvability of 1-cocycles with coefficients in the sheaf of sections of a bundle; in particular, the solvability conditions in the inhomogeneous problem determine obstructions to the solvability of 1-cocycles, i.e. the first cohomology group. Using this connection, we can apply the methods of boundary value problems to vector bundles. The results enable us to elucidate the role of boundary value problems in the general theory of Riemann surfaces.     

四元数分析中正则函数与非齐次n阶方程(■~n F)/(■z~n)=f的某些边值问题

主要讨论了四元数空间中正则函数与非齐次n阶方程(■~n F)/(■z~n)=f在超球上的Dirichlet问题和双圆柱上具有任意整数指标的Riemann-Hilbert问题,给出了可解条件和解的积分表示式.     

Integral equations in a diffraction problem on a locally inhomogeneous medium interface

We study the diffraction of an E-polarized field on a locally inhomogeneous interface of transparent media. We prove the unique solvability of the boundary value diffraction problem and obtain integral representations of the solution. We derive a system of integral equations equivalent to the original boundary value problem and prove a solvability theorem for this system.