The convolution method for fourier expansions: The case of a crack (screen) in an inhomogeneous space

We consider boundary value problems in the space R _n for the equation

A method of Fourier series for solution of problems in piecewise inhomogeneous domains with rectilinear crack (screen)

In the framework of the theory of harmonic functions, potentials of steady state processes (heat conduction, filtration, or electrostatics) in the piecewise inhomogeneous plane separated by a rectilinear strongly permeable crack or by a weakly permeable screen into two half-planes with quadratic permeability functions are constructed. The motion is induced by given singular points of the potential (sources, sinks, etc.). Compact formulas that directly express potentials in these domains in terms of harmonic functions are obtained; the resulting functions map the set of harmonic functions to the set of potentials conserving the type of singularities.     

Solution of boundary-value problems with generalized transmission conditions of the type of a crack or a screen

We obtain formulas for solutions of boundary-value problems similar to classical ones in cylindrical domains under additional generalized transmission conditions of the type of a strongly permeable crack or a weakly permeable screen. The obtained solutions are operations that affect a known function only in one variable. By a composition of the mentioned operators we solve boundary-value problems with generalized transmission conditions on intersecting surfaces.     

The factorization method for cracks in inhomogeneous media

We consider the inverse scattering problem of determining the shape and location of a crack surrounded by a known inhomogeneous media. Both the Dirichlet boundary condition and a mixed type boundary conditions are considered. In order to avoid using the background Green function in the inversion process, a reciprocity relationship between the Green function and the solution of an auxiliary scattering problem is proved. Then we focus on extending the factorization method to our inverse shape reconstruction problems by using far field measurements at fixed wave number. We remark that this is done in a non intuitive space for the mixed type boundary condition as we indicate in the sequel.     

Approximate method for solving the boundary value problem for a parabolic equation with inhomogeneous transmission conditions of nonideal contact type

A linear parabolic equation in a disconnected domain with inhomogeneous transmission conditions of the nonideal contact type is studied. A generalized formulation of the problem is considered. An analogue of the Galerkin method is proposed for solving the problem, and the stability of the method is investigated. This makes it possible to prove existence and uniqueness theorems for the equation under different assumptions on the data smoothness.     

The Urbanik generalized convolutions in the non-commutative probability and a forgotten method of constructing generalized convolution

The paper deals with the notions of weak stability and weak generalized convolution with respect to a generalized convolution, introduced by Kucharczak and Urbanik. We study properties of such objects and give examples of weakly stable measures with respect to the Kendall convolution. Moreover, we show that in the context of non-commutative probability, two operations: the q-convolution and the (q,1)-convolution satisfy the Urbanik??s conditions for a generalized convolution, interpreted on the set of moment sequences. The weak stability reveals the relation between two operations.     

Study of a boundary value transmission problem for two-dimensional flows in a piecewise anisotropic inhomogeneous porous layer

We consider a boundary value transmission problem for two-dimensional filtration flows in an anisotropic porous layer consisting of adjacent domains in which the media have essentially different conductivities (permeability and thickness). In general, the layer conductivity is specified by a nonsymmetric second rank tensor whose components are modeled by continuously differentiable functions of coordinates. To study the problem, we use two complex planes, the physical plane and an auxiliary plane, which are related by a homeomorphic (one-to-one and continuous) transformation satisfying an equation of the Beltrami type. On the physical plane, we pose a transmission problem for a rather complicated elliptic system of equations. This problem is reduced on the auxiliary plane to canonical form, which dramatically simplifies the analysis of the problem. Then the problem is reduced to a system of boundary singular integral equations with generalized kernels of the Cauchy type, which are expressed via the fundamental solutions of the main equations. The boundary value transmission problem studied here can be used as a mathematical model of processes arising in the recovery of fluids (water and oil) from natural soil formations of complicated geological structure.     

A new non-iterative singular sources method for the reconstruction of piecewise constant media.

Summary. We describe a novel non-iterative method for the reconstruction of a piecewise constant inhomogeneous medium in acoustic scattering, which we call the singular sources method. The basic idea of the method is to use the behaviour of the scattered field for singular incident fields (multipoles) to calculate the size of the refractive index n at some point z₀ on the boundary of the support of the scatterer and then eliminate this value from the data by subtracting a known piecewise constant background medium. The paper includes the theory for the singular sources method to locate the unknown support of an inhomogeneous medium for a known inhomogeneous background medium. Also, we give a new uniqueness proof for the reconstruction of a piecewise constant medium in two or three dimensions, using techniques that differ from those used to prove previous well-known results.Mathematics Subject Classification (2000): 35J05, 45Q05, 47A52, 78A46, 81U40Revised version received August 6, 2003     

The Nother and Riemann-Roch type theorems for piecewise algebraic curve

A piecewise algebraic curve is a curve determined by the zero set of a bivariate spline function. In this paper, the Nother type theorems for Cμpiecewise algebraic curves are obtained. The theory of the linear series of sets of places on the piecewise algebraic curve is also established. In this theory, singular cycles are put into the linear series, and a complete series of the piecewise algebraic curves consists of all effective ordinary cycles in an equivalence class and all effective singular cycles which are equivalent specifically to any effective ordinary cycle in the equivalence class. This theory is a generalization of that of linear series of the algebraic curve. With this theory and the fundamental theory of multivariate splines on smoothing cofactors and global conformality conditions, and the results on the general expression of multivariate splines, we get a formula on the index, the order and the dimension of a complete series of the irreducible Cμpiecewise algebraic curves and the degree, the genus and the smoothness of the curves, hence the Riemann-Roch type theorem of the Cμpiecewise algebraic curve is established.     

Estimates and asymptotic expansions for condenser p-capacities. The anisotropic case of segments

We provide estimates and asymptotic expansions of condenser p-capacities and focus on the anisotropic case of (line) segments.

After preliminary results, we study p-capacities of points with respect to asymptotic approximations, positivity cases and convergence speed of descending continuity. We introduce equidistant condensers to point out that the anisotropy caused by a segment in the p-Laplace equation is such that the Pólya-Szegö rearrangement inequality for Dirichlet type integrals yields a trivial lower bound. Moreover, when p > N , one cannot build an admissible solution for a segment, however small its length may be, by extending the case of a punctual obstacle.

Our main contribution is to provide a lower bound to the N -dimensional condenser p-capacity of a segment, by means of the N -dimensional and of the (N ?1)-dimensional condenser p-capacities of a point. The positivity cases follow for p-capacities of segments. Our method could be extended to obstacles with codimensions ≥ 2 in higher dimensions, such as surfaces in ?⁴.

Introducing elliptical condensers, we obtain an estimate and the asymptotic expansion for the condenser 2-capacity of a segment in the plane. The topological gradient of the 2-capacity is not an appropriate tool to separate curves and obstacles with non-empty interior in 2D. In the case p ≠ 2, elliptical condensers should prove useful to obtain further estimates of p-capacities of segments.     

Existence of solutions for dual singular integral equations with convolution kernels in case of non-normal type

This paper is devoted to the study of dual singular integral equations with convolution kernels in the case of non-normal type. Via using the Fourier transforms, we transform such equations into Riemann boundary value problems. To solve the equation, we establish the regularity theory of solvability. The general solutions and the solvable conditions of the equation are obtained. Especially, we investigate the asymptotic property of solutions at nodes. This paper will have a significant meaning for the study of improving and developing complex analysis, integral equations and Riemann boundary value problems.     

Homogenization of boundary value problems for monotone operators in perforated domains with rapidly oscillating boundary conditions of fourier type

We deal with homogenization problem for nonlinear elliptic and parabolic equations in a periodically perforated domain, a nonlinear Fourier boundary conditions being imposed on the perforation border. Under the assumptions that the studied differential equation satisfies monotonicity and 2-growth conditions and that the coefficient of the boundary operator is centered at each level set of unknown function, we show that the problem under consideration admits homogenization and derive the effective model. Bibliography: 24 titles.     

Galerkin finite element method for generalized Forchheimer equation of slightly compressible fluids in porous media

We consider the generalized Forchheimer flows for slightly compressible fluids. Using Muskat's and Ward's general form of Forchheimer equations, we describe the fluid dynamics by a nonlinear degenerate parabolic equation for the density. We study Galerkin finite elements method for the initial boundary value problem. The existence and uniqueness of the approximation are proved. A prior estimates for the solutions in , time derivative in and gradient in , with a∈(0,1) are established. Error estimates for the density variable are derived in several norms for both continuous and discrete time procedures. Numerical experiments using backward Euler scheme confirm the theoretical analysis regarding convergence rates. Copyright © 2017 John Wiley & Sons, Ltd.     

A procedure for computing successive approximations in the asymptotic space-time (ST) ray method of Gaussian beams. II. (An isotropic inhomogeneous elastic medium)

The method of spatial rays is applied in the construction of the formulas for the asymptotic expansions of the Gaussian beam type in an elastic medium. The coefficients for higher-order approximations of the expansions constructed are subject to a recurrent process. The process is considered for theP (orS) component of the displacement vector in an inhomogeneous elastic medium taken as an example.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 88–100, 1989.     

An analytic method for the initial value problem of the electric field system in vertical inhomogeneous anisotropic media

The time-dependent system of partial differential equations of the second order describing the electric wave propagation in vertically inhomogeneous electrically and magnetically biaxial anisotropic media is considered. A new analytical method for solving an initial value problem for this system is the main object of the paper. This method consists in the following: the initial value problem is written in terms of Fourier images with respect to lateral space variables, then the resulting problem is reduced to an operator integral equation. After that the operator integral equation is solved by the method of successive approximations. Finally, a solution of the original initial value problem is found by the inverse Fourier transform.     

The local Rankin-Selberg convolution for GL(n): divisibility of the conductor

Let F be a non-Archimedean local field and an integer. Let be irreducible supercuspidal representations of GL with . One knows that there exists an irreducible supercuspidal representation of GL, with , such that the local constants (in the sense of Jacquet, Piatetskii-Shapiro and Shalika) are distinct. In this paper, we show that, when is an unramified twist of , one may here takem dividingn and , for a prime divisor ofn depending on and the order of : in particular, , where is the least prime divisor of . This follows from a result giving control of certain divisibility properties of the conductor of a pair of supercuspidal representations. Received: 11 November 2000 / Accepted: 15 January 2001 / Published online: 23 July 2001     

The partial regularity for minimizers of splitting type variational integrals under general growth conditions i. the autonomous case

We consider splitting type variational problems with general growth conditions and prove the partial regularity (and the full regularity in 2D) of minimizers in the case of x-dependence. The results obtained generalize the results of Bildhauer and Fuchs concerning such problems with power growth conditions. Bibliography: 17 titles.