Investigation of boundary value problems for plane-parallel fluid flows in an anisotropic porous medium

We pose and consider the first and second boundary value problems and the transmission boundary value problem for plane-parallel steady flows in an anisotropic porous medium characterized by the permeability tensor, which is not necessarily symmetric. If the anisotropic medium is homogeneous, then the solutions of the problems in the case of canonical boundaries (a straight line or an ellipse) can be found in closed form, and in the case of arbitrary smooth boundaries, the study of these problems can be reduced with the use of Cauchy type integrals to the solution of inhomogeneous integral equations of the second kind. These problems are mathematical models of topical practical problems that arise, for example, in fluid (water or oil) recovery from natural soil strata of complicated geological structure.     

Singular integrals with generalized Cauchy kernels for the velocity field in boundary value problems of filtration in an anisotropic inhomogeneous porous layer

We study two-dimensional stationary and nonstationary boundary value problems of fluid filtration in an anisotropic inhomogeneous porous layer whose conductivity is modeled by a not necessarily symmetric tensor. For the velocity field, we introduce generalized singular Cauchy and Cauchy type integrals whose kernels are expressed via the leading solutions of the main equations and have a hydrodynamic interpretation. We obtain the limit values of a Cauchy type generalized integral (Sokhotskii-Plemelj generalized formulas). This permits one to develop a method for solving boundary value problems for the filtration velocity field. The idea of the method and its efficiency are illustrated for the boundary value problem of filtration in adjacent layers of distinct conductivities and the problem of the evolution of liquid interface.     

Two boundary value problems for a strongly anisotropic inhomogeneous elastic ring

The second boundary value problem (displacements are given on the boundary) and the improper mixed problem for a cylindrically orthotropic ring are studied. It is assumed that the coefficients of elasticity are continuously differentiable functions of the coordinates and depend on a small parameter in a specific manner. The form of the dependence of the coefficients on the small parameter is selected in such a way that in the case of constant coefficients it describes bonding of the ring by two families of very rigid fibers located along the radius vectors and concentric circles, where the stiffness of the fiber families is of identical order. Consequently, the coefficients of elasticity are represented in the form of products of constants which will later be called provisionally the “stiffnesses”, and functions of the coordinates. It is assumed that the stiffnesses in the radial and circumferential directions are equal and exceed and shear stiffness considerably. The asymptotic form of the solution of the boundary value problems under consideration is constructed when the ratio between the shear stiffness and the stiffness in the radial direction is used as the small parameter. In the case of the second boundary value problem the limit boundary value problem is described by a hyperbolic system of equations and is not solvable uniquely, since one of the families of characteristics is parallel to the boundary. When constructing the asymptotic form the necessity arises to average the coefficients of elasticity with respect to the circumferential coordinate. In this respect, there is an analogy with the results obtained in /1/ where the boundary value problem was studied for a second-order elliptic equation.     

Generalized cauchy singular integral for the boundary values of two-dimensional flows in an anisotropic-inhomogeneous layer of a porous medium

We consider the main boundary value problems of two-dimensional stationary flows in an anisotropic-inhomogeneous layer with an arbitrary (not necessarily symmetric) permeability tensor. We present Cauchy integrals and Cauchy type integrals whose kernels can be expressed via the fundamental solutions of the main equations and have a hydrodynamic meaning. This permits one to develop the method of singular integral equations for solving two-dimensional boundary value problems. The considered problems can be used as mathematical models, in particular, for the extraction of fluids (water, oil) from natural layers of soil with complicated geological structure.     

An anisotropic inverse boundary value problem

We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ? Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ_*γ which will produce the same voltage and current measurements on ? Ω. We prove the converse in two dimensions (i.e., if γ₁ and γ₂ produce the same boundary measurements, then γ₁, = Ψ_*γ₂ for an appropriate Ψ) for conductivities which are near a constant.     

The third boundary value problem in potential theory for domains with a piecewise smooth boundary

The paper investigates the third boundary value problem for the Laplace equation by the means of the potential theory. The solution is sought in the form of the Newtonian potential (1), (2), where is the unknown signed measure on the boundary. The boundary condition (4) is weakly characterized by a signed measure the corresponding operator on the space of signed measures on the boundary of the investigated domain G. If there is 0 such that the essential spectral radius of is smaller than || (for example, if G R ³ is a domain with a piecewise smooth boundary and the restriction of the Newtonian potential on G is a finite continuous functions) then the third problem is uniquely solvable in the form of a single layer potential (1) with the only exception which occurs if we study the Neumann problem for a bounded domain. In this case the problem is solvable for the boundary condition for which (G) = 0.     

On a double boundary layer in a nonlinear boundary value problem

The question of the isotopy of a quasiconformal mapping and its special aspects in dimension greater than 2 are considered. It is shown that an arbitrary quasiconformal mapping of a ball has an isotopy to the identity map such that the coefficient of quasiconformality (dilatation) of the mapping varies continuously and monotonically. In contrast to the planar case, in dimension higher than 2 such an isotopy is not possible in an arbitrary domain. Examples showing specific features of the multidimensional case are given. In particular, they show that even when such an isotopy exists, it is not always possible to perform an isotopy so that the coefficient of quasiconformality approaches 1 monotonically at each point in the source domain.     

On the discrete boundary value problem for anisotropic equation

In this paper we consider the discrete anisotropic boundary value problem using critical point theory. Firstly we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one nontrivial solution. Secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two nontrivial solutions.     

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.     

Solvability of an inhomogeneous boundary value problem for steady MHD equations

In this paper, we consider the steady MHD equations with inhomogeneous boundary conditions for the velocity and the tangential component of the magnetic field. Using a new construction of the magnetic lifting, we obtain existence of weak solutions under sharp assumption on boundary data for the magnetic field.     

A Galerkin method with modified piecewise polynomials for solving a second-order boundary value problem

Summary The classical Ritz-Galerkin method is applied to a linear, second-order, self-adjoint boundary value problem. The coefficient functions of the operator exhibit a piecewise smooth behaviour characteristic of some physical situations. A trial function is constructed using a modified quintic smooth Hermite space , in order to meet some desired regularity conditions for the approximate solution. A collocation technique is used to reduce the amount of computational work. Known convergence properties for the projection method are recalled which, in this particular case, are illustrated by a series of numerical experiments.     

On a certain boundary value problem arising in shrinking sheet flows

This paper presents an analysis of the boundary value problem resulting from the magnetohydrodynamic (MHD) viscous flow influenced by a shrinking sheet with suction for the cases of two-dimensional (m = 1) and axisymmetric (m = 2) shrinking. The influences of the parameter m as well as the effects of suction parameter s and Hartmann number M² on similar entrainment velocity f(∞) and flow characteristics are studied. To this purpose, the resulting nonlinear ordinary differential equation is solved numerically using the 4th order Runge-Kutta method in combination with a shooting procedure. The obtained results elucidate reliability and efficiency of the technique from which interesting features between the skin friction coefficient f″(0) and the entrainment velocity f(∞) as function of the mass transfer parameter s can also be obtained.     

Translating solutions to the second boundary value problem for curvature flows

 We consider the flow of strictly convex hypersurfaces driven by curvature functions subject to the second boundary condition and show that they converge to translating solutions. We also discuss translating solutions for Hessian equations. Received: 10 May 2001 / Revised version: 30 January 2002     

On the second boundary value problem for a class of fully nonlinear flows II

This article is a continuation of an earlier work (Huang and Ye in Int Math Res Not, 2017.  https://doi.org/10.1093/imrn/rnx278), where the long time existence and convergence for some special cases of parabolic type special Lagrangian equations were given. The long time existence and convergence of the flow are obtained for all cases in this article. In particular, we can prescribe the second boundary value problems for a family of special Lagrangian graphs.