Optimal control of the layer size in the problem of equilibrium of elastic bodies with overlapping domains

We consider the problem of equilibrium of a two-layer elastic body. The first of the layers contains a crack,while the second is a circle centered at one of the crack tips. The round layer is glued by its boundary to the first layer. The unique solvability is proved for this problem in the nonlinear formulation. An optimal control problem is also considered. The radius a of the second layer is chosen as a varying parameter under assumption that a takes positive values from a closed interval. It is shown that there are a value of a minimizing the functional that characterizes how potential energy depends on the crack length and a value of a minimizing the functional that characterizes the opening of the crack.     

Approximation of the solution and its derivative for the singularly perturbed Black-Scholes equation with nonsmooth initial data

A problem for the black-Scholes equation that arises in financial mathematics is reduced, by a transformation of variables, to the Cauchy problem for a singularly perturbed parabolic equation with the variables x, t and a perturbation parameter ɛ, ɛ ∈ (0, 1]. This problem has several singularities such as the unbounded domain, the piecewise smooth initial function (its first-order derivative in x has a discontinuity of the first kind at the point x = 0), an interior (moving in time) layer generated by the piecewise smooth initial function for small values of the parameter ɛ, etc. In this paper, a grid approximation of the solution and its first-order derivative is studied in a finite domain including the interior layer. On a uniform mesh, using the method of additive splitting of a singularity of the interior layer type, a special difference scheme is constructed that allows us to ɛ-uniformly approximate both the solution to the boundary value problem and its first-order derivative in x with convergence orders close to 1 and 0.5, respectively. The efficiency of the constructed scheme is illustrated by numerical experiments. The text was submitted by the authors in English.     

Thin-layer version of the moment equations in the boundary layer problem

The two-dimensional problem of a hypersonic kinetic boundary layer developing on a thin body in the case of a monatomic gas is considered. The model of the flow arises from the kinetic theory of gases and, within its accuracy, i.e., in the approximation of a hypersonic boundary layer, takes into account the strong nonequilibrium of the flow with respect to translational degrees of freedom. A method for representing the solution of the problem in terms of the solution of a similar classical (Navier-Stokes) hypersonic boundary layer problem is described. For the kinetic version of the problem, it is shown that the shear stress and the specific heat flux on the body surface are equal to their counterparts in the Navier-Stokes boundary layer.     

The order of convergence in the stefan problem with vanishing specific heat

The paper is concerned with the two-phase Stefan problem with a small parameter ϵ, which coresponds to the specific heat of the material. It is assumed that the initial condition does not coincide with the solution for t = 0 of the limit problem related to ε = 0. To remove this discrepancy, an auxiliary boundary layer type function is introduced. It is proved that the solution to the two-phase Stefan problem with parameter ϵ differs from the sum of the solution to the limit Hele–Shaw problem and a boundary layer type function by quantities of order O(ϵ). The estimates are obtained in H?lder norms. Bibliography: 13 titles.     

Boundary Integral Solution of the Time-Fractional Diffusion Equation

Here we consider initial boundary value problem for the time–fractional diffusion equation by using the single layer potential representation for the solution. We derive the equivalent boundary integral equation. We will show that the single layer potential admits the usual jump relations and discuss the mapping properties of the single layer operator in the anisotropic Sobolev spaces. Our main theorem is that the single layer operator is coercive in an anisotropic Sobolev space. Based on the coercivity and continuity of the single layer operator we finally show the bijectivity of the operator in a certain range of anisotropic Sobolev spaces.      

On the Parameter-Uniform Convergence of Exponential Spline Interpolation in the Presence of a Boundary Layer

The paper is concerned with the problem of generalized spline interpolation of functions having large-gradient regions. Splines of the class C², represented on each interval of the grid by the sum of a second-degree polynomial and a boundary layer function, are considered. The existence and uniqueness of the interpolation L-spline are proven, and asymptotically exact two-sided error estimates for the class of functions with an exponential boundary layer are obtained. It is established that the cubic and parabolic interpolation splines are limiting for the solution of the given problem. The results of numerical experiments are presented.     

The three-dimensional problem of the axisymmetric deformation of an orthotropic spherical layer

A 3D problem of the deformation of an elastic orthotropic spherical layer that is subjected to normal pressure applied to its outer and inner surfaces is analyzed. Asymptotic first-order approximation solutions are obtained for a slightly orthotropic layer for which the elastic moduli in the meridional and circumferential directions have similar values. The solutions that are obtained are used for analyzing the scleral shell under intraocular pressure; however, they can also be used for solving the inverse problem of analyzing the stress–strain state of a human eye during intravitreal injections. The influence that the meridional and circumferential elastic moduli have on the magnitudes of changes in the relative layer thickness and in the length of the anteroposterior eye axis due to elevated intraocular pressure is studied.     

On the global existence and uniqueness of solutions to the nonstationary boundary layer system

In this paper, we study the problem of boundary layer for nonstationary flows of viscous incompressible fluids. There are some open problems in the field of boundary layer. The method used here is mainly based on a transformation which reduces the boundary layer system to an initial-boundary value problem for a single quasilinear parabolic equation. We prove the existence of weak solutions to the modified nonstationary boundary layer system. Moreover, the stability and uniqueness of weak solutions are discussed.     

An adapted Galerkin method for the resolution of Dirichlet and Neumann problems in a polygonal domain

The Neumann problem for Laplace's equation in a polygonal domain is associated with the exterior Dirichlet problem obtained by requiring the continuity of the potential through the boundary. Then the solution is the simple layer potential of the charge q on the boundary. q is the solution of a Fredholm integral equation of the second kind that we solve by the Galerkin method. The charge q has a singular part due to the corners, so the optimal order of convergence is not reached with a uniform mesh. We restore this optimal order by grading the mesh adequately near the corners. The interior Dirichlet problem is solved analogously, by expressing the solution as a double layer potential.     

Mathematical model of fuel layer degradation when the laser target is heated by thermal radiation in the reactor working chamber

We propose a mathematical model of the changes occurring in the geometrical properties of the deuterium–tritium layer on the laser target in the process of its insertion into the reactor working chamber. The model is a parabolic equation of general form in spherical coordinates with nonlinear boundary conditions on a moving boundary. We show that under physically justified assumptions this problem may be regarded as a Stefan problem for a singularly perturbed parabolic equation. The first terms of the solution series are written out. Numerical calculations of the fuel layer degradation time are presented for a real target.     

Seepage from a channel in an irrigated soil layer in the presence of an inclusion in the underlying highly permeable zero-head layer

We investigate the boundary-value problem describing plane steady seepage through a soil layer into an underlying zero-head layer capped by an impervious section. Uniform infiltration is assumed on the free surface. The solution of the problem is used to construct a direct computer algorithm; the computation results are illustrated with tables and diagrams.Translated from Vychislitei'naya i Prikladnaya Matematika, No. 62, pp. 52–56, 1987.     

Solution of the Neumann problem for the Laplace equation

For fairly general open sets it is shown that we can express a solution of the Neumann problem for the Laplace equation in the form of a single layer potential of a signed measure which is given by a concrete series. If the open set is simply connected and bounded then the solution of the Dirichlet problem is the double layer potential with a density given by a similar series.     

Propagation of TM waves in a layer with arbitrary nonlinearity

A boundary value problem for Maxwell’s equations describing propagation of TM waves in a nonlinear dielectric layer with arbitrary nonlinearity is considered. The layer is located between two linear semi-infinite media. The problem is reduced to a nonlinear boundary eigenvalue problem for a system of second-order nonlinear ordinary differential equations. A dispersion equation for the eigenvalues of the problem (propagation constants) is derived. For a given nonlinearity function, the dispersion equation can be studied both analytically and numerically. A sufficient condition for the existence of at least one eigenvalue is formulated.     

Efficient numerical algorithm for solving the gravimetry problem of finding a lateral density in a layer: Parallel implementation

The paper is devoted to developing the new time- and memory-efficient algorithm BiCGSTABmem for solving the inverse gravimetry problem of determination of a variable density in a layer using the gravitational data. The problem is in solving the linear Fredholm integral equation of the first kind. After discretization of the domain and approximation of the integral operator, this problem is reduced to solving a large system of linear algebraic equations. It is shown that the matrix of coefficients is the Toeplitz-block-Toeplitz one in the case of the horizontal layer. For calculating and storing the elements of this matrix, we construct an efficient method, which significantly reduces the required memory and time. For the case of the curvilinear layer, we construct a method for approximating the parts of the matrix by a Toeplitz-block-Toeplitz one. This allows us to exploit the same efficient method for storing and processing the coefficient matrix in the case of a curvilinear layer. To solve the system of linear equations, we constructed the parallel algorithm on the basis of the stabilized biconjugated gradient method with using the Toeplitz-block-Toeplitz structure of the matrix. We implemented the BiCGSTAB and BiCGSTABmem algorithms for the Uran cluster supercomputer using the hybrid MPI + OpenMP technology. A model problem with synthetic data was solved for a large grid. It was shown that the new BiCGSTABmem algorithm reduces the computation time in comparison with the BiCGSTAB. Scalability of the parallel algorithm was studied.     

A nonlinear optimal control problem for the nonsteady temperature of a layer with constraints on the thermal stress

We propose a numerical method of constructing the optimal heating regime for a thermally stressed unbounded layer with constraints on the control and thermal stresses. Solving the nonlinear optimization problem for rapidity is reduced to solving the inverse problem of thermoelasticity. The results of numerical studies are presented. Translated fromMatematichni Metody i Fiziko-Mekhanichni Polya, Vol. 38, 1995.     

Asymptotic of the Solution of the Contact Problem for a Thin Elastic Plate and a Viscoelastic Layer

The contact problem for a thin elastic rigid plate described by the elasticity equations and a viscoelastic layer is solved. The ratio of the thicknesses of the plate and the layer is a small parameter, while the ratio of the Young’s moduli of the layer and the plate is proportional to the cube of this parameter. The asymptotic expansion of the solution is constructed. A theorem on the estimate of the error of asymptotic approximation is formulated. Such problem appears in geophysics, in modeling of the Earth crust–magma interaction.     

The linearization method in the inverse problem of determination of low-permeability zones in an oil-bearing layer

The possibility of determination of low-permeability zones in an oil-bearing layer from bore-pressure measurements is studied. The formulation of the problem on the pressure distribution in the layer is considered in the case when the pressure is invariable across the layer. The problem is reduced to an integral equation. The linearization of the inverse problem of determination of low-permeability zones is considered. An iterative technique is proposed for solution of this problem. The technique involves reduction of the inverse problem to an operator equation.     

The diffusion of a discontinuity of the shear stress at the boundary of a viscoplastic half-plane

For the problem of the diffusion of a discontinuity of the shear stress at the boundary of a half-plane, which is a special case of the general problem of the diffusion of a vortex layer, the classes of media and types of assignment of boundary conditions for which self-similar solutions exist are discussed. For a viscoplastic medium in a half-plane the problem reduces to the problem in a layer of time-variable thickness, the solution of which does not possess the property of analyticity. The long-term asymptotic of this problem are investigated. In the case where, at an accessible boundary, it is possible simultaneously to measure both the shear stress and the horizontal velocity, an algorithm is proposed for finding a quantity that is difficult to measure, A namely, the thickness of the zone of viscoplastic flow.     

The stress state of the finite cylinder with the circular cracks

The axysimmetrical torsion problem for a finite cylinder with an arbitrary quantity of the cylindrical layers is solved. The cylinder is weaked by the parallel circular cracks in the first internal layer. The stated boundary valued problem problem is reduced to the system of the integro-duferential equations solving with the help of the orthogonal polinomials method. The stress intensity factors (SIF) are obtained. The dependences of SIF values from the cracks' sizes, their location, and ratio of the layers' shear moduluses are concretized for the case of the two layers. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solutions for the problem of boundary layer formation in a pseudo-plastic fluid: The case of gradual acceleration

We consider the development of the nonstationary boundary layer about a body that gradually starts to move in a resting fluid. Under certain conditions, we construct the solutions for the problem of formation of boundary layer in a pseudo-plastic fluid. The method used here is mainly based on a transformation which reduces the boundary layer system to a boundary value problem for a single quasilinear parabolic equation.