An anti-plane shear problem

We consider an anti-plane shear of an elastic cylinder with a non-convex stored energy function. So, we look for solutions of a non-convex problem of the calculus of variations with Dirichlet boundary conditions. We give sufficient conditions on the boundary data to get existence or non-existence results for this non-convex problem. We also prove some uniqueness results for the relaxed problem associated with the initial problem.     

Inverse electromagnetic scattering for a locally perturbed perfectly conducting plate

We consider a simple but fully three-dimensional inverse problem to determine the shape of a local perturbation of a perfectly conducting plate from far-field measurements of time harmonic electromagnetic fields. For this purpose we reformulate the model problem as an exterior Maxwell problem for a symmetric domain, and prove an equivalence between the model problem and its reformulation. Then, linear sampling method is applied to solve the reformulated problem. We illustrate the feasibility of this method by some numerical examples.     

On some diffusive waves in non-linear heat conduction

We address the non-linear heat conduction in the presence of absorption for the case of spherical symmetry geometry. The non-linear model is based on both a temperature-dependent thermal conductivity and a non-linear generalization of the Fourier law. The governing equation belongs to a class of degenerate parabolic equations. We obtain similarity solutions in closed form for the Cauchy problem corresponding to an instantaneous point source problem. We investigate the non-linear effects on the propagation of the temperature distrubances. We find that in certain cases the temperature distribution displays travelling wave characteristics. The solution for the Cauchy problem is recovered by considering a suitable first boundary value problem.     

Singularities in auxetic elastic bimaterials

In this paper we study the effects of negative Poisson's ratios on elastic problems containing singularities. Materials with a negative Poisson's ratio are termed auxetic. We present a brief review of such materials. The elasticity problem of a bimateral wedge is presented, then two particular cases of this problem are investigated: the free-edge problem and the interface crack problem. We study the effect on the stress singularity due to one portion of the bimaterial becoming auxetic. We find that the auxetic material has a significant effect on the singularity order, even causing the singularity to vanish for certain values of the elastic constants.     

Multipoint problem for higher-order parabolic equations in a parallelepiped

We study the well-posedness of a problem for a Petrovskii-parabolic equation with coefficients depending on the space coordinates and with multipoint conditions with respect to the time variable. We establish conditions for the existence and uniqueness of the classical solution of the problem. For the proof of the existence of a solution of the problem, the method of divided differences is used. We prove a metric-type theorem on lower bounds for the small denominators that appear in the construction of the solution.     

Optimal Microstructures Drag Reducing Mechanism of Riblets

We consider an optimal shape design problem of periodically distributed three-dimensional microstructures on surfaces of swimming bodies in order to reduce their drag. Our model is restricted to the flow in the viscous sublayer of the boundary layer of a turbulent flow. The costs for the optimization problem are very high because the three-dimensional flow equations have to be solved several times. We avoid this problem by approximations: the microscopic optimization problem is reduced applying homogenization. Considering a special geometry (riblets) the resulting so-called macroscopic optimization problem can be additionally reduced to a two-dimensional problem. We analyze the drag reducing mechanism of riblets which are believed to be optimal structures. Therefore we perform direct simulations on the total rough channel for different shapes of microstructures: riblets and fully three-dimensional structures.     

On the Numerical-Analytic Investigation of Parametrized Problems with Nonlinear Boundary Conditions

We consider a parametrized boundary-value problem containing an unknown parameter both in nonlinear ordinary differential equations and in nonlinear boundary conditions. By using a suitable change of variables, we reduce the original problem to a family of problems with linear boundary conditions plus certain nonlinear algebraic determining equations. We construct a numerical-analytic scheme suitable for studying the solutions of the transformed boundary-value problem.     

On one class of differential equations of fractional order

We consider the Darboux problem for a differential equation of fractional order that contains a regularized mixed derivative. Sufficient conditions for the existence and uniqueness of a solution of this problem are obtained in the class of continuous functions. We also propose a method for finding an approximate solution of this problem and prove the convergence of this method.     

Computation of aeroelastic transfer functions of aircraft elements under vibrations in a compressible flow

We consider the problem on the computation of aeroelastic transfer functions of an airfoil under vibrations in a compressible flow with the actual natural frequencies of the airfoil taken into account. The problem is reduced to a Neumann type boundary value problem for the scalar Helmholtz equation.We estimate the reliability of the computation results.     

Axisymmetric strain of a thick-walled pipe made of a highly elastic material

We reduce the plane strain problem to a nonlinear elasticity problem for inhomogeneous bodies by choosing a new form of the elastic potential, whose parameters are determined from the data known in the literature. By using the geometric linearization method, we reduce that problem to a sequence of linear elasticity problems for inhomogeneous bodies. We obtain an analytic solution of the corresponding linear elasticity problem in the case of an arbitrary continuously differentiable dependence of the shear modulus on the radial coordinate. We determine the pipe stress-strain state and parameters in the case of finite and large strains for given sets of initial data and estimate the accuracy of the solution thus obtained.     

Problem of interaction of thin rigid needle-shaped inclusions located between different elastic materials

We study a piecewise-homogeneous elastic plane composed of two half-planes with different elastic parameters and two thin rigid needle-shaped inclusions located between them. One inclusion is rigidly connected with the environment, and the other inclusion is not, while contacting with it like a smooth rigid punch. We consider the plane deformed state generated by stresses given at infinity. The problem is reduced to a combination of a matrix Riemann boundary-value problem from the theory of analytic functions and a matrix Hilbert problem, which can be solved in terms of integrals through the reduction to two separate scalar Riemann boundary-value problems on a twosheeted Riemann surface.We explicitly obtain the complex potentials of the composite elastic plane, the stress intensity factors near the tips of the inclusion, and the rotation angles of the inclusions. We also present numerical examples illustrating how the stresses near the inclusions depend on the elastic and geometric parameters of the problem.     

On the construction of asymptotic solutions of two-point boundary-value problems for degenerate singularly perturbed systems of differential equations

We consider the problem of the existence of a solution of a two-point boundary-value problem for degenerate singularly perturbed linear systems of differential equations. We obtain asymptotic formulas for this solution.     

On two-scale homogenized equations of one-dimensional nonlinear thermoviscoelasticity with rapidly oscillating nonsmooth data

We study the initial-boundary value problem for a system of quasilinear equations of one-dimensional nonlinear thermoviscoelasticity with rapidly oscillating nonsmooth coefficients and initial data. We rigorously justify the passage to the corresponding limit initial-boundary value problem for a system of two-scale homogenized integro-differential equations, including the existence theorem for the limit problem. The results are global with respect to the time interval and the data.     

On the Investigation of One Semiexplicit System of Differential Equations in the Case of a Variable Pencil of Matrices

We investigate a semiexplicit Cauchy problem for a system of ordinary differential equations in the case of a variable pencil of matrices. We determine sufficient conditions of existence of solutions and consider the problem of their number.     

A numerical approximation of the thermal coupling of fluids and solids

In this article we analyze the problem of the thermal coupling of fluids and solids through a common interface. We state the global thermal problem in the whole domain, including the fluid part and the solid part. This global thermal problem presents discontinuous physical properties that depend on the solution of auxiliary problems on each part of the domain (a fluid flow problem and a solid state problem). We present a domain decomposition strategy to iteratively solve problems posed in both subdomains and discuss some implementation aspects of the algorithm. This domain decomposition framework is also used to revisit the use of wall function approaches used in this context. Copyright © 2008 John Wiley & Sons, Ltd.     

On a non-linear moving boundary problem for a diffusion–convection equation

We study a one-dimensional free boundary problem for a non-linear diffusion–convection equation whose diffusivity is heterogeneous in space as well as being non-linear. Under the Bäcklund transformation the problem is reduced to an associated free boundary problem. We prove the existence and uniqueness, local in time, of the solution by using the Friedman Rubinstein integral representation method and the Banach contraction theorem.     

Hamilton Jacobi Equations with Obstacles

We consider a problem in the theory of optimal control proposed for the first time by Bressan. We characterize the associated minimum time function using tools from geometric measure theory and we obtain, as a corollary, an existence theorem for a related variational problem.     

Analysis of right closed thin-walled prismatic shells supported by stringers with geometric and physical nonlinearity taken into account

We consider thin-walled right-angle closed prismatic shells with rigid contour of the transverse cross-section. Such shells underlie the schemes used in the analysis of various thin-walled spatial structures. The use of nonlinear physical and geometric relations in the computations permits numerically obtaining the strength margin of the corresponding structures. In the present paper, we propose methods for obtaining a boundary value problem and analyzing such shells with nonlinear factors taken into account; the problem is presented as a system of linear differential equations with variable coefficients. We show that, within the approach proposed, this boundary value problem has a fixed structure independent of the special form of nonlinearity. The entire variety of problems of static analysis of right-angle prismatic shells with nonlinear factors taken into account can be reduced to solving this boundary value problem. Methods for taking a specific nonlinearity into account are treated as various methods for obtaining expressions for the variable coefficients in the matrices of the boundary value problem. We present methods for solving this boundary value problem numerically; these methods are independent of the specific form of the nonlinearity.     

Boundary-value problems for differential equations unresolved with respect to the derivative in the space of bounded number sequences

We study a nonlinear countable-point boundary-value problem for a differential equation unresolved with respect to the derivative. This equation and a nonlinear boundary condition are defined in the Banach space of bounded number sequences. We study the reducibility of the posed problem to a multipoint boundary-value problem in a finite-dimensional space. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 391–415, July–September, 2007.     

Control of homogeneous shear flow of multimode Maxwell fluids

We consider a homogeneous parallel shear flow of a multimode Maxwell fluid. This problem results in a set of ordinary differential equations for the stresses. In this system, we view the shear rate as a control and consider the problem of steering the system to a given state of stress. The objective is to steer the system from given initial stresses to a final state of stress, allowing the shear rate to vary in an arbitrary fashion. We show that this problem is related to a calculus of variations problem. For the case of two modes, we obtain a characterization of the set of achievable streses.