Generalized green operator for an impulsive boundary-value problem with switchings

We establish constructive existence conditions and construct a generalized Green operator for the construction of solutions of a Noetherian linear boundary-value problem for a system of ordinary differential equations with switchings and pulse action in critical and noncritical cases. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 1, pp. 51–65, January–March, 2007.     

Asymptotic expansion of solutions of a singularly perturbed boundary-value problem

For linear singularly perturbed systems of ordinary differential equations, we construct an asymptotic expansion of a solution by using the method of boundary functions. Using pseudoinverse matrices and projections, we find all terms of the asymptotic expansion in the noncritical case.Translated from Neliniini Kolyvannya, Vol. 7, No. 2, pp. 155–168, April–June, 2004.     

A second-order multivalued boundary-value problem and impulsive neutral functional differential inclusions in Banach spaces

In this paper, by using the new fixed-point theorem of O'Regan and Precup and a noncompact measure, we study the existence of solutions of a second-order multivalued boundary-value problem in Banach spaces and the existence of a mild solution for impulsive neutral functional differential inclusions in Banach spaces. The compactness conditions and the upper-semicontinuity conditions for multivalued integral operators are weakened in this paper. Published in Neliniini Kolyvannya, Vol. 11, No. 2, pp. 191–207, April–June, 2008.     

Bifurcation of non-negative solutions for an elliptic system

In the paper, we consider a nonlinear elliptic system coming from the predator-prey model with diffusion. Predator growth-rate is treated as bifurcation parameter. The range of parameter is found for which there exists nontrivial solution via the theory of bifurcation from infinity, local bifurcation and global bifurcation.     

Approximate optimal stabilization of solutions of a parabolic boundary-value problem by bounded control

For the problem of the optimal stabilization of a process described by a parabolic boundary-value problem, we construct and substantiate an approximate averaged feedback control (synthesis). __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 2, pp. 264–279, April–June, 2006.     

Domain of convergence of an iterative procedure for an autonomous boundary-value problem

We find an estimate for the range of values of a small parameter for which the convergence of an iterative procedure for the construction of solutions of an autonomous weakly nonlinear Noether boundary-value problem for a system of ordinary differential equations in the critical case is preserved. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 416–432, July–September, 2006.     

Periodic solutions of an impulsive system with two discontinuity points

We establish conditions for the existence of a 2π-periodic solution of an impulsive system with two discontinuity points at nonfixed times.     

Analytical solutions of the boundary-value problem of nonstationary flow of viscoplastic medium between two plates

Summary Nonstationary flow of a viscoplastic medium between two parallel plates is considered for the case of a varying pressure gradient. The problem is reduced to the Stephan problem, with the condition on the boundary separating the flow domain from the quasi-rigid domain. Four multiparameter families of exact solutions are found. The first family describes the flow decelerations up to a full stop. The second family determines the development of the flow from the state of rest as the pressure gradient increases. The third family describes the development of the flow for the case where (1) the pressure gradient is constant and exceeds the threshold value related to the yield stress, (2) the upper plate does not move, and (3) the lower plate moves with a constant acceleration. Finally, the fourth family determines the flow retardation, when the pressure gradient is constant and is less than the threshold value. The decrease in the flow of the viscoplastic medium can be achieved for certain values of parameters by increasing the quasi-rigid domain, whereas the viscoplastic flow remains unchanged. Received 7 October 1998; accepted for publication 8 April 1999     

Acceleration of convergence of an iteration procedure for a weakly nonlinear boundary-value problem

We construct a new iteration procedure for finding solutions of a Noetherian weakly nonlinear boundary-value problem for a system of ordinary differential equations in the noncritical case. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 127–132, January–March, 2006.     

A class of exact solutions of a nonlinear boundary-value problem for fluid flow in a rotating cylinder

The flow of an ideal incompressible weightless fluid that fills a rotating cylinder is investigated. The rotation axis of the cylinder is outside it and parallel to the cylinder generator, and the form of the cylinder section is determined in the process of solution of the problem. In the paper, a class of exact solutions of the problem is obtained in terms of elementary functions for different angular velocities of the cylinder. In these solutions, the flow field is formed by two straight vortex filaments parallel to the cylinder generator. The intensities of the vortex filaments are determined by the angular velocity . Investigations of ideal fluid flow in rotating vessels were begun already in the last century by Stokes and Zhukovskii [1]. The subject has been reviewed in monographs [2, 3].Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No, 1, pp. 71–75, January–February, 1984.     

On the problem of integrability of the axisymmetric boundary-value problem of dynamics for an inhomogeneous anisotropic finite cylinder

A closed solution is obtained for the axisymmetric boundary-value problem of dynamics for a finite cylinder with exponential elasticity and inertial inhomogeneity and a certain relationship between elastic constants on the basis of correlations of the linear theory of elasticity of an anisotropic inhomogeneous body. The boundary conditions are arbitrary on the curvilinear surface and are given in mixed form on the ends. The method of finite integral transforms is employed. Specific cases for cylinders of transverscly isotropic and isotropic homogeneous material are discussed. Institute of Architecture and Civil Engineering, Samara, Russia. Translated from Prikladnaya Mekhanika, Vol. 35, No. 4, pp. 19–29, April, 1999.     

Closed solutions of boundary-value problems of coupled thermoelasticity

Coupled equations of thermoelasticity take into account the effect of nonuniform heating on the medium deformation and that of the dilatation rate on the temperature distribution. As a rule, the coupling coefficients are small and it is assumed, sometimes without proper justification, that the effect of the dilatation rate on the heat conduction process can be neglected. The aim of the present paper is to construct analytical solutions of some model boundary-value problems for a thermoelastic bounded body and to determine the body characteristic dimensions and the medium thermomechanical moduli forwhich it is necessary to take into account that the temperature and displacement fields are coupled. We consider some models constructed on the basis of the Fourier heat conduction law and the generalized Cattaneo-Jeffreys law in which the heat flux inertia is taken into account. The solution is constructed as an expansion in a biorthogonal system of eigenfunctions of the nonself-adjoint operator pencil generated by the coupled equations of motion and heat conduction. For the model problem, we choose a special class of boundary conditions that allows us to exactly determine the pencil eigenvalues.     

Averaging of initial-value and boundary-value problems for one class of oscillatory impulsive systems

We give a new classification of fixed times of pulse action (uniform, functional, limiting, and quantitatively limiting). Several results obtained earlier for oscillatory systems with uniform and limiting times of pulse action are generalized to similar systems with functional times of pulse action. Namely, we obtain estimates, which are exact with respect to a small parameter ε, for the deviation of solutions and their partial derivatives for original and averaged initial-value, boundary-value, and multipoint problems. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 68–84, January–March, 2006.     

Seepage-limitation optimization of the shape of an irrigation channel by the inverse boundary-value problem method

A study is made of the problem of finding the optimal shape of the section of an irrigation channel from the point of view of minimizing the seepage loss; the inverse boundary-value problem method [1] is used.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 3, pp. 74–80, May–June, 1984.     

Cosserat spectrum of the first boundary-value problem of elasticity

The three-dimensional problem of calculating the Cosserat spectrum of the first boundaryvalue problem of elasticity for a body of revolution is considered. Using grids containing 900 and 3600 nodes, it is shown that the sequence of eigenvalues for a sphere found by E. Cosserat and F. Cosserat does not describe the entire range of eigenvalues.     

Conditions for bifurcation of solutions of degenerate boundary-value problems

We obtain conditions for the bifurcation of solutions of linear degenerate Noetherian boundary-value problems with small parameter under the assumption that the unperturbed degenerate differential system can be reduced to the central canonical form.     

Autonomous noetherian boundary-value problem in the critical case

We establish constructive conditions for the existence of solutions of an autonomous Noetherian weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case and develop a modified iterative procedure for finding its solutions.