Bifurcation of solutions of an impulsive boundary-value problem

We establish constructive conditions for the bifurcation of solutions and construct an iteration procedure for finding solutions of a linear Noether boundary-value problem for a system of ordinary differential equations with pulse action in the critical case. We obtain an estimate for the range of values of a small parameter for which the iteration procedure converges. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 21–31, January–March, 2007.     

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.     

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.     

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 boundary-value problem for a nonlinear heat-conduction equation

Methods for the numerical solution of the boundary-value problem are comidered. One method is related to the method of successive approximations and the other employs the collocation method [1]. A relationship between the latter method and the Ritz and Galerkin methods [2] is shown. An application of the collocation method to the nonstationary problem is given. The approximate solution is represented in analytic form. A way of finding the absolute error of the approximate solution is given.     

THE MIXED BOUNDARY-VALUE PROBLEM FOR NON-LOCAL ASYMMETRIC ELASTICITY

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅｒｅｅｘｉｓｔｓｔｈｅａｒｇｕｍｅｎｔｂｅｔｗｅｅｎＡｔｋｉｎｓｏｎ（ｓｅｅ［１～４］）ａｎｄＥｒｉｎｇｅｎａｎｄｃｏ＿ｗｏｒｋｅｒｓ（ｓｅｅ［５～７］）ｏｖｅｒｔｈｅｎｏｎ＿ｌｏｃａ...     

Averaging of a boundary-value problem for a multifrequency system with deviating argument

Using the averaging method, we investigate the solvability of a boundary-value problem for a multifrequency system with deviating argument and integral boundary conditions. We obtain an estimate for the difference between solutions of the original and averaged problems. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 4, pp. 519–527, October–December, 2007.     

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.     

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.     

Approximate bounded synthesis for one weakly nonlinear boundary-value problem

We construct and substantiate an approximate control in the form of feedback for the problem of approximate bounded synthesis with semidefinite quality criterion for a parabolic equation containing a nonlinear term that depends regularly on a small parameter.     

Regular integral equations for the second boundary-value problem of the bending of an anisotropic elastic plate

Two systems of Fredholm equations of the second kind are constructed for the solution of the second boundary-value problem of the bending of an anisotropic plate (a normal bending moment and a generalized shear force are specified on the boundary of the simply-connected domain) under the assumption of validity of the Kirchhoff-Love hypotheses. Correct equilibrium conditions are specified for the examined boundary-value problem. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 3, pp. 108–119, May–June, 2005.     

Solution of the inverse boundary-value problem of aerohydrodynamics with allowance for a boundary layer

The problem of constructing the contour of a wing profile in a viscous (incompressible or compressible) flow from the velocity distribution, given in terms of the arc abscissa, is solved in the approximation of boundary layer theory. The solvability conditions are obtained. Numerical calculations are carried out. Wing profile contours are constructed from velocity distributions that ensure the nonseparation of the flow. The effect of viscosity and compressibility on the solution of the problem is considered.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 28–32, July–August, 1989.The authors are grateful to G. Yu. Stepanov for useful discussions.     

Singular perturbation solution of boundary-value problem for a secord-order differential-difference equation

In this paper, the method of two-variables expansion is used to construct boundary layer terms of asymptotic solution of the boundary-value problem for a second-order DDE. The n-order formal asymptotic solution is obtained and the error is estimated. Thus the existence of uniformly valid asymptotic solution is proved.     

Generalized thermoelastic problem of an infinite body with a spherical cavity under dual-phase-lags

The aim of the present contribution is the determination of the thermoelastic temperatures, stress, displacement, and strain in an infinite isotropic elastic body with a spherical cavity in the context of the mechanism of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord–Shulman (2TLS) model and two-temperature dual-phase-lag (2TDP) model of thermoelasticity are combined into a unified formulation with unified parameters. The medium is assumed to be initially quiescent. The basic equations are written in the form of a vector matrix differential equation in the Laplace transform domain, which is then solved by the state-space approach. The expressions for the conductive temperature and elongation are obtained at small times. The numerical inversion of the transformed solutions is carried out by using the Fourier-series expansion technique. A comparative study is performed for the thermoelastic stresses, conductive temperature, thermodynamic temperature, displacement, and elongation computed by using the Lord–Shulman and dual-phase-lag models.     

Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator

A numerical method, based on the design of two artificial neural networks, is presented in order to approximate the viscosity and density features of fluids from the eigenvalues of the Stokes operator. The finite element method is used to solve the direct problem by training a first artificial neural network. A nonlinear map of eigenvalues of the Stokes operator as a function of the viscosity and density of the fluid under study is then obtained. This relationship is later inverted and refined by training a second artificial neural network, solving the aforementioned inverse problem. Numerical examples are presented in order to show the effectiveness and the limitations of this methodology.