Singular-perturbation method for nonlinear second-order evolution inclusions with Volterra operators

We consider a class of second-order operator differential inclusions with Volterra-type operators. Using the singular-perturbation method, we study the problem of the existence of a solution of the Cauchy problem for these inclusions. Important a priori estimates are obtained for solutions and their derivatives. We give an example that illustrates the proposed approach to the analysis of the problem under consideration. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 27–43, January–March, 2009.     

Non-smooth modelling of electrical systems using the flux approach

The non-smooth modelling of electrical systems, which allows for idealised switching components, is described using the flux approach. The formulations and assumptions used for non-smooth mechanical systems are adopted for electrical systems using the position–flux analogy. For the most important non-smooth electrical elements, like diodes and switches, set-valued branch relations are formulated and related to analogous mechanical elements. With the set-valued branch relations, the dynamics of electrical circuits are described as measure differential inclusions. For the numerical solution, the measure differential inclusions are formulated as a measure complementarity system and discretised with a difference scheme, known in mechanics as time-stepping. For every time-step a linear complementarity problem is obtained. Using the example of the DC–DC buck converter, the formulation of the measure differential inclusions, state reduction and their numerical solution using the time-stepping method is shown for the flux approach.     

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.     

Averaging of impulsive differential inclusions with Hukuhara derivative

We substantiate the method of complete or partial averaging for differential inclusions that contain the Hukuhara derivative and are subject to pulse influence at fixed times. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 3, pp. 416–432, July–September, 2007.     

Nonconvex-valued impulsive functional differential inclusions with variable times

We use the Schaefer fixed-point theorem, combined with the Bressan-Colombo selection theorem for lower-semicontinuous multivalued operators with nonempty closed and decomposable values, for the investigation of the existence of solutions for first-and second-order impulsive functional differential inclusions with variable times. Published in Neliniini Kolyvannya, Vol. 10, No. 4, pp. 443–463, October–December, 2007.     

Prediction of the effective elastic properties of spheroplastics by the generalized self-consistent method

The problem of predicting the effective elastic properties of composites with prescribed random location and radius variation in spherical inclusions is solved using the generalized self-consistent method. The problem is reduced to the solution of the averaged boundary-value problem of the theory of elasticity for a single inclusion with an inhomogeneous transition layer in a medium with desired effective elastic properties. A numerical analysis of the effective properties of a composite with rigid spherical inclusions and a composite with spherical pores is carried out. The results are compared with the known solution for the periodic structure and with the solutions obtained by the standard self-consistent methods. Perm’ State Technical University, Perm’ 614600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 186–190, May–June, 1999.     

Stress-strain state of an anisotropic plate with an elliptic hole and thin rigid inclusions

The stress-strain state of an anisotropic plate containing an elliptic hole and thin, absolutely rigid, curvilinear inclusions is studied. General integral representations of the solution of the problem are constructed that satisfy automatically the boundary conditions on the elliptic-hole contour and at infinity. The unknown density functions appearing in the potential representations of the solution are determined from the boundary conditions at the rigid inclusion contours. The problem is reduced to a system of singular integral equations which is solved by a numerical method. The effects of the material anisotropy, the degree of ellipticity of the elliptic hole, and the geometry of the rigid inclusions on the stress concentration in the plate are studied. The numerical results obtained are compared with existing analytical solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 173–180, July–August, 2007.     

Periodic viable trajectories of differential inclusions

In this paper, the periodic viable trajectories of differential inclusions are discussed. Firstly, a simplified property of differential inclusions is given. Then, an existence theorem of periodic viable trajectories of differential inclusions in a finite dimensional space is proved. With the above results and Galerkin’s approximation, an existence theorem of periodic viable trajectories of partial differential inclusions in a Hilbert space is proved.     

Conditions for the existence of a solution of a Noether boundary-value problem for a second-order system

We consider a weakly nonlinear boundary-value problem for a system of second-order ordinary differential equations. We find a sufficient condition for the existence of at least one solution of this problem and propose a convergent iterative algorithm for the determination of its solution. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 3, pp. 368–375, July–September, 2006.     

New approach to the investigation of the existence of periodic solutions of systems of differential equations and their construction

Methods developed for the solution of general equations with restrictions are applied to the construction of periodic solutions of systems of differential equations. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 55–70, January–March, 2007.     

Analyzing the stress–strain state of thick cylindrical shells

Two approaches to the analysis of the stress–strain state of thick cylindrical shells are elaborated. The shell is divided by concentric cross-sectional circles into several coaxial cylindrical shells. Each of these shells has its own curvature determined on its midline. The stress–strain state of the original shell is described by satisfying the interface conditions between the component shells. The distribution of unknown functions throughout the thickness is determined by finding the analytic solution of a system of differential equations in the first approach and is approximated by polynomial functions in the second approach. The stress–strain state of thick shells is analyzed. It is revealed that the effect of reduction becomes stronger with increasing curvature     

Construction of asymptotic formulas for solutions of singularly perturbed degenerate systems of differential equations

We obtain an asymptotic solution of the Cauchy problem for a singularly perturbed degenerate system of differential equations in the case of a singular limit pencil of matrices. Translated from Neliniini Kolyvannya, Vol. 11, No. 3, pp. 408–420, July–September, 2008.     

Instability of solutions for certain nonlinear vector differential equations of fourth order

The main purpose of this paper is to give a result with explanatory example that deals directly with the instability of the trivial solution of a certain nonlinear vector differential equation of the fourth order. The result established here improves and includes a well-known instability result established for a scalar nonlinear differential equation of the fourth order. Published in Neliniini Kolyvannya, Vol. 12, No. 1, pp. 120–129, January–March, 2009.     

On the global stability of a nonlinear functional differential equation with pulse action

We give sufficient conditions for the global stability of the zero solution of a functional differential equation with pulse action and with nonlinear function satisfying the conditions of negative feedback and sublinear growth. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 258–269, April–June, 2007.     

Mathematical model of a heterogeneous medium consisting of a matrix and spherical inclusions

A mathematical model of a heterogeneous medium consisting of an elastoplastic matrix and elastic spherical inclusions is constructed. The model takes into account plastic zones, which appear in the vicinity of the inclusions. It is shown that when the effect of plastic zones is taken into account, the averaged “moduli” of volume compression, shear, and yield strength depend not only on the volume concentrations of the inclusions, but also on the mean pressure in the medium. Institute of Theoretical and Applied Mechanics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 4, pp. 170–178, July–August, 1999.     

Conditions for the unique solvability of the initial-value problem for linear second-order differential equations with argument deviations

We establish new conditions under which the initial-value problem for a system of linear second-order differential equations with argument deviations has a unique solution, which depends monotonically on additive perturbations of the problem. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 4, pp. 535–547, October–December, 2006.     

On approximate solution of boundary-value problems by the least squares method

Using the least squares method, we construct a new iterative procedure for finding solutions of a weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case in the form of an expansion of a solution in a generalized Fourier polynomial in the neighborhood of the generating solution. We obtain an estimate for the range of values of the small parameter for which this iterative procedure converges to the required solution. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 554–573, October–December, 2008.     

Critical case for singularly perturbed Noether boundary-value problems

We construct an asymptotic expansion of a solution for singularly perturbed linear systems of ordinary differential equations of the Noether type in the critical case. We successively determine all terms of the asymptotic expansion by the method of boundary functions and pseudoinverse matrices. __________ Translated from Neliniini Kolyvannya, Vol. 11, No. 1, pp. 45–54, January–March, 2007.     

Global stability of solutions of nonstationary monotone differential equations with impulsive action in the pseudolinear form

We consider a generalization of the principle of comparison for pseudolinear differential equations with impulsive perturbations in a Banach space. On the basis of these results, we establish the conditions for the global stability in a cone of the trivial solution for the analyzed class of equations. The obtained results are used for the investigation of stability in the Takagi–Sugeno models.     

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded and semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of semi-rigid connected Reddy–Bickford beam with variable cross-section on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed.