An approach to deducing approximate solutions to the Cauchy problem for nonlinear differential equations

A new technique for the construction of numerical methods based on continued fractions is proposed. A characteristic feature of these algorithms is the fact that for certain values of the parameters it is possible to obtain both novel and traditional (explicit and implicit) numerical methods for the solution of the Cauchy problem for ordinary differential equations. Two-sided formulas are proposed by means of which it is possible to obtain on each integration step not only upper and lower approximations to the exact solution, but also information concerning the magnitude of the leading term of the error without the need for additional calculations of the right-hand side of the initial differential equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 12, pp. 1695–1701, December, 1992.     

Contact Interactions between Individual Fibers of Cellulose and Its Derivatives: Mechanism of Cationic Surfactant Action

The effect of cationic surfactants (cetyltrimethylammonium bromide and tetrabutylammonium iodide) and cationic polyelectrolyte polyethylenimine on the coefficient of friction and adhesion force p(during shear and tensile testing, respectively) at the point contacts between the individual fibers of cellulose, viscose, and cellulose acetate was studied. The comparison of these data with the results of -potential and adsorption measurements enabled us to relate the nonmonotonic dependence of parameters and pon the surfactant concentration to the changes in electrosurface properties and the hydrophilicity or hydrophobicity of the fibers. The specific free energy of interaction U[mJ/m²] was estimated for cellulose acetate fibers.     

Mathematical modeling of heat-conduction processes in randomly inhomogeneous bodies using Feynman diagrams

This work is devoted to the mathematical modeling of heat-conduction processes in multiphase bodies with randomly inhomogeneous structure. A nonstationary initial boundary-value problem is formulated on the basis of the Fourier law for a whole body. The technique of Feynman diagrams is applied to the investigation of averaged temperature fields. We obtain the Dyson equation and a nonlocal heatconduction equation for the temperature field averaged over an ensemble of phase configurations in multiphase randomly inhomogeneous bodies. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 178–184, January–March, 2008.     

Region of applicability of applied theories in problems of the stability of bars and plates with low shear stiffness subjected to uniaxial compression

The question of the region of applicability of applied theories of bending of bars and plates in stability problems is discussed in relation to the case of uniaxial compression. The critical loads given by these theories are compared with the recently obtained solution [4, 5, 7–10] based on the three-dimensional linearized equations of the theory of elasticity [6]. It is established that for the problems in question the Timoshenko and Ambartsumyan theories are accurate enough for engineering purposes over the entire practical region of variation of the geometric and physical parameters of the bar (plate), whereas the use of the Euler-Bernoulli and Kirchhoff-Love theories may lead to unacceptable errors.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev; Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga; Lvov Polytechnic Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1124–1126, November–December, 1969.     

Creep stability of anisotropic shallow spherical shells with account for transverse shear strains

The deformation properties of glass-reinforced plastics are approximated by the relations for a homogeneous anisotropic material, only the shear creep strains being taken into account, since the normal forces, coinciding with the direction of the reinforcement, produce much less creep. Deflections commensurable with the thickness of the shell are considered (geometric nonlinearity), but the strain components are assumed small. The nonlinear relations obtained for this problem have been solved (for a particular case) on a computer. At loads less than the long-time critical value the growth of deflections in time is damped; at large loads the rate of growth of deflections increases.Mekhanika Polimerov, Vol. 2, No. 6, pp. 875–885, 1966