Kulinich G. L.
On the Stabilization of a Solution of the Cauchy Problem for One Class of Integro-Differential Equations
Ukr. Mat. Zh. - 2004. - 56, № 12. - pp. 1699 – 1706
We consider a solution of the Cauchy problem u(t, x), t > 0, x ∈ R 2, for one class of integro-differential equations. These equations have the following specific feature: the matrix of the coefficients of higher derivatives is degenerate for all x. We establish conditions for the existence of the limit lim t→∞ u(t, x) = v(x) and represent the solution of the Cauchy problem in explicit form in terms of the coefficients of the equation.
On Stabilization of Energy of a Conservative System Perturbed by a Random Process of “White-Noise” Type in the Itô Form
Ukr. Mat. Zh. - 2001. - 53, № 10. - pp. 1429-1435
We investigate the problem of deterministic control over the behavior of the total energy of the simplest conservative nonlinear system with one degree of freedom without friction in the case of random perturbations by a process of the “white-noise” type in the Itô form. These perturbations act under a fixed angle to the vector of phase velocity of the conservative system.
Ukr. Mat. Zh. - 2000. - 52, № 9. - pp. 1251-1256
For inhomogeneous systems of Itô stochastic differential equations, we introduce the notion of local invariance of surfaces and the notion of local first integral. We obtain results that give the general possibility of finding invariant surfaces and functionally independent first integrals of stochastic differential equations.
Qualitative analysis of the influence of random perturbations of “white-noise” type applied along the vector of phase velocity on a harmonic oscillator with friction
Ukr. Mat. Zh. - 1997. - 49, № 1. - pp. 35–46
We consider representations in the phase plane for the harmonic oscillator with friction under random perturbations applied along the vector of phase velocity. We investigate the behavior of the amplitude, phase, and total energy of the damped oscillator.
Limit theorems for one-dimensional nonhomogeneous stochastic diffusion equations under irregular dependence of the coefficients on a parameter
Ukr. Mat. Zh. - 1990. - 42, № 4. - pp. 435–443
Ukr. Mat. Zh. - 1968. - 20, № 3. - pp. 396–400
Ukr. Mat. Zh. - 1967. - 19, № 2. - pp. 119–125