Filtration formulas for solutions of nonlinear equations with random right-hand sides

Explicit filtration formulas are obtained for the solutions of nonlinear differential equations with random right-hand sides. In the case of a Gaussian random process, these formulas are simplified.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 586–596, May, 1994.     

Fractional differential equations with worsening right-hand sides

We suggest methods for studying fractional differential equations with Caputo and Riemann-Liouville derivatives. Existence and uniqueness theorems are proved for the Cauchy problem for differential equations with worsening operators in scales of Banach spaces.     

Stability of differential systems with rational right-hand sides

We prove theorems which establish estimates for the domain of stability of a differential system with rational right-hand side. We also construct estimates of the convergence of solutions of the system.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 9, pp. 1295–1299, September, 1995.     

The averaging principle for differential equations with right-hand sides having constant delay

We study the limit behavior as0 of the solutions of the equation ,x (t)=(t) fort [–h,0], and the limit behavior of normalized deviation of the solution of this equation from the solution of the averaged equation under the assumption that the right-hand side satisfies a Lipschitz condition with a random constant and possesses bounded moments of some order.Translated fromTeoriya Sluchaínykh Protsessov, Vol. 14, pp. 10–15, 1986.     

Periodic solutions of singularly-perturbed systems of differential equations with impulse effect

Summary The present paper considers the problem for existence, uniqueness and asymptotic representation of periodic solutions of some singularly-perturbed systems of differential equations with impulses. The study is carried out with the help of a modification of the boundary functions method, adequately worked out to match the specific issues of the considered problem.

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Zusammenfassung Wir betrachten das Problem der Existenz, Eindeutigkeit und der asymptotischen Darstellung periodischer Lösungen singulär gestörter Systeme von Differentialgleichungen mit Impulsen. Die Studie wird mittels speziell dem Problem angepaßter Modifikation der Randwertfunktionsmethode ausgeführt.

     

Periodic solutions for systems of second order differential equations

We consider vector differential equtions x″ + g(t, x, x′) = f(t), x?Rⁿ, and show the existence of periodic solutions under conditions which show that growth conditions on g required in earlier results are not needed.     

One-step methods of any order for ordinary differential equations with discontinuous right-hand sides

Summary The right-hand sides of a system of ordinary differential equations may be discontinuous on a certain surface. If a trajectory crossing this surface shall be computed by a one-step method, then a particular numerical analysis is necessary in a neighbourhood of the point of intersection. Such an analysis is presented in this paper. It shows that one can obtain any desired order of convergence if the method has an adequate order of consistency. Moreover, an asymptotic error theory is developed to justify Richardson extrapolation. A general one-step method is constructed satisfying the conditions of the preceding theory. Finally, a simplified Newton iteration scheme is used to implement this method.     

Second-order moment equations for a system of differential equations with random right-hand side

We present a method for the derivation of second-order moment equations for solutions of a system of nonlinear equations that depends on a finite-valued semi-Markov or Markov process. For systems of linear differential equations with random coefficients, the case where the inhomogeneous part contains white noise is considered.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 5, pp. 687–690, May, 2004.     

Minimax estimates for functionals of solutions of wave equations with time-periodic right-hand sides

Minimax mean-square estimates for the velocity potential in the steady regime, as well as for the intensities of harmonic ponts sources, are obtained from the results of inaccurate measurements of the potential. Bibliography: 4 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 77, 1993, pp 87–96.     

Existence and boundary behavior of solutions of Hessian equations with singular right-hand sides

In this paper, we establish the existence of viscosity solutions of Hessian equations with singular right-hand sides and obtain the asymptotic boundary behavior of solutions. The asymptotic results generalize those for Poisson equations and Monge-Ampère equations, and are more precise than obtained from Hopf lemma.     

Asymptotic representation of solutions of regularly perturbed systems of differential equations with nonclassical right-hand side

The problem of the B-asymptotic representation, with respect to a small parameter, of solutions of a regularly perturbed system of differential equations with impulse action on surfaces and of differential equations with discontinuous righthand side is considered. Theorems concerning the B-analytic dependence of solutions on the small parameter are proved. Algorithms for calculating the coefficients of the expansion are developed.Translated from Ukrainskii Maternaticheskii Zhurnal, Vol. 43, No. 10, pp. 1298–1304, October, 1991.     

Integral manifolds of systems of differential equations with a random right-hand side in a Banach space

The concept of an integral manifold of a system of differential equations with a random right-hand side is introduced. The problem of the existence of an integral manifold of a certain class of differential equations in a Banach space and several of its properties are investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 1, pp. 17–21, January, 1992.