The scaling method for free interface problems in ordinary differential equations

A method is proposed for the solution of boundary value problems in which the locations of the interfaces are unknown. Numerical results are obtained for the steady state solution of a shafttype furnace.     

On a Kneser-type problem for a system of ordinary differential equations

We establish sufficient conditions for the solvability of a Kneser-type problem concerning the existence of a monotonic solution of a system of ordinary differential equations satisfying an initial condition.Translated from Matematicheskie Zametki, Vol. 15, No. 6, pp. 897–906, June, 1974.     

Global Lipschitz continuity of free boundaries in the one-phase Stefan problem

Summary It is proved that the free boundary t=s(x)in the one-phase Stefan problem is uniformly Lipschitz continuous in the strip 0tT for any T>0with a Lipschitz coefficient depending only on the specified data under some conditions.This work was supported by C.N.R. Italy.     

A nonlocal problem for singular linear systems of ordinary differential equations

A system of linear ordinary differential equations is examined on an infinite half-interval. This system is supplemented by the boundedness condition for solutions and a nonlocal linear condition specified by the Stieltjes integral. A method for approximating the resulting problem by a problem posed on a finite interval is proposed, and the properties of the latter are investigated. A numerically stable method for solving this problem is examined. This method uses an auxiliary boundary value problem with separated boundary conditions.     

On then-th order iterative ordinary linear differential equations

Summary It is shown that a change of the basis in the solution space of a second order linear differential equation induces a covariant change in the solution space of the corresponding iterative equation. Also studied is the problem to what extent a solution of an iterative equation determines the equation.Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth.     

On conjugacy of high-order linear ordinary differential equations

It is shown that the differential equation

On two comparison tests for linear second-order ordinary differential equations

We prove two comparison tests for second-order linear differential equations. The well-known Sturm comparison theorem is a straightforward corollary of the first of them. The second (integral) test permits one to use specific integral relations between the coefficients of two equations to prove the oscillation of one of the equations assuming the oscillation of the other.     

On the theory of linear systems of ordinary differential equations

Author's abstract of dissertation for the attainment of the scientific degree of Doctor of the Physical and Mathematical Sciences. Dissertation defended May 24, 1968 at the sitting of the Council of the Mechanical and Mathematical Faculty of the M. V. Lomonosov State University.Official opponents: Doctor of the Physical and Mathematical Sciences D. V. Anosov, Doctor of the Physical and Mathematical Sciences Professor R. é Vinograd, and Doctor of the Physical and Mathematical Sciences Professor Ya. G. Sinai.Translated from Matematicheskie Zametki, Vol. 4, No. 4, pp. 483–490, October, 1968.     

Differentiable dependence upon the data in a one-phase Stefan problem

It is well known that the free boundary of the one-phase Stefan problem (1.1–5) depends continuously on the boundary data [1]. In this paper we prove that, in addition, the solution operator S which, to each g. assigns the corresponding free boundary, is continuously Frechet differentiable and we give the defining formulas of the Frechet derivative.     

Singular numbers of a boundary-value problem on the halfline for a linear system of ordinary differential equations

Novosibirsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 4, pp. 5–12, July–August, 1989.     

Weighted boundary problem for a system of linear ordinary differential equations with singularities. I

Institute of Mathematics and Cybernetics, Academy of Sciences of the Lithuanian SSR. Translated from Litovskii Matematicheskii Sbornik (Lietuvos Matematikos Rinkinys), Vol. 29, No. 1, pp. 109–122, January–March, 1989.     

On periodic solutions of linear degenerate second-order ordinary differential equations

We consider a scalar linear second-order ordinary differential equation whose coefficient of the second derivative may change its sign when vanishing. For this equation, we obtain sufficient conditions for the existence of a periodic solution in the case of arbitrary periodic inhomogeneity.     

On the refined integral method for the one-phase Stefan problem with time-dependent boundary conditions

Refined integral heat balance is developed for Stefan problem with time-dependent temperature applied to exchange surface. The method is applied to phase change in the half-plane and ordinary differential equation is obtained for the solid/liquid interface. The results are compared to those obtained by heat balance integral, perturbation and numerical methods.     

A nonlinear singular eigenvalue problem for a linear system of ordinary differential equations with redundant conditions

A nonlinear eigenvalue problem for a linear system of ordinary differential equations is examined on a semi-infinite interval. The problem is supplemented by nonlocal conditions specified by a Stieltjes integral. At infinity, the solution must be bounded. In addition to these basic conditions, the solution must satisfy certain redundant conditions, which are also nonlocal. A numerically stable method for solving such a singular overdetermined eigenvalue problem is proposed and analyzed. The essence of the method is that this overdetermined problem is replaced by an auxiliary problem consistent with all the above conditions.