ISSN 1817-2172, рег. Эл. № ФС77-39410, ВАК

Differential Equations and Control Processes
(Differencialnie Uravnenia i Protsesy Upravlenia)

On a Local Property of One-dimensional Linear Differential Equation of the Second Order


Sofia Stanislavovna Rasova

Institute of Problems in Mechanical Engineering (IPME) RAS
PhD in physics and mathematics,
senior scientific collaborator

Boris Pavlovich Harlamov

Doctor of Siences (physics and mathematics),
head of laboratory of IPME RAS


We consider one-dimensional linear second order differential equation and the solution of the Dirichlet problem of this equation on an interval of the unknown function values. Two partial solutions of this problem on a small symmetric neighborhood of an interior point of this interval are investigated. The first solution has values 1 on the left edge, and 0 on the right edge of the neighborhood. The second one has 0 and 1 correspondingly. It is shown that properties of these partial solutions when the length of the neighborhood tends to 0 characterize the initial equation completely. Namely, two initial coefficients of the decomposition of each partial solution with respect to the neighborhood length are proportional to corresponding coefficients of this equation. To prove this theorem we use the generalized Green formula and the generalized semigroup property of the family of the Dirichlet problem solutions.



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