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Владикавказский математический журнал

Vladikavkaz Mathematical Journal

Южный математический институт - филиал Федерального государственного бюджетного учреждения науки Федерального научного центра «Владикавказский научный центр Российской академии наук» (ЮМИ ВНЦ РАН)

Chernoff Approximations of the Solution of Linear ODE with Variable Coefficients

Черновские аппроксимации решения линейного ОДУ с переменными коэффициентами

10.46698/a3908-1212-5385-q

17999

12 2025

27 4 124 135

https://vmj.ru/eng/archive/detail.php?ELEMENT_ID=18033&SECTION_ID=647

Ремизов

И. Д.

Remizov

I. D.

ivremizov@yandex.ru

Лаборатория топологических методов в динамике НИУ ВШЭ, Россия, 603155, Нижний Новгород, Большая Печерская, 25/12 National Research University Higher School of Economics, 25/12, Room 224, Bol. Pecherskaya St., Nizhny Novgorod 603155, Russia

The Chernoff approximation method is a powerful and flexible tool of functional analysis, which allows in many cases to express \(\exp(tL)\) in terms of variable coefficients of a linear differential operator \(L\). In this paper, we prove a theorem that allows us to apply this method to find the resolvent of \(L\). Our theorem states that the Laplace transforms of Chernoff approximations of a \(C_0\)-semigroup converge to the resolvent of the generator of this semigroup. We demonstrate the proposed method on a second-order differential operator with variable coefficients. As a consequence, we obtain a new representation of the solution of a~nonhomogeneous linear ordinary differential equation of the second order in terms of functions that are coefficients of this equation, playing the role of parameters of the problem. For the Chernoff function, based on the shift operator, we give an estimate for the rate of convergence of approximations to the solution.

Метод черновских аппроксимаций является мощным и гибким инструментом функционального анализа, позволяющим во многих случаях выразить \(\exp(tL)\) через переменные коэффициенты линейного дифференциального оператора \(L\). В данной работе доказывается теорема, позволяющая применять этот метод для нахождения резольвенты оператора \(L\). Наша теорема утверждает, что преобразования Лапласа аппроксимаций Чернова \(C_0\)-полугруппы сходятся к резольвенте генератора этой полугруппы. Мы демонстрируем предложенный метод на дифференциальном операторе второго порядка с переменными коэффициентами. В качестве следствия мы получаем новое представление решения неоднородного линейного обыкновенного дифференциального уравнения второго порядка в терминах функций, являющихся коэффициентами этого уравнения, играющих роль параметров задачи. Для функции Чернова на основе оператора сдвига мы даем оценку скорости сходимости приближений к решению.

полугруппы операторов резольвента оператора линейное ОДУ с переменными коэффициентами представление решения черновские аппроксимации

operator semigroups resolvent operator linear ODE with variable coefficients representation of the solution Chernoff approximations

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