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Скорость сходимости черновских аппроксимаций решений эволюционных уравнений
Математические заметки. 2020. Т. 108. № 3. С. 463-468.
This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.
Vedenin A., Voevodkin V., Galkin V. et al., Mathematical notes 2020 Vol. 108 No. 3 P. 451-456
Short communication is presented without abstract ...
Added: December 29, 2021
Vedenin A., Galkin V., Karatetskaia E. et al., / Cornell University. Series arXiv "math". 2020.
This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations. ...
Added: October 12, 2019
Vedenin A., Remizov I., / Cornell University. Series math "arxiv.org". 2020.
Abstract. The method of Chernoff approximation was discovered by Paul Chernoff in 1968 and now is a powerful and flexible tool of contemporary functional analysis. This method is different from grid-based approach and helps to solve numerically the Cauchy problem for evolution equations, e.g., for heat equation and for more general parabolic second-order partial differential equations ...
Added: December 14, 2020
Remizov I., Journal of Functional Analysis 2016 Vol. 270 No. 12 P. 4540-4557
For a densely defined self-adjoint operator $\mathcal{H}$ in Hilbert space $\mathcal{F}$ the operator $\exp(-it\mathcal{H})$ is the evolution operator for the Schr\"odinger equation $i\psi'_t=\mathcal{H}\psi$, i.e. if $\psi(0,x)=\psi_0(x)$ then $\psi(t,x)=(\exp(-it\mathcal{H})\psi_0)(x)$ for $x\in Q.$ The space $\mathcal{F}$ here is the space of wave functions $\psi$ defined on an abstract space $Q$, the configuration space of a quantum system, ...
Added: March 3, 2018
Beklaryan A., Beklaryan L., Дифференциальные уравнения 2017 Т. 53 № 2 С. 148-159
The Cauchy problem for the homogeneous linear functional-differential equation of a pointwise type, defined on the line, is considered. In the case of one-dimensional equation we formulated the theorem of existence and uniqueness of solutions with estimating of its order of growth. This research is carried out within the formalism based on group peculiarities of ...
Added: February 12, 2017
Dragunova K., Гаращенкова А. А., Remizov I., / Cornell University. Series arXiv "math". 2021.
Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients. For many classes of equations such approximations have already been constructed, however, the speed of their convergence to the exact solution has not been properly studied. ...
Added: December 16, 2021
Galkin O., Remizov I., Математические заметки 2022 Т. 111 № 2 С. 297-299
Despite the fact that Chernoff's theorem has been published for more than 50 years ago and since then it has been actively used (including, for example, for building of the Smolyanov surface measure), only a small number of results are known on the rate of convergence of Chernoff approximations. In this message we announce some ...
Added: October 14, 2021
Galkin O., Remizov I., / Cornell University. Series math "arxiv.org". 2021. No. 2104.01249.
Эта статья посвящена изучению скорости сходимости черновских приближений к сильно непрерывным однопараметрическим полугруппам. Мы приводим естественные примеры, для которых эта сходимость: произвольно высока; произвольно медленна; выполняется в сильной операторной топологии, но не выполняется в нормальной операторной топологии. Мы также доказываем общую теорему, которая дает оценку сверху для скорости стремления к нулю нормы остатка, появляющегося в черновских приближениях. Мы ...
Added: October 14, 2021
Solvability Problems for a Linear Homogeneous Functional-Differential Equation of the Pointwise Type
Beklaryan A., Beklaryan L. A., Differential Equations 2017 Vol. 53 No. 2 P. 145-156
The Cauchy problem for a linear homogeneous functional-differential equation of the pointwise type defined on a straight line is considered. Theorems on the existence and uniqueness of the solution in the class of functions with a given growth are formulated for the case of the one-dimensional equation. The study is performed using the group peculiarities ...
Added: March 6, 2017
V. I. Bogachev, T. I. Krasovitskii, S. V. Shaposhnikov, Doklady Mathematics 2020 Vol. 102 No. 3 P. 464-467
We give a solution to the Kolmogorov problem on uniqueness of probability solutions to a parabolic Fokker–Planck–Kolmogorov equation. ...
Added: October 31, 2022
Matveenko V. D., Korolev A. V., Zhdanova M. O., Прикладная математика и вопросы управления 2017 № 2 С. 55-64
Мы изучаем игровое равновесие в модели с производством в сети с двумя типами агентов, обладающих разной продуктивностью. Каждый агент может инвестировать часть своего начального запаса в первом из двух временных периодов; потребление во втором периоде зависит от его инвестиций и продуктивности, так же как и от инвестиций его соседей в сети. Мы вводим формализацию понятия ...
Added: August 8, 2017
Gorshkov O., Колпаков И. Ю., Афанасенко К. А., Академия педагогических идей «Новация» 2017 № 11 С. 146-153
Conditions of solvability of the Cauchy problem for one differential equation of the first order which hasn't been resolved by rather derivative are found in work. Conditions of solvability of the Cauchy problem were received with application theorem of the Leray–Schauder's type. ...
Added: October 15, 2020
Korolev A. V., М. : Юрайт, 2017
Данный учебник дает читателю необходимые знания по теории обыкновенных дифференциальных уравнений, теории разностных уравнений и теории стохастических дифференциальных уравнений. Все три названные раздела одинаково необходимы для полноценного экономического образования. Учебник состоит из теоретической и практической частей. В практической части курса подробно и обстоятельно демонстрируются на конкретных примерах методы решения различных задач. В результате изучения данного ...
Added: August 9, 2017
Dragunova K., Никбахт Н., Remizov I., Журнал Средневолжского математического общества 2023 Т. 25 № 4 С. 255-272
Chernoff approximations are a flexible and powerful tool of functional analysis, which can be used, in particular, to find numerically approximate solutions of some differential equations with variable coefficients. Such approximations have already been constructed for many classes of equations, however, the question of the rate of convergence of approximations has not even been raised ...
Added: November 10, 2023
Remizov I., Applied Mathematics and Computation 2018 Vol. 328 P. 243-246
We present a general method of solving the Cauchy problem for a linear parabolic partial differential equation of evolution type with variable coefficients and demonstrate it on the equation with derivatives of orders two, one and zero. The method is based on the Chernoff approximation procedure applied to a specially constructed shift operator. It is ...
Added: May 25, 2018
Zlotnik Alexander, Zlotnik Ilya, Computational Methods in Applied Mathematics 2015 Vol. 15 No. 2 P. 233-245
We consider the Cauchy problem for the 1D generalized Schrὅdinger equation on the whole axis. To solve it, any order finite element in space and the Crank-Nicolson in time method with the discrete transpa\-rent boundary conditions (TBCs) has recently been constructed. Now we engage the global Richardson extrapolation in time to derive the high order ...
Added: March 3, 2015
Remizov I., Journal of Mathematical Physics 2019 Vol. 60 No. 7 P. 1-8
We present a general method of solving the Cauchy problem for multidimensional parabolic (diffusion type) equation with variable coefficients which depend on spatial variable but do not change over time. We assume the existence of the C0-semigroup (this is a standard assumption in the evolution equations theory, which guarantees the existence of the solution) and ...
Added: June 28, 2019
Galkin O., Galkina S., / Cornell University. Series math "arxiv.org". 2020. No. arXiv:2012.07174.
This short communication (preprint) is devoted to mathematical study of evolution equations that are important for mathematical physics and quantum theory; we present new explicit formulas for solutions of these equations and discuss their properties. The results are given without proofs but the proofs will appear in the longer text which is now under preparation.
In ...
Added: December 13, 2020
Roeckner M., Shaposhnikov S., Bogachev V., Journal of Evolution Equations 2013 Vol. 13 No. 3 P. 577-593
We prove a new uniqueness result for highly degenerate second-order parabolic equations on the whole space. A novelty is also our class of solutions in which uniqueness holds. ...
Added: October 15, 2014
Florido Calvo F. A., Remizov I., / Cornell University. Series arXiv "math". 2021.
Dynamics of closed quantum systems on curves, surfaces and more general manifolds is governed by the Schroedinger equation with time-independent Hamiltonian. Solving Cauchy problem for this equation provides full information on the future and the past of the system if we know the state of the system at the initial moment of time t=0. However, ...
Added: December 16, 2021
Veretennikov A., Veretennikova M., / Cornell University. Series "Working papers by Cornell University". 2019.
New convergence rate asymptotic bound for a class of homogeneous Markov chains is established. ...
Added: November 14, 2019
Zlotnik A., Lomonosov T., Doklady Mathematics 2018 Vol. 98 No. 2 P. 458-463
An explicit two-time-level and spatially symmetric finite-difference scheme approximating the 1D quasi-gasdynamic system of equations is studied. The scheme is linearized about a constant solution, and new necessary and sufficient conditions for the L2 -dissipativity of solutions of the Cauchy problem are derived, including, for the first time, the case of a nonzero background velocity ...
Added: September 12, 2018
Pavel S. Prudnikov, / Cornell University. Series math "arxiv.org". 2020.
Paul Chernoff in 1968 proposed his approach to approximations of one-parameter operator semigroups while trying to give a rigorous mathematical meaning to Feynman's path integral formulation of quantum mechanics. In early 2000's Oleg Smolyanov noticed that Chernoff's theorem may be used to obtain approximations to solutions of initial-value problems for linear partial differential equations (LPDEs) ...
Added: December 16, 2020
Remizov I., Potential analysis 2020 Vol. 52 P. 339-370
In the paper we derive two formulas representing solutions of Cauchy problem for two Schrödinger equations: one-dimensional momentum space equation with polynomial potential, and multidimensional position space equation with locally square integrable potential. The first equation is a constant coefficients particular case of an evolution equation with derivatives of arbitrary high order and variable coefficients ...
Added: September 30, 2018