Calculation of nth Derivative with Minimum Error Based on Function’s Measurement

A. S. Kochurov; Кочуров А. С.; A. S. Demidov; Демидов А. С.

doi:10.31857/S0044466923090077

Calculation of nth Derivative with Minimum Error Based on Function’s Measurement

Authors: Kochurov A.S.¹^,2, Demidov A.S.¹^,3^,4
Affiliations:
1. Moscow State University
2. Moscow Center of Fundamental and Applied Mathematics
3. Moscow Institute of Physics and Technology
4. Peoples’ Friendship University of Russia (RUDN University)
Issue: Vol 63, No 9 (2023)
Pages: 1428-1437
Section: General numerical methods
URL: https://rjonco.com/0044-4669/article/view/664978
DOI: https://doi.org/10.31857/S0044466923090077
EDN: https://elibrary.ru/XMHVDC
ID: 664978

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Abstract

A solution of the problem that arises in all cases where it is required to approximately calculate the derivatives of an a priori smooth function by its experimental discrete values is proposed. The problem is reduced to finding an “optimal” step of difference approximation. This problem has been studied by many mathematicians. It turned out that to find an optimal approximation step for the kth-order derivative, it is required to know a highly accurate estimate of the modulus of the derivative of order k+1. The proposed algorithm, which gives such an estimate, is applied to the problem of thrombin concentration, which determines the dynamics of blood coagulation. This dynamics is represented by plots and provides a solution of the thrombin concentration problem, which is of interest to biophysicists.

Keywords

reconstruction of nth derivative, thrombin concentration, optimal approximation step, estimate of modulus of higher-order derivatives

About the authors

A. S. Kochurov

Moscow State University; Moscow Center of Fundamental and Applied Mathematics

Email: kchrvas@yandex.ru
119234, Moscow, Russia; 119991, Moscow, Russia

A. S. Demidov

Moscow State University; Moscow Institute of Physics and Technology; Peoples’ Friendship University of Russia (RUDN University)

Author for correspondence.
Email: demidov.alexandre@gmail.com
119234, Moscow, Russia; 123098, Moscow; 115419, Moscow, Russia

References

Dunster J.L., Gibbins J.M., Panteleev M.A., Volpert V. Modeling thrombosis in silico: Frontiers, challenges, unresolved problems and milestones // Physics of Life Reviews. 2018. Vol. 26–27. P. 57–95. https://doi.org/10.1016/j.plrev.2018.02.005
Panteleev M.A., Dashkevich N.M., Ataullakhanov F.I. Hemostasis and thrombosis beyond biochemistry: roles of geometry, flow and diffusion // Thrombosis Research. 2015. Vol. 136. No 4. P. 699–711. Epub 2015 Jul 29. Review. PubMed PMID: 26278966.https://doi.org/10.1016/j.thromres.2015.07.025
Атауллаханов Ф.И., Лобанова Е.С., Морозова О.Л., Шноль Э.Э., Ермакова Е.А., Бутылин А.А., Заикин А.Н. Сложные режимы распространения возбуждения и самоорганизация в модели свертывания крови // Успехи физ. наук. 2007. Т. 177. № 1. С. 87–104.
Арестов В.В., Акопян Р.Р. Задача Стечкина о наилучшем приближении неограниченного оператора ограниченными и родственные ей задачи // Тр. Ин-та матем. и механ. УрО РАН. 2020. Т. 26. № 4. С. 7‒31.
Стечкин С.Б. Неравенства между нормами производных произвольной функции // Acta scient. math. 1965. Vol. 26. № 3–4. P. 225–230.
Арестов В.В. О наилучшем приближении операторов дифференцирования // Матем. заметки. 1967. Т. 1. № 2. С. 149–154.
Буслаев А.П. О приближении оператора дифференцирования // Матем. заметки. 1981. Т. 29. № 5. С. 372–378.
Бабенко К.И. Основы численного анализа. Москва-Ижевск: НИЦ “Регулярная и хаотическая динамика”, 2002. С. 3–847.