по
Journal Menu
> Issues > Rubrics > About journal > Authors > About the Journal > Requirements for publication > Council of Editors > Peer-review process > Peer-review in 24 hours: How do we do it? > Policy of publication. Aims & Scope. > Article retraction > Ethics > Copyright & Licensing Policy > Publication in 72 hours: How do we do it? > Digital archiving policy > Open Access Policy > Open access publishing costs > Article Identification Policy > Plagiarism check policy
Journals in science databases
About the Journal

Публикация за 72 часа - теперь это реальность!
При необходимости издательство предоставляет авторам услугу сверхсрочной полноценной публикации. Уже через 72 часа статья появляется в числе опубликованных на сайте издательства с DOI и номерами страниц.
По первому требованию предоставляем все подтверждающие публикацию документы!
MAIN PAGE > Back to contents
Cybernetics and programming
Reference:

On the influence of the method of approximation of an unknown function on the stability of numerical methods for solving the anomalous diffusion equation
Litvinov Vladimir Andreevich

PhD in Physics and Mathematics

Docent, the department of Informatics and Special Technology, Barnaul Law Institute of the Ministry of Internal Affairs of Russia

656038, Russia, Altaiskii krai, g. Barnaul, ul. Chkalova, 49

lva201011@yandex.ru
Другие публикации этого автора
 

 

Abstract.

The subject of the research is numerical algorithms for solving fractional partial differential equations. The object of the study is the stability of several algorithms for the numerical solution of the anomalous diffusion equation. Algorithms based on the difference representation of the fractional Riemann-Liuville derivative and the Caputo derivative for various orders of accuracy are considered. A comparison is made of the results of numerical calculations using the analyzed algorithms for a model problem with the exact solution of the anomalous diffusion equation for various orders of the fractional derivative with respect to the spatial coordinate. The results of the work were obtained on the basis of the analysis of the constructed difference schemes for stability, the conducted numerical experiments and a comparative analysis of the data obtained. The main conclusions of the study are the advantage of using the approximation of the fractional Caputo derivative compared to using the difference scheme for the fractional Riemann-Liouville derivative in the numerical solution of the anomalous diffusion equation. The paper also indicates the importance of choosing the method of difference approximation of the second derivative, which is a derivative of the Caputo.

Keywords: derivative Of Caputo, approximation, diffusion, difference scheme, algorithm, stability, fractional derivative, numerical method, Riemann-Liouville derivative, differential equation

DOI:

10.25136/2306-4196.2019.2.29201

Article was received:

11-03-2019


Review date:

12-03-2019


Publish date:

27-05-2019


This article written in Russian. You can find full text of article in Russian here .

References
1.
Uchaikin V.V. Metod drobnykh proizvodnykh. – Ul'yanovsk: Izd-vo «Artishok», 2008. – 512 s.
2.
Changpin Li, Fanhai Zeng Numerical Methods for Fractional Calculus.– Taylor & Francis Group.– 2015. – 300 p.
3.
Lukashchuk S.Yu. Dvukhsetochnye parallel'nye algoritmy dlya resheniya drobno-differentsial'nykh uravnenii anomal'noi diffuzii Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya: Vychislitel'naya matematika i informatika, 2012, №47(306) – S.83–98.
4.
Ivashchenko D.S. Chislennye metody resheniya pryamykh i obratnykh zadach dlya uravneniya diffuzii drobnogo poryadka po vremeni: diss. kand. fiz.mat. nauk. – Tomsk: Institut matematiki SO RAN, 2008. – 187 s.
5.
Litvinov V.A. Variatsionnoe interpolirovaniya reshenii drobnykh differentsial'nykh uravnenii Izvestiya vysshikh uchebnykh zavedenii. Fizika, 2018, t. 261, №8(728). – S.39–44.
6.
Uchaikin V.V., Litvinov V.A. Nelineinaya teoriya vozmushchenii na osnove variatsionnogo printsipa: model'nye primery. Izvestiya vysshikh uchebnykh zavedenii. Prikladnaya nelineinaya dinamika, 2018, t. 26, № 6. – S. 82–98.
7.
Samko S.G., Kilbas A.A., Marichev O.I. Integraly i proizvodnye drob-nogo poryadka i nekotorye ikh prilozheniya. – Minsk: Nauka i tekhnika, 1987. – 688 s.
8.
Kalitkin N.N. Chislennye metody: ucheb. posobie. –2-e izd., ispravlennoe. – SPb.: BKhV-Peterburg, 2011. – 592 s.
Link to this article

You can simply select and copy link from below text field.


Other our sites:
Official Website of NOTA BENE / Aurora Group s.r.o.
"History Illustrated" Website