Fractional derivatives as weighted average of historical values: an application to COVID-19 in Brazil

Autores

  • Michele Martins Lopes Instituto de Matemática, Estatística e Computação Científica / Universidade Estadual de Campinas
  • Francielle Santo Pedro Universidade Federal de São Paulo
  • José Paulo Carvalho dos Santos Instituto de Ciências Exatas / Universidade Federal de Alfenas
  • Daniel Sánchez Ibá´ñez Centro de Docencia en Ciencias Basicas para Ingeniería / Universidad Austral de Chile
  • Estevão Esmi Instituto de Matemática, Estatística e Computação Científica / Universidade Estadual de Campinas
  • Laécio Carvalho de Barros Instituto de Matemática, Estatística e Computação Científica / Universidade Estadual de Campinas

DOI:

https://doi.org/10.21167/cqdv22n22022275284

Palavras-chave:

Memory effect, Caputo derivative, Riemann-Liouville derivative, Statistical expectation, COVID-19.

Resumo

The memory effect is an interesting tool that can be seen in fractional differential equations. To show this clearly, in this paper we prove that the Caputo derivative of a function 𝑓, as well as the Riemann-Liouville integral and derivative, are proportional to a weighted average of the historical values of 𝑓 or 𝑓 ′. For this, we use the statistical expectation of functions, whose random variable follows a beta distribution. Moreover, through the respective probability density functions, for each operator we specify the weight of the historical values of the function to determine its current value, according to the values of the fractional order of the derivative. Furthermore, to prove the effectiveness of the memory effect to describe real phenomena, we compared a classic model with its fractional version to model COVID-19 in Brazil.

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Publicado

28-09-2022

Como Citar

LOPES, M. M.; PEDRO, F. S.; SANTOS, J. P. C. dos; IBÁ´ÑEZ, D. S.; ESMI, E.; BARROS, L. C. de. Fractional derivatives as weighted average of historical values: an application to COVID-19 in Brazil. C.Q.D. - Revista Eletrônica Paulista de Matemática, Bauru, v. 22, n. 2, p. 275–284, 2022. DOI: 10.21167/cqdv22n22022275284. Disponível em: https://sistemas.fc.unesp.br/ojs/index.php/revistacqd/article/view/389. Acesso em: 13 nov. 2024.