Unexpected behavior of Caputo fractional derivative
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Unexpected behavior of Caputo fractional derivative Lucas Kenjy Bazaglia Kuroda1 · Arianne Vellasco Gomes2 · Robinson Tavoni1 · Paulo Fernando de Arruda Mancera1 · Najla Varalta2 · Rubens de Figueiredo Camargo3
Received: 3 August 2015 / Revised: 7 December 2015 / Accepted: 14 December 2015 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract This paper discusses the modeling via mathematical methods based on fractional calculus, using Caputo fractional derivative. From the fractional models associated with harmonic oscillator, logistic equation and Malthusian growth, an unexpected behavior of the Caputo fractional derivative is discussed. The difference between the rate of variation and the order of the Caputo fractional derivative is explained.
Communicated by Paulo Fernando de Arruda Mancera and Igor Freire. RFC thanks CNPq—National Counsel of Technological and Scientific Development (455920/2014-1) and PFAM thanks FAPESP—São Paulo Research Foundation (2013/08133-0).
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Rubens de Figueiredo Camargo [email protected] Lucas Kenjy Bazaglia Kuroda [email protected] Arianne Vellasco Gomes [email protected] Robinson Tavoni [email protected] Paulo Fernando de Arruda Mancera [email protected] Najla Varalta [email protected]
1
Departamento de Bioestatística, Instituto de Biociências, UNESP, Distrito de Rubião Junior, Botucatu, SP 18618-689, Brazil
2
Faculdade de Ciências, UNESP, Av. Eng. Luiz Edmundo Carrijo Coube, 14-01, Bairro Vargem Limpa, Bauru, SP 17033-360, Brazil
3
Departamento de Matemática, Faculdade de Ciências, UNESP, Av. Eng. Luiz Edmundo Carrijo Coube, 14-01, Bairro Vargem Limpa, Bauru, SP 17033-360, Brazil
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L. K. B. Kuroda et al.
Keywords Caputo fractional derivative · Fractional modeling · Fractional calculus · Fractional harmonic oscillator · Fractional logistic equation Mathematics Subject Classification
26A33
1 Introduction The obtaining of a differential equation whose solution describes well the reality brings great difficulty. In Albert Einstein words “One thing I have learned in a long life: that all our science, measured against reality, is primitive and childlike—and yet it is the most precious thing we have”. Usually, the closer we are to perfectly describe a real problem, the bigger are the number of variables involved and the complexity of the equations. In this sense, the Non-Integer Order Calculus, traditionally known as Fractional Calculus (FC),1 which is the branch of mathematics that deals with the study of integrals and derivatives of non-integer orders, has played an outstanding role (Machado et al. 2011). Several mathematicians and applied researchers have obtained important results and generalizations from modeling real processes using FC (Arafa et al. 2016; Camargo et al. 2009a, 2012; Camargo and de Oliveira 2015; Debnath 2003; Mainardi 2009; Ortigueira and Machado 2015; Podlubny 1999; Soubhia et al. 2010). Considering a differential equation that describes a specific phenomenon, a common way to use fraction
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