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Unexpected behavior of Caputo fractional derivative

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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.

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Notes

  1. In fact, the name Fractional Calculus is not accurate, since the order of an integral and a derivative can be real and also complex.

  2. For convenience, is defined that \(I^0f(t)=f(t)\).

  3. It follows from the definition that if \(\alpha =m\), so \(D^m f(t)=I^{m-m}\,D^m f(t)=I^0\,D^mf(t)=D^mf(t)\), that is, the usual derivative is a particular case.

  4. Indeed, this argument is not accurate, and it will be explained in the last section.

  5. Since for \(1<\alpha \le 2\), in Eq. (8), \(m=2\).

References

  • Arafa AAM, Rida SZ, Khalil M (2012) Fractional modeling dynamics of HIV and CD4+ T-cells during primary infection. Nonlinear Biomed Phys 6:1–7

    Article  Google Scholar 

  • Arafa AAM, Hanafy IM, Gouda MI (2016) Stability analysis of fractional order HIV infection of \(^+\)T cells with numerical solutions. J Fract Calc Appl 7(1)

  • Camargo RF, de Oliveira EC (2015) Cálculo fracionário. Editora Livraria da Física, São Paulo

    Google Scholar 

  • Camargo RF, Chiacchio AO, de Oliveira EC (2008) Differentiation to fractional orders and the fractional telegraph equation. J Math Phys 49(033505)

  • Camargo RF, Charnet R, Charnet R, de Oliveira EC (2009) On some fractional Green’s functions. J Math Phys 50(043514):112

  • Camargo RF, Chiacchio AO, de Oliveira EC (2009) Solution of the fractional Langevin equation and the Mittag–Leffler functions. J Math Phys 50(063507)

  • Camargo RF, de Oliveira EC, Vaz Jr. J (2009) On anomalous diffusion and the fractional generalized Langevin equation for a harmonic oscillator. J Math Phys 50(123518)

  • Camargo RF, de Oliveira EC, Vaz J Jr (2012) On the generalized Mittag–Leffler function and its application in a fractional telegraph equation. Math Phys Anal Geom 15(1):1–16

    Article  MathSciNet  MATH  Google Scholar 

  • de Oliveira EC, Machado JT (2014) A review of definitions for fractional derivatives and integrals. Math Probl Eng (238459):6

  • Debnath L (2003) Recent applications of fractional calculus to science and engineering. Int J Math Math Sci 2003(54):3413–3442

    Article  MathSciNet  MATH  Google Scholar 

  • Diethelm K (2010) The analysis of fractional differential equations. Springer, Heidelberg

    Book  MATH  Google Scholar 

  • Diethelm K, Ford NJ, Freed AD, Luchko Yu (2005) Algorithms for the fractional calculus selection of numerical methods. Comput Methods Appl Mech Eng 194(6–8):743773

  • Elsadany AA, Matouk AE (2014) Dynamical behaviors of fractional-order Lotka Volterra predator prey model and its discretization. J Appl Math Comput 49(1):269–283

  • El-Sayed AMA, El-Mesiry AEM, El-Saka HAA (2007) One the fractional-order logistic equation. Appl Math Lett 20:817–823

    Article  MathSciNet  MATH  Google Scholar 

  • El-Sayed AMA, Rida SZ, Arafa AAM (2009) On the solutions of time-fractional bacterial chemotaxis in a diffusion gradient chamber. Int J Nonlinear Sci 7:485–492

    MathSciNet  Google Scholar 

  • Gutierrez RE, Rosario JM, Machado JT (2010) Fractional order calculus: basic concepts and engineering applications. Math Probl Eng 2010(375858):19

  • Hilfer R (2000) Applications of fractional calculus in physics, vol 128. World Scientific, Singapore

    Book  MATH  Google Scholar 

  • Khalia R, Horania MA, Yousefa A, Sabadheb M (2014) A new definition of fractional derivative. J Comput Appl Math 264:65–70

    Article  MathSciNet  Google Scholar 

  • Kolwankar KM, Gangal AD (1996) Fractional differentiability of nowhere differentiable functions and dimensions. Chaos 6(4):19

    Article  MathSciNet  MATH  Google Scholar 

  • Li M, Lim SC, Chen S (2011) Exact solution of impulse response to a class of fractional oscillators and its stability. Math Probl Eng 2011(657839):9

  • Machado JAT, Kiryakova V, Mainardi F (2011) Recent history of fractional calculus. Commun Nonlinear Sci Numer Simul 16(3):1140–1153

    Article  MathSciNet  MATH  Google Scholar 

  • Mainardi F (2009) Fractional calculus and waves in linear viscoelasticity. Imperial College Press, London

    Google Scholar 

  • Mainardi F, Gorenflo R (2000) On Mittag–Leffler-type functions in fractional evolution process. J Comput Appl Math 118:283–299

    Article  MathSciNet  MATH  Google Scholar 

  • Malinowska AB, Odzijewicz T, Torres DFM (2015) Advanced methods in the fractional calculus of variations. In: Springer briefs in applied sciences and technology

  • Matigon D (1996) Stability results for fractional differential equations with applications to control processing. In: Computational engineering in systems applications, Lille, pp 963–968

  • Matouk AE (2010) Dynamical behaviors, linear feedback control and synchronization of the fractional order Liu system. J Nonlinear Syst Appl 3:135–140

    Google Scholar 

  • Matouk AE (2015) Chaos synchronization of a fractional-order modified Van der Pol Duffing system via new linear control, backstepping control and Takagi Sugeno fuzzy approaches. Complexity

  • Matouk AE, Elsadany AA, Ahmed E, Agiza HN (2015) Dynamical behavior of fractional-order Hastings Powell food chain model and its discretization. Commun Nonlinear Sci Numer Simul 27(1–3):153–167

  • Mittag-Leffler GM (1903) Sur la nouvelle fonction \(E_\alpha (z)\). C R Acad Sci 137:554–558

    MATH  Google Scholar 

  • Ortigueira MD, Machado JAT (2015) What is a fractional derivative? J Comput Phys 293:4–13

    Article  MathSciNet  MATH  Google Scholar 

  • Podlubny I (1999) Mathematics in science and engineering, vol 198., Fractional differential equationsAcademic Press, San Diego

    MATH  Google Scholar 

  • Podlubny I (2002) Geometric and physical interpretation of fractional integration and fractional differentiation. Fract Calc Appl Anal 5(4):367–386

    MathSciNet  MATH  Google Scholar 

  • Sabatier J, Argrawal OP, Machado JAT (2007) Advances in fractional calculus: theoretical developments and applications in physics and engineering. Springer, New York

    Book  Google Scholar 

  • Soubhia AL, Camargo RF, de Oliveira EC, Vaz J Jr (2010) Theorem for series in three-parameter Mittag-Leffler function. Fract Calc Appl Anal 13:9–20

    MathSciNet  MATH  Google Scholar 

  • Tavassoli MH, Tavassoli A, Rahimi MRO (2013) The geometric and physical interpretation of fractional order derivatives of polynomial functions. Differ Geom Dyn Syst 15:93–104

    MathSciNet  MATH  Google Scholar 

  • Varalta N, Gomes AV, Camargo RF (2014) A prelude to the fractional calculus applied to tumor dynamic. TEMA 15(2):211–221

    MathSciNet  Google Scholar 

  • Verhulst PF (1838) Notice sur la loi que la population poursuit dans son accroissement. Corresp Math Phys 10:113–121

    Google Scholar 

  • Wiman A (1905) Über den Fundamental Satz in der Theorie der Funktionen \(E_{\alpha } (x)\). Acta Math 29:191–201

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors thanks the research group MApliC for the important and productive discussions.

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Authors and Affiliations

  1. Departamento de Bioestatística, Instituto de Biociências, UNESP, Distrito de Rubião Junior, Botucatu, SP, 18618-689, Brazil

    Lucas Kenjy Bazaglia Kuroda, Robinson Tavoni & Paulo Fernando de Arruda Mancera

  2. Faculdade de Ciências, UNESP, Av. Eng. Luiz Edmundo Carrijo Coube, 14-01, Bairro Vargem Limpa, Bauru, SP, 17033-360, Brazil

    Arianne Vellasco Gomes & Najla Varalta

  3. Departamento de Matemática, Faculdade de Ciências, UNESP, Av. Eng. Luiz Edmundo Carrijo Coube, 14-01, Bairro Vargem Limpa, Bauru, SP, 17033-360, Brazil

    Rubens de Figueiredo Camargo

Authors
  1. Lucas Kenjy Bazaglia Kuroda
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  2. Arianne Vellasco Gomes
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  3. Robinson Tavoni
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  4. Paulo Fernando de Arruda Mancera
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  5. Najla Varalta
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  6. Rubens de Figueiredo Camargo
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Corresponding author

Correspondence to Rubens de Figueiredo Camargo.

Additional information

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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Kuroda, L.K.B., Gomes, A.V., Tavoni, R. et al. Unexpected behavior of Caputo fractional derivative. Comp. Appl. Math. 36, 1173–1183 (2017). https://doi.org/10.1007/s40314-015-0301-9

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  • DOI: https://doi.org/10.1007/s40314-015-0301-9

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