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Two-Sided Fractional Derivatives

  • Manuel Duarte Ortigueira
Chapter
Part of the Lecture Notes in Electrical Engineering book series (LNEE, volume 84)

Abstract

In previous chapters the causal and anti-causal fractional derivatives were presented. An application to shift-invariant linear systems was studied. Those derivatives were introduced into four steps: 1. Use as starting point the Grünwald–Letnikov differences and derivatives. 2. With an integral formulation for the fractional differences and using the asymptotic properties of the Gamma function obtain the generalised Cauchy derivative. 3. The computation of the integral defining the generalised Cauchy derivative is done with the Hankel path to obtain regularised fractional derivatives. 4. The application of these regularised derivatives to functions with Laplace transform, we obtain the Liouville fractional derivative and from this the Riemann–Liouville and Caputo, two-step derivatives.

Keywords

Fractional Derivative Gamma Function Integration Path Integer Order Fractional Difference 
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References

  1. 1.
    Henrici P (1991) Applied and computational complex analysis, vol 2. Wiley, New York, pp 389–391zbMATHGoogle Scholar
  2. 2.
    Okikiolu GO (1966) Fourier transforms of the operator H ?. Proc Camb Philos Soc 62:73–78MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Andrews GE, Askey R, Roy R (1999) Special functions. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  4. 4.
    Ortigueira MD (2004) From differences to differintegrations. Fract Calc Appl Anal 7(4):459–471MathSciNetzbMATHGoogle Scholar
  5. 5.
    Ortigueira MD (2006) A coherent approach to non integer order derivatives. Signal Process Special Sect Fract Calc Appl Signals Syst 86(10):2505–2515zbMATHGoogle Scholar
  6. 6.
    Ortigueira MD, Serralheiro AJ (2006) A new least-squares approach to differintegration modelling. Signal Process Special Sect Fract Calc Appl Signals Syst 86(10):2582–2591zbMATHGoogle Scholar
  7. 7.
    Ortigueira MD (2006) Riesz potentials and inverses via centred derivatives. Int J Math Math Sci 2006:1–12. Article ID 48391Google Scholar
  8. 8.
    Samko SG, Kilbas AA, Marichev OI (1987) Fractional integrals and derivatives—theory and applications. Gordon and Breach Science Publishers, New YorkGoogle Scholar
  9. 9.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar
  10. 10.
    Podlubny I (1999) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Academic Press, San DiegozbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Faculdade de Ciências/Tecnologia da UNLUNINOVA and DEECaparicaPortugal

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