Two-Sided Fractional Derivatives
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.
KeywordsFractional Derivative Gamma Function Integration Path Integer Order Fractional Difference
- 7.Ortigueira MD (2006) Riesz potentials and inverses via centred derivatives. Int J Math Math Sci 2006:1–12. Article ID 48391Google Scholar
- 8.Samko SG, Kilbas AA, Marichev OI (1987) Fractional integrals and derivatives—theory and applications. Gordon and Breach Science Publishers, New YorkGoogle Scholar