TY - CHAP

T1 - Two-sided fractional derivatives

AU - Ortigueira, Manuel Duarte

PY - 2011

Y1 - 2011

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79959693820&partnerID=8YFLogxK

U2 - 10.1007/978-94-007-0747-4_5

DO - 10.1007/978-94-007-0747-4_5

M3 - Chapter

AN - SCOPUS:79959693820

SN - 9789400707467

T3 - Lecture Notes in Electrical Engineering

SP - 101

EP - 121

BT - Fractional Calculus for Scientists and Engineers

ER -