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 -