TY - JOUR
T1 - Complex Grünwald–Letnikov, Liouville, Riemann–Liouville, and Caputo derivatives for analytic functions
AU - Ortigueira, Manuel Duarte
AU - DEE Group Author
PY - 2011/1/1
Y1 - 2011/1/1
N2 - The well-known Liouville, Riemann–Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grünwald–Letnikov derivative. Particularly, starting from a complex formulation of the Grünwald–Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann–Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann–Liouville and Caputo derivatives with the Grünwald–Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.
AB - The well-known Liouville, Riemann–Liouville and Caputo derivatives are extended to the complex functions space, in a natural way, and it is established interesting connections between them and the Grünwald–Letnikov derivative. Particularly, starting from a complex formulation of the Grünwald–Letnikov derivative we establishes a bridge with existing integral formulations and obtained regularised integrals for Liouville, Riemann–Liouville, and Caputo derivatives. Moreover, it is shown that we can combine the procedures followed in the computation of Riemann–Liouville and Caputo derivatives with the Grünwald–Letnikov to obtain a new way of computing them. The theory we present here will surely open a new way into the fractional derivatives computation.
U2 - 10.1016/j.cnsns.2011.02.022
DO - 10.1016/j.cnsns.2011.02.022
M3 - Article
VL - 16
SP - 4174
EP - 4182
JO - Communications In Nonlinear Science And Numerical Simulation
JF - Communications In Nonlinear Science And Numerical Simulation
SN - 1007-5704
IS - NA
ER -