Begell House Inc.
Critical Reviewsâ„˘ in Biomedical Engineering
CRB
0278-940X
35
6
2007
INITIALIZATION, CONCEPTUALIZATION, AND APPLICATION IN THE GENERALIZED (FRACTIONAL) CALCULUS
447-553
Carl F.
Lorenzo
National Aeronautics and Space Administration, Glenn Research Center, Cleveland, Ohio, USA
Tom T.
Hartley
Department of Electrical and Computer Engineering, University of Akron, USA
This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.