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Guido Maione (Technical University of Bari, Italy), Antonio Punzi (Technical University of Bari, Italy) and Kang Li (Queen’s University of Belfast, UK)

Copyright: © 2013
|Pages: 29

DOI: 10.4018/978-1-4666-2666-9.ch010

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TopFractional Calculus (FC) is a topic older than three centuries. Namely, its birth can be dated back to an exchange of letters and ideas between Leibniz and marquis de L’Hôpital in 1695. In this correspondence, de L’Hôpital asked to Leibniz what could be the value and meaning of a non-integer (fractional) order derivative _{}, more specifically with *ν* = 0.5. This could extend the classical integer order derivative _{}, with *n*∈ℕ, to a more general case, in which the order of differentiation *ν* could be a fractional number. Considering *n*∈ℤ and *ν* < 0, an integer order integral could be extended to a fractional order one. Leibniz gave the result and answered on September 30, 1695, that *“It will lead to an apparent paradox, from which one day useful consequences will be drawn”*.

Since then many mathematicians and scientists (Euler, Abel, Lacroix, Fourier, Lagrange, Laplace, Riemann, Liouville, Kellang, Grünwald, Letnikov, Caputo, etc.) have formulated and investigated formal properties of non-integer order differentiation and integration. As an example, Heaviside said that *“there is a universe of mathematics lying in between the complete differentiations and integrations”*. At his time, he faced the problem of a rigorous justification for the square root operation of a partial differentiation operator *p*. After algebraic manipulations, he obtained *p ^{α}*, with

The idea of fractional derivative or integral can be described in different ways. The most popular definitions are due to Riemann-Liouville, to Grünwald-Letnikov, and to Caputo. The Riemann-Liouville basic definition of fractional order integral generalizes the repeated integration in the Cauchy formula:

However, evaluation of Equation 3 requires initial values of the fractional order derivatives that are of difficult interpretation and measurement.

The Caputo definition avoids physical interpretation of initial conditions in Laplace transform of the fractional order derivative operator by writing:

.Namely, the Laplace transform requires the knowledge of the initial values of standard integer order derivatives:

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