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Unformatted text preview: GEOMETRIC AND PHYSICAL INTERPRETATION OF FRACTIONAL INTEGRATION AND FRACTIONAL DIFFERENTIATION * Igor Podlubny Dedicated to Professor Francesco Mainardi, on the occasion of his 60-th birthday Abstract A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., in- tegration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral. Mathematics Subject Classification : 26A33, 26A42, 83C99, 44A35, 45D05 Key Words and Phrases : fractional derivative, fractional integral, frac- tional calculus, geometric interpretation, physical interpretation * Partially supported by Grant 1/7098/20 of the Slovak Grant Agency for Science (VEGA) and by the Cambridge Colleges Hospitality Scheme grant (2000). 368 I. Podlubny 1. Introduction It is generally known that integer-order derivatives and integrals have clear physical and geometric interpretations, which significantly simplify their use for solving applied problems in various fields of science. However, in case of fractional-order integration and differentiation, which represent a rapidly growing field both in theory and in applications to real- world problems, it is not so. Since the appearance of the idea of differ- entiation and integration of arbitrary (not necessary integer) order there was not any acceptable geometric and physical interpretation of these op- erations for more than 300 years. The lack of these interpretations has been acknowledged at the first international conference on the fractional calculus in New Haven (USA) in 1974 by including it in the list of open problems . The question was unanswered, and therefore repeated at the subsequent conferences at the University of Strathclyde (UK) in 1984  and at the Nihon University (Tokyo, Japan) in 1989 . The round-table discussion [13, 10, 14] at the conference on transform methods and special functions in Varna (1996) showed that the problem was still unsolved, and since that time the situation, in fact, still did not change. Fractional integration and fractional differentiation are generalizations of notions of integer-order integration and differentiation, and include n-th derivatives and n-fold integrals ( n denotes an integer number) as particular cases. Because of this, it would be ideal to have such physical and geomet- ric interpretations of fractional-order operators, which will provide also a link to known classical interpretations of integer-order differentiation and integration....
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