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Fractional calculus, in allowing integrals and derivatives of any positive real order (the term “fractional” is kept only for historical reasons), can be considered a branch of mathematical analysis which deals with integro-differential equations where the integrals are of convolution type and exhibit (weakly singular) kernels of power-law type. It has a history of at least three hundred years because it can be dated back to the letter from G.W. Leibniz to G.A. de L’Hôpital and J. Wallis, dated 30 September 1695, in which the meaning of the one-half order derivative was first discussed and were made some remarks about its possibility. Subsequent mention of fractional derivatives was made, in some context or the other by L. Euler (1730), J.L. Lagrange (1772), P.S. Laplace (1812), S.F. Lacroix (1819), J.B.J. Fourier (1822), N.H. Abel (1823), J. Liouville (1832), B. Riemann (1847), H.L. Greer (1859), H. Holmgren (1865), A.K. Grünwald (1867), A.V. Letnikov (1868), N.Ya. Sonin (1869), H. Laurent (1884), P.A. Nekrassov (1888), A. Krug (1890), O. Heaviside (1892), S. Pincherle (1902), H. Weyl (1919), P. Lévy (1923), A. Marchaud (1927), H.T. Davis (1936), A. Zygmund (1945), M. Riesz (1949), W. Feller (1952), just to cite some relevant contributors up the mid of the last century, see e.g. [1,2]. Recently, a poster illustrating the major contributors during the period 1695-1970 has been published .
- 1.K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations (Wiley, New York, 1993)Google Scholar
- 2.R. Gorenflo, F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, edited by A. Carpinteri, F. Mainard (Springer Verlag, Wien, 1997), p. 223 [e-print in http://arxiv.org/abs/0805.3823]Google Scholar
- 4.F. Mainardi, An historial perspective on fractional calculus in linear viscoelasticity, [e-print http://arxiv.org/abs/007.2959]Google Scholar
- 5.F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticiy (Imperial College Press, London, 2010)Google Scholar