SOME PIONEERS OF THE APPLICATIONS OF FRACTIONAL CALCULUS Duarte Val´erio, Jos´e Tenreiro Machado, Virginia Kiryakova Abstract In the last decades fractional calculus (FC) became an area of intensive research and development. This paper goes back and recalls important pio- neers that started to apply FC to scientific and engineering problems during the nineteenth and twentieth centuries. Those we present are, in alphabet- ical order: Niels Abel, Kenneth and Robert Cole, Andrew Gemant, Andrey N. Gerasimov, Oliver Heaviside, Paul L´evy, Rashid Sh. Nigmatullin, Yuri N. Rabotnov, George Scott Blair. Key Words and Phrases Fractional calculus, applications, pioneers, Abel, Cole, Gemant, Gerasimov, Heaviside, L´evy, Nigmatullin, Rabotnov, Scott Blair 1. Introduction In 1695 Gottfried Leibniz asked Guillaume l’Hˆopital if the (integer) order of derivatives and integrals could be extended. Was it possible if the order was some irrational, fractional or complex number? “Dream commands life” and this idea motivated many mathematicians, physicists and engineers to develop the concept of fractional calculus (FC). Dur- ing four centuries many famous mathematicians contributed to the theo- retical development of FC. We can list (in alphabetical order) some im- portant researchers since 1695 (see details at [1, 2, 3], and posters at http://www.math.bas.bg/∼fcaa): • Abel, Niels Henrik (5 August 1802 - 6 April 1829), Norwegian math- ematician • Al-Bassam, M. A. (20th century), mathematician of Iraqi origin • Cole, Kenneth (1900 - 1984) and Robert (1914 - 1990), American physicists • Cossar, James (d. 24 July 1998), British mathematician • Davis, Harold Thayer (5 October 1892 - 14 November 1974), Amer- ican mathematician • Djrbashjan, Mkhitar Mkrtichevich (11 September 1918 - 6 May 1994), Armenian (also Soviet Union) mathematician; family name transcripted also as Dzhrbashian, Jerbashian; short CV can be found in Fract. Calc. Appl. Anal. 1, No 4 (1998), 407–414 • Erd´elyi, Arthur (2 October 1908 - 12 December 1977), Hungarian- born British mathematician • Euler, Leonhard (15 April 1707 - 18 September 1783), Swiss math- ematician and physicist • Feller, William (Vilim) (7 July 1906 - 14 January 1970), Croatian- American mathematician • Fourier, Jean Baptiste Joseph (21 March 1768 - 16 May 1830), French mathematician and physicist • Gelfand, Israel (Isra¨ıl) Moiseevich (2 September 1913 - 5 October 2009), Soviet mathematician (of Jewish origin, born in Russian Em- pire in a southern Ukrainian town of Okny, moved later to the USA) • Gemant, Andrew (1895 - 1983), American physicist • Gerasimov, Andrey Nikolaevich (20th century), Russian (then So- viet) physicist working in the field of mechanics • Gru¨nwald, Anton K. (1838 - 1920), German mathematician • Hadamard, Jacques Salomon (8 December 1865 - 17 October 1963), French mathematician • Hardy, Godfrey Harold (7 February 1877 - 1 December 1947), Eng- lish mathematician • Heaviside, Oliver (18 May 1850 - 3 February 1925), English electri- cal engineer, mathematician, and physicist • Holmgren, Hjalmar J. (1822 - 1885), Swedish mathematician • de l’Hoˆpital, Guillaume Franc¸ois Antoine (1661 - 2 February 1704), French mathematician • Kilbas, Anatoly Aleksandrovich (20 July 1948 - 28 June 2010), Be- larusian mathematician; biographical and memorial notes can be found in Fract. Calc. Appl. Anal. 11, No 4 (2008), an issue ded- icated to his 60th anniversary, and in 13, No 2 (2010), 221–223; a founding editor of this journal with a lot of publications therein • Kober, Hermann (1888 - 4 October 1973), mathematician born in Poland, studied and worked in Germany, later in Great Britain • Krug, Anton (19th - 20th century), German mathematician Lacroix, • Sylvestre Fran¸cois (28 April 1765 - 24 May 1843), French mathematician • Lagrange, Joseph-Louis (25 January 1736 - 10 April 1813), Italian- French mathematician and astronomer • Laplace, Pierre-Simon (23 March 1749 - 5 March 1827), French mathematician and astronomer • Laurent, Paul Matthieu Hermann (2 September 1841 - 19 February 1908), French mathematician • Leibniz, Gottfried Wilhelm (1 July 1646 - 14 November 1716), Ger- man mathematician and philosopher • Letnikov, Aleksey Vasil’evich (1 January 1837 - 27 February 1888), Russian mathematician; for his contributions, see in [4] • L´evy, Paul Pierre (15 September 1886 - 15 December 1971), French mathematician • Liouville, Joseph (24 March 1809 - 8 September 1882), French math- ematician • Littlewood, John Edensor (9 June 1885 - 6 September 1977), British mathematician • Love, Eric Russel (31 March 1912 - 7 August 2001), British-Austra- lian mathematician • Marchaud, Andr´e (1887 - 1973), French mathematician • Mikola´s, Miklo´s (20th century), Hungarian mathematician • Mittag-Leffler, Magnus Gustaf (Go¨sta) (16 March 1846 - 7 July 1927), Swedish mathematician • Montel, Paul Antoine Aristide (29 April 1876 - 22 January 1975), French mathematician • Nagy, B´ela Sz˝okefalvi (29 July 1913 - 22 December 1998), Hungar- ian mathematician • Nekrasov, Pavel Alekseevich (13 February 1853 - 20 December 1924), Russian mathematician • Newton, (Sir) Isaac (4 January 1643 - 31 March 1727), English physicist, mathematician, astronomer, natural philosopher, alche- mist, and theologian • Nigmatullin, Rashid Shakirovich (5 January 1923 - 7 July 1991), Tatarstan, Russian Federation (then Soviet Union) scientist in the field of radio-engineering, radioelectronics, electrochemistry, etc. • Pincherle, Salvatore (11 March 1853 - 10 July 1936), Italian math- ematician • Post, Emil Leon (11 February 1897 - 21 April 1954), Polish-American mathematician • Rabotnov, Yury Nikolaevich (24 February 1914 - 15 May 1985), Russian (then Soviet) scientist in the field of mechanics • Riemann, Georg Friedrich Bernhard (17 September 1826 - 20 July 1866), German mathematician • Riesz, Marcel (16 November 1886 - 4 September 1969), Hungarian mathematician • Ross, Bertram (1917 - 27 October 1993), American mathematician • Scott Blair, George William (1902 - 1987), British chemist • Shilov, Georgiy Evgen’evich (3 February 1917 - 17 January 1975), Soviet mathematician • Sneddon, Ian Naismith (8 December 1919 - 4 November 2000), Scot- tish mathematician • Sonine, Nikolay Yakovlevich (22 February 1849 - 27 February 1915), Russian mathematician; for his contributions, see in [4] • Tardy, Placido (23 October 1816 - 2 November 1914), Italian math- ematician • Wallis, John (23 November 1616 - 28 October 1703), English math- ematician • Weierstrass, Karl Theodor Wilhelm (31 October 1815 - 19 February 1897), German mathematician • Weyl, Hermann Klaus Hugo (9 November 1885 - 8 December 1955), German mathematician • Widder, David Vernon (25 March 1898 - 8 July 1990), American mathematician • Zygmund, Antoni (25 December 1900 - 30 May 1992), Polish-born American mathematician During the last two decades practical implementations emerged for the FC theory and it is now recognized to be an important tool to describe phenomena that classical integer-order calculus neglects, see e.g. [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Nevertheless, it is important to remember and pay honor to the pioneers initiating the first applications of FC to scientific and engineering problems, some of them found below in alphabetical order. Our paper is not intended to have a detailed biographical description, but solely to highlight some of the main contributions in the area of applicable FC. Also, as these authors worked and published before the Internet era and digitalization of literature, for some of them only a few data could be found, and other names are dropped for the same reason. 2. Niels Abel Niels Henrik Abel was born in 1802 in Nedstrand, Norway. He became a proficient mathematician when he was a teenager, and died when he was only twenty six, with tuberculosis, in 1829. Much of his important work in Analysis became known only after his death: a paper of his on integration was lost, Gauß did not care to read a letter he sent proving a result on fifth- order equations, Cauchy misplaced a paper of his on elliptic equations, and a treatise he wrote on that subject was only discovered after his death. He got a professorship appointment only two days after he died, so he lived in straitness all his life, [18]. C Figure 1. Niels Henrik Abel (1802-1829) His work in Mathematics, particularly in group theory and elliptic func- tions, is outstanding, especially considering how early in life he died. His connection to FC comes from an application of fractional integrals he made to the problem in Physics which is finding a tautochrone curve, i.e. a curve y(x) such that a mass m at rest on any of its points will always take the same time T to slide down without friction till reaching the final point (x, y) = (0, 0), irrespective of the initial height h at which it begins to move because of the acceleration of gravity g, [19, 20, 6]. While it is known since Huygen’s 1659 work on pendulums that the tautochrone curve is the cycloid (Figure 2), this was found from geometrical arguments; Abel’s solution, based upon Mathematical Analysis, leads to a fractional integral. Because of the conservation of mechanical energy (due to the nonexis- tence of friction), the initial energy at rest mgh is conserved when the mass is at an arbitrary height y and has the mechanical energy mgy + 1 mV 2, and so its velocity V is given by 2 fro an arbitrary to the y ce tra elled m y end of the curve is h y x 0 Figure 2. The cycloid − dR(y) dy and so V = dy dt . Equaling this to V above we get and from here, we arrive at We want T (h) to be independent of h: thus, the integral in (2.4) can have any value of y as its upper limit of integration. Let us rewrite it as = where the last equality is a consequence of the law of exponents. Making √ a = 2gT we can get, from (2.2), π 1 The parametric equations of a cycloid generated by a circle of radius r, centred on (0, r), rotating an angle θ on y = 2r, are which is the same as (2.7). As the curve begins at the origin, we can obtain an integral representation for the curve given by 3. Kenneth Cole and Robert Cole Kenneth Stewart Cole (Figure 3, left) was born on 10 July 1900 in Ithaca, New York, USA, [21, 22, 23].