Abstract
This opening lecture is intended to serve as a propaedeutic for the papers to be presented at this conference whose nonhomogeneous audience includes scientists, mathematicians, engineers and educators. This expository and developmental lecture, a case study of mathematical growth, surveys the origin and development of a mathematical idea from its birth in intellectual curiosity to applications. The fundamental structure of fractional calculus is outlined. The possibilities for the use of fractional calculus in applicable mathematics is indicated. The lecture closes with a statement of the purpose of the conference.
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References
Leibnitz, G.W., Leibnitzen's Mathematische Schriften, Hildesheim, Germany: Georg Olm, 1962, v. 2, pp. 301–302.
Lacroix, S.F., Traité du Calcul Différentiel et du Calcul Intégral, Paris: Mme. VeCourcier, 1819, Tome Troisiéme, seconde édition, pp. 409–410.
Spanier, Jerome and Oldham, Keith B., The Fractional Calculus, New York: Academic Press, 1974.
Abel, Niels Henrik, "Solution de quelques problèmes a`l'aide d'intégrales définies," Oeuvres Complètes, Christiania, 1881, tome première, 16–18.
Liouville, Joseph, "Mémoire sur quelques Quéstions de Géometrie et de Mécanique, et sur un nouveau genre de Calcul pour résoudre ces Quéstions," Journal de l'École Polytechnique, 1832, tome XIII, XXIe cahier, pp. 1–69.
A more detailed discussion of Liouville's first and second definitions and also of their connection with the Riemann definition can be found in The Development of the Gamma Function and A Profile of Fractional Calculus, by Bertram Ross, New York University dissertation, 1974, Chapter V, pp. 142–210. University Microfilms, Ann Arbor, Mich., #74-17154, PO #45122.
Debnath, Lokenath and Speight, T.B., "On Generalized Derivatives," Pi Mu Epsilon Journal, v. 5, 1971, ND 5, pp. 217–220, East Carolina University.
Details will be found in "A Chronological Bibliography of Fractional Calculus with Commentary," by Bertram Ross in The Fractional Calculus [3], pp. 3–15, and in [6].
Davis, Harold Thayer, The Theory of Linear Operators, Bloomington, Indiana: The Principia Press, 1936, p. 20.
See [6], pp. 158–162.
The first one to apply Dirichlet's method to kernels of the form (x-t)v is Wallie Abraham Hurwitz in 1908. Cited by Whittaker and Watson, A Course in Modern Analysis, 4th edition, 1963, p. 76.
I am indebted to Dr. Kenneth S. Miller, Riverside Research Institute, New York City, for this contribution.
This approach was recommended by George F. Carrier, Harvard University.
Love, Eric Russell, "Fractional Derivatives of Imaginary Order," The Journal of the London Mathematical Society, Volume III (Second Series), 1971, pp. 241–259.
Farrell, Orin J. and Ross, Bertram, Solved Problems in Analysis, New York: Dover Publications, 1971, 279. First published in 1963, New York: The Macmillan Co.
Brenke, W.C., "An Application of Abel's Integral Equation," American Mathematical Monthly, 1922, v. 29, 58–60.
Bernoulli's equation is strictly valid for steady, frictionless flow in a stream tube. It is used, however, in engineering for flows with friction by modification of solutions with a suitable friction factor.
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Ross, B. (1975). A brief history and exposition of the fundamental theory of fractional calculus. In: Ross, B. (eds) Fractional Calculus and Its Applications. Lecture Notes in Mathematics, vol 457. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0067096
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DOI: https://doi.org/10.1007/BFb0067096
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