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Fractional Calculus

Integral and Differential Equations of Fractional Order

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Fractals and Fractional Calculus in Continuum Mechanics

Part of the book series: International Centre for Mechanical Sciences ((CISM,volume 378))

Abstract

In these lectures we introduce the linear operators of fractional integration and fractional differentiation in the framework of the Riemann-Liouville fractional calculus. Particular attention is devoted to the technique of Laplace transforms for treating these operators in a way accessible to applied scientists, avoiding unproductive generalities and excessive mathematical rigor. By applying this technique we shall derive the analy ical solutions of the most simple linear integral and differential equations of fractional order. We shall show the fundamental role of the Mittag-Leffler function, whose properties are reported in an ad hoc Appendix. The topics discussed here will be: (a) essentials of Riemann-Liouville fractional calculus with basic formulas of Laplace transforms, (b) Abel type integral equations of first and second kind, (c) relaxation and oscillation type differential equations of fractional order.

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Authors and Affiliations

  1. Free University of Berlin, Berlin, Germany

    R. Gorenflo

  2. University of Bologna, Bologna, Italy

    F. Mainardi

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  1. R. Gorenflo
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  2. F. Mainardi
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Editor information

Editors and Affiliations

  1. Polytechnic of Turin, Italy

    A. Carpinteri

  2. University of Bologna, Italy

    F. Mainardi

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Gorenflo, R., Mainardi, F. (1997). Fractional Calculus. In: Carpinteri, A., Mainardi, F. (eds) Fractals and Fractional Calculus in Continuum Mechanics. International Centre for Mechanical Sciences, vol 378. Springer, Vienna. https://doi.org/10.1007/978-3-7091-2664-6_5

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  • DOI: https://doi.org/10.1007/978-3-7091-2664-6_5

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  • Print ISBN: 978-3-211-82913-4

  • Online ISBN: 978-3-7091-2664-6

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