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CISM LECTURE NOTES
International Centre for Mechanical Sciences
Palazzo del Torso, Piazza Garibaldi, Udine, Italy
FRACTIONAL CALCULUS:
Integral and Differential Equations of Fractional Order
Rudolf GORENFLO and Francesco MAINARDI
Department of Mathematics Department of Physics
Free University of Berlin University of Bologna
Arnimallee 3 Via Irnerio 46
D-14195 Berlin, Germany I-40126 Bologna, Italy
gorenflo@math.fu-berlin.de mainardi@bo.infn.it
T X PRE-PRINT 54 pages : pp. 223-276
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ABSTRACT..........................p.223
1. INTRODUCTION TO FRACTIONAL CALCULUS . . . . . . . . p. 224
2.FRACTIONALINTEGRALEQUATIONS ............ p.235
3. FRACTIONAL DIFFERENTIAL EQUATIONS: 1-st PART . . . . p. 241
4. FRACTIONAL DIFFERENTIAL EQUATIONS: 2-nd PART . . . . p. 253
CONCLUSIONS .......................p.261
APPENDIX:THEMITTAG-LEFFLERTYPEFUNCTIONS . . . p. 263
REFERENCES ........................p.271
The paper is based on the lectures delivered by the authors at the CISM Course
Scaling Laws and Fractality in Continuum Mechanics: A Survey of the Methods based
on Renormalization Group and Fractional Calculus, held at the seat of CISM, Udine,
from 23 to 27 September 1996, under the direction of Professors A. Carpinteri and
F.Mainardi.
This T X pre-print is a revised version (December 2000) of the chapter published in
E
A. Carpinteri and F. Mainardi (Editors): Fractals and Fractional Calculus
in Continuum Mechanics, Springer Verlag, Wien and New York 1997, pp.
223-276.
Such book is the volume No. 378 of the series CISM COURSES AND LECTURES
[ISBN 3-211-82913-X]
i
c
1997, 2000 Prof. Rudolf Gorenflo - Berlin - Germany
c
1997, 2000 Prof. Francesco Mainardi - Bologna - Italy
fmcism1x.tex, fmrg1x.tex = versions in plain T X, 54 pages.
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ii
R.Gorenflo and F.Mainardi 223
FRACTIONAL CALCULUS:
Integral and Differential Equations of Fractional Order
Rudolf GORENFLO and Francesco MAINARDI
Department of Mathematics Department of Physics
Free University of Berlin University of Bologna
Arnimallee 3 Via Irnerio 46
D-14195 Berlin, Germany I-40126 Bologna, Italy
gorenflo@math.fu-berlin.de mainardi@bo.infn.it
ABSTRACT
In these lectures we introduce the linear operators of fractional integration and frac-
tional 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 gen-
eralities and excessive mathematical rigor. By applying this technique we shall derive
the analytical 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.
2000MathematicsSubjectClassification: 26A33,33E12,33E20,44A20,45E10,45J05.
Thisresearch was partially supported by Research Grants of the Free University of
Berlin and the University of Bologna. The authors also appreciate the support given
by the National Research Councils of Italy (CNR-GNFM) and by the International
Centre of Mechanical Sciences (CISM).
224 Fractional Calculus: Integral and Differential Equations of Fractional Order
1. INTRODUCTION TO FRACTIONAL CALCULUS
1.1 Historical Foreword
Fractional calculus is the field of mathematical analysis which deals with the
investigation and applications of integrals and derivatives of arbitrary order. The
term fractional is a misnomer, but it is retained following the prevailing use.
The fractional calculus may be considered an old and yet novel topic. It is an
old topic since, starting from some speculations of G.W. Leibniz (1695, 1697) and
L. Euler (1730), it has been developed up to nowadays. A list of mathematicians,
who have provided important contributions up to the middle of our century, includes
P.S. Laplace (1812), J.B.J. Fourier (1822), N.H. Abel (1823-1826), J. Liouville (1832-
1873), B. Riemann (1847), H. Holmgren (1865-67), A.K. Gru¨nwald (1867-1872), A.V.
Letnikov (1868-1872), H. Laurent (1884), P.A. Nekrassov (1888), A. Krug (1890), J.
Hadamard (1892), O. Heaviside (1892-1912), S. Pincherle (1902), G.H. Hardy and
J.E. Littlewood (1917-1928), H. Weyl (1917), P. L´evy (1923), A. Marchaud (1927),
H.T. Davis (1924-1936), A. Zygmund (1935-1945), E.R. Love (1938-1996), A. Erd´elyi
(1939-1965), H. Kober (1940), D.V. Widder (1941), M. Riesz (1949).
However, it may be considered a novel topic as well, since only from a little more
than twenty years it has been object of specialized conferences and treatises. For the
first conference the merit is ascribed to B. Ross who organized the First Conference
on Fractional Calculus and its Applications at the University of New Haven in June
1974, and edited the proceedings, see [1]. For the first monograph the merit is
ascribed to K.B. Oldham and J. Spanier, see [2], who, after a joint collaboration
started in 1968, published a book devoted to fractional calculus in 1974. Nowadays,
the list of texts and proceedings devoted solely or partly to fractional calculus and its
applications includes about a dozen of titles [1-14], among which the encyclopaedic
treatise by Samko, Kilbas & Marichev [5] is the most prominent. Furthermore, we
recall the attention to the treatises by Davis [15], Erd´elyi [16], Gel’fand & Shilov
[17], Djrbashian [18, 22], Caputo [19], Babenko [20], Gorenflo & Vessella [21], which
contain a detailed analysis of some mathematical aspects and/or physical applications
of fractional calculus, although without explicit mention in their titles.
In recent years considerable interest in fractional calculus has been stimulated
by the applications that this calculus finds in numerical analysis and different areas
of physics and engineering, possibly including fractal phenomena. In this respect
A. Carpinteri and F. Mainardi have edited the present book of lecture notes and
entitled it as Fractals and Fractional Calculus in Continuum Mechanics.Forthe
topic of fractional calculus, in addition to this joint article of introduction, we have
contributed also with two single articles, one by Gorenflo [23], devoted to numerical
methods, and one by Mainardi [24], concerning applications in mechanics.
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