Lecturer: Francesco MAINARDI, Università di Bologna, Bologna, Italy Webpage: http://www.fracalmo.org/mainardi/index.htm   Title: AN INTRODUCTION TO FRACTIONAL CALCULUS Date and time: Mon, March 11 to Fri, March 15, 2013, 9:00 to 11:00 Abstract: Fractional calculus, in allowing integrals and derivatives of any positive order (the term ”fractional” kept only for historical reasons), can be considered a branch of mathematical physics which mainly deals with integro-differential equations, where integrals are of convolution form with weakly singular kernels of power law type. In recent decades fractional calculus has won more and more interest in applications in several fields of applied sciences. The purpose of these lecture notes is to provide the essentials of fractional calculus and outline its role in providing simplest evolution processes related to relaxation, oscillation, diffusion and wave propagation phenomena. The treatment mainly reflects the research activity and style of the author in the related scientific areas during the last decades. The students are required to be acquainted with integration in the complex plane, Fourier and Laplace transforms, asymptotic series, Eulerian functions (Gamma and Beta) .The essential notions of linear viscoelasticity, and higher transcendental functions of the Mittag-Leffler and Wright type are given by the author. Bibliography:

[[1] F. Mainardi: "Fractional Calculus and Waves in Linear Viscoelasticity", Imperial College Press, London (2010), pp. 340, ISBN 978-1-84816-329-4, see: http://www.icpress.co.uk/mathematics/p614.html

[2] F. Mainardi, Yu. Luchko and G. Pagnini : "The fundamental solution of the space-time fractional diffusion equation", Fractional Calculus and Applied Analysis, Vol. 4, No 2, 153-192 (2001). [E-print http://arxiv.org/abs/cond-mat/0702419]

[3] F. Mainardi : "Fractional calculus, some basic problems in continuum and statistical mechanics", in A. Carpinteri and F. Mainardi (Editors), Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien (1997), pp. 291-348. Vol. no 378, series CISM Courses and Lecture Notes, (ISBN 3-211-82913-X) [Advanced School held at CISM, Udine, Italy, 23-27 September 1996] [E-print http://arxiv.org/abs/1201.0863]

[4] R. Gorenflo and F. Mainardi : "Fractional calculus, integral and differential equations of fractional order”, in A. Carpinteri and F. Mainardi (Editors),\Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, Wien (1997), pp. 223-276. Vol. no 378, series CISM Courses and Lecture Notes, (ISBN 3-211-82913-X) [Advanced School held at CISM, Udine, Italy, 23-27 September 1996] [E-print http://arxiv.org/abs/0805.3823]

[5] F. Mainardi: "Fractional relaxation-oscillation and fractional diffusion-wave phenomena", Chaos, Solitons and Fractals, Vol. 7, No 9, pp. 1461-1477 (1996).