Plenary Lecture
Fractional Calculus: What Is It? And What Is It for?
Professor Francesco Mainardi
Department of Physics
University of Bologna
&
INFN
Italy
E-mail: francesco.mainardi@bo.infn.it
Abstract: In 1695 L’Hospital inquired to Leibniz what meaning could be given to the symbol d^n y/dx^n when n = 1/2. In a letter dated September 30, 1695 Leibniz replied “It will lead to a paradox, from which one day useful consequences will be drawn”. This discussion lead to a new branch of mathematics which deals with derivatives and integrals of arbitrary order and is known as Fractional Calculus. Of course this is a misnomer kept only for historical reasons. It can be considered as a branch of mathematical analysis which deals with integro-differential operators and equations where the integrals are of convolution type and exhibit (weakly singular) kernels of power-law type. It is strictly related to the theory of pseudo-differential operators. Fractional differential and integral equations have gained considerable popularity and importance during the past three decades. The main advantage of the fractional calculus is that provides excellent instruments for the description of memory and non local properties of various materials and processes. The list of applications is huge and includes, just to cite a few, Visco-elasticity, Electrical Circuits, Control theory, intermediate phenomena between Diffusion and Wave propagation, Biology, Bioengineering, Image processing, Finance, Stochastic processes….. Below a partial list of books published since 2010 on theory and application of Fractional Calculus.
- Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity, Imperial College Press, London and World Scientific, Singapore.
- Diethelm, K. (2010). The Analysis of Fractional Differential Equations, Springer, Berlin.
- Tarasov, V.E. (2010). Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media, Springer, Berlin.
- Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J. (2012). Fractional Calculus: Models and Numerical Methods, World Scientific, Singapore.
- Uchaikin, V.V. (2013). Fractional Derivatives for Physicists and Engineers. Vol. I: Background and Theory; Vol. II Applications, Springer, Berlin/Higher Education Press, Beijing.
- Herrmann, R. (2014) Fractional Calculus. An introduction for Physicsts, 2-nd Ed. World Scientific, Singapore.
- Atanackovic, T.M., Pilipovic, S., Stankovic, B. and Zorica, D. (2014). Fractional Calculus with Applications in Mechanics. Vol I: Vibrations and Diffusion Processes; Vol. II :Wave Propagation, Impact and Variational Principles, John Wiley & Sons, Hoboken, NJ.
- Daftardar-Gejji, V., Editor (2014). Fractional Calculus. Theory and Applications, Narosa, Delhi.
- Gorenflo, R., Kilbas, A., Mainardi, F. and Rogosin, S. (2014). Mittag-Leffler Functions, Related Topics and Applications, Springer Verlag, Berlin.
Brief Biography of the Speaker: Presently Francesco MAINARDI is retired professor of Mathematical Physics from the University of Bologna where has taught this course since 40 years. His fields of research concern several topics of applied mathematics, including diffusion and wave problems, asymptotic methods, integral transforms, special functions, fractional calculus and non-Gaussian stochastic processes. At present his H-index is > 40
For a full biography, list of references on author's papers and books see: Home Page: http://www.fracalmo.org/mainardi/index.htm and http://scholar.google.com/citations?user=UYxWyEEAAAAJ&hl=en&oi=ao