Essentials of Fractional Calculus

Essentials of fractional calculus are presented. Different kinds of integral and differential operators of fractional order are discussed. The notion of the Riemann–Liouville fractional integralLiouville fractional integral is introduced as a natural generalization of the repeated integral written in a convolution-type form. The Riemann–Liouville fractional derivative is defined as left-inverse to the Riemann–Liouville fractional integral. The Caputo fractional derivative and the Riesz fractional operators (including the fractional Laplace operator) are considered. The cumbersome aspects of space-fractional differential operators disappear when one computes their Fourier integral transforms. In solutions of various types of equations with fractional operators, there appear the Mittag-Leffler functionsMittag-Leffler function and the Wright functionWright function. The main properties of the Mittag-Leffler functionsMittag-Leffler function and the Wright function are presented.