AN INTRODUCTION TO FRACTIONAL CALCULUS

The following sections are included:

• Various Approaches to the Fractional Calculus

  • Some history

  • Basic definitions; Riemann–Liouville and Weyl approaches

  • Three examples

  • Approaches by Hadamard, by contour integration and other methods

• Leibniz Rule and Applications; Semigroups of Operators

  • Fractional Leibniz rule for functions

  • Fractional Landau-Kallman-Rota-Hille inequalities for operators

  • The behaviour of semigroup operators at zero and infinity with rates

• Liouville-Grünwald, Marchaud and Riesz Fractional Derivatives

  • Liouville-Grünwald derivatives and their chief properties

  • A crucial proposition; basic theorems

  • The point-wise Liouville-Grünwald fractional derivative

  • Extensions and applications of the Liouville-Grünwald calculus

  • The Marchaud fractional derivative

  • Equivalence of the Weyl and Marchaud fractional derivatives

  • Riesz derivatives on ℝ

• Various Applications

  • Integral representations of special functions

  • Stirling functions of the first kind

  • Stirling functions of the second kind

  • Euler functions

  • Eulerian numbers E(α, k) for α ∈ ℝ

  • The Bernoulli functions B_α(x) for α ∈ ℝ

  • Ordinary and partial differential equations and other applications

• Integral Transforms and Fractional Calculus

  • Fourier transforms

  • Mellin transforms

  • Laplace transforms and characterizations of fractional derivatives

• References