Threefold Introduction to Fractional Derivatives

Contents:

• 2 Threefold Introduction to Fractional Derivatives
  • 2.1 Historical Introduction to Fractional Derivatives
    • 2.1.1 Leibniz
    • 2.1.2 Euler
    • 2.1.3 Paradoxa and Problems
    • 2.1.4 Liouville
    • 2.1.5 Fourier
    • 2.1.6 Grünwald
    • 2.1.7 Riemann
  • 2.2 Mathematical Introduction to Fractional Derivatives
    • 2.2.1 Fractional Integrals
      • 2.2.1.1 Iterated Integrals
      • 2.2.1.2 Riemann-Liouville Fractional Integrals
      • 2.2.1.3 Weyl Fractional Integrals
      • 2.2.1.4 Riesz Fractional Integrals
      • 2.2.1.5 Fractional Integrals of Distributions
      • 2.2.1.6 Integral Transforms
      • 2.2.1.7 Fractional Integration by Parts
      • 2.2.1.8 Hardy-Littlewood Theorem
      • 2.2.1.9 Additivity
    • 2.2.2 Fractional Derivatives
      • 2.2.2.1 Riemann-Liouville Fractional Derivatives
      • 2.2.2.2 General Types of Fractional Derivatives
      • 2.2.2.3 Marchaud-Hadamard Fractional Derivatives
      • 2.2.2.4 Weyl Fractional Derivatives
      • 2.2.2.5 Riesz Fractional Derivatives
      • 2.2.2.6 Grünwald-Letnikov Fractional Derivatives
      • 2.2.2.7 Fractional Derivatives of Distributions
      • 2.2.2.8 Fractional Derivatives at Their Lower Limit
      • 2.2.2.9 Fractional Powers of Operators
      • 2.2.2.10 Pseudodifferential Operators
    • 2.2.3 Eigenfunctions
  • 2.3 Physical Introduction to Fractional Derivatives
    • 2.3.1 Basic Questions
    • 2.3.2 Fractional Space
    • 2.3.3 Fractional Time
      • 2.3.3.1 Basic Questions
      • 2.3.3.2 Time Evolution
      • 2.3.3.3 Continuity
      • 2.3.3.4 Homogeneity
      • 2.3.3.5 Causality
      • 2.3.3.6 Fractional Time Evolution
      • 2.3.3.7 Infinitesimal Generator
      • 2.3.3.8 Remarks
    • 2.3.4 Identification of from Models
      • 2.3.4.1 Bochner-Levy Fractional Diffusion
      • 2.3.4.2 Montroll-Weiss Fractional Diffusion
      • 2.3.4.3 Continuous Time Random Walks
• A Tables
• B Function Spaces
• C Distributions
• References