Fractional Calculus

The following sections are included:

• Contents

• Preface

The following sections are included:

• Fourier and Laplace Transforms

• Special Functions and Their Properties
  • The Gamma function and related special functions
  • Hypergeometric functions
  • Mittag-Leffler functions

• Fractional Operators
  • Riemann-Liouville fractional integrals and fractional derivatives
  • Caputo fractional derivatives
  • Liouville fractional integrals and fractional derivatives. Marchaud derivatives
  • Generalized exponential functions
  • Hadamard type fractional integrals and fractional derivatives
  • Fractional integrals and fractional derivatives of a function with respect to another function
  • Grünwald-Letnikov fractional derivatives

The following sections are included:

• The Approximation of Fractional Differential and Integral Operators
  • Methods based on quadrature theory
  • Grünwald-Letnikov methods
  • Lubich’s fractional linear multistep methods

• Direct Methods for Fractional Ordinary Differential Equations
  • The basic idea
  • Quadrature-based direct methods

• Indirect Methods for Fractional Ordinary Differential Equations
  • The basic idea
  • An Adams-type predictor-corrector method
  • The Cao-Burrage-Abdullah approach

• Linear Multistep Methods

• Other Methods

• Methods for Terminal Value Problems

• Numerical Methods for Multi-Term Fractional Differential Equations and Multi-Order Fractional Differential Systems

• The Extension to Fractional Partial Differential Equations
  • General formulation of the problem
  • Examples

The following sections are included:

• Methods for Ordinary Differential Equations
  • Dealing with non-locality: The finite memory principle, nested meshes, and the approaches of Deng and L_i
  • Parallelization of algorithms
  • When and when not to use fractional linear multistep formulas
  • The use of series expansions
  • The generalized Adams methods as an efficient tool for multi-order fractional differential equations
  • Two classes of singular equations as application examples

• Methods for Partial Differential Equations
  • The method of lines
  • Backward difference formulas for time-fractional parabolic and hyperbolic equations
  • Other methods
  • Methods for equations with space-fractional operators

The following sections are included:

• Introduction

• Stirling Functions of the First Kind s(α, κ)
  • Equivalent definitions
  • Multiple sum representations. The Riemann Zeta function

• General Stirling Functions s(α, β) with Complex Arguments
  • Definition and main result
  • Differentiability of the s(α, β); The zeta function encore
  • Recurrence relations for s(α, β)

• Stirling Functions of the Second Kind S(α, κ)
  • Stirling functions S(α, κ), α ≥ 0, and their representations by Liouville and Marchaud fractional derivatives
  • Stirling functions S(α, κ), α < 0, and their representations by Liouville fractional integrals
  • Stirling functions S(α, κ), α ∈ ℂ, and their representations
  • Stirling functions S(α, κ), α ∈ ℂ, and recurrence relations
  • Further properties and first applications of Stirling functions S(α, κ), α ∈ ℂ
  • Applications of Stirling functions S(α, κ) (α ∈ ℂ) to Hadamard-type fractional operators

• Generalized Stirling Functions of the Second Kind S(n, β; ∈) and S(n, β)
  • Definition and some basic properties
  • Main properties

• The Generalized Stirling Functions S(α, β; ∈) and S(α, β)
  • Basic properties
  • Representations by Liouville fractional operators
  • First application
  • Special examples

• Connections Between the Stirling Functions of First and Second Kind
  • Coincidence relations
  • Results from sampling analysis
  • Generalized orthogonality properties
  • The s(α, κ) connecting two types of fractional derivatives
  • The representation of a general fractional difference operator via s(α, κ)

The following sections are included:

• Fractional Euler-Lagrange Equations
  • Introduction and survey of results
  • Fractional Euler-Lagrange equations for discrete and continuous systems
    • Fractional Euler-Lagrange equations for discrete systems
  • Fractional Lagrangian formulation of field systems
  • Fractional Euler-Lagrange equations with delay
    • Riemann-Liouville fractional Euler-Lagrange equations with delay
    • Caputo fractional Euler-Lagrange equations with delay
  • Fractional discrete Euler-Lagrange equations
  • Fractional Lagrange-Finsler geometry
  • Applications
    • Fractional variational principles with Riesz derivatives
    • Multi-order and multi-term fractional variational formulations with Hilfer derivatives
    • A fractional Lagrangian approach of Schrödinger equations
    • Fractional Lagrangians which differ by a fractional Riesz derivative
    • Euler-Lagrange equations in fractional space
    • Multi time fractional Lagrangian equations
    • Fractional Faddeev-Jackiw formalism
    • Fractional variational calculus with generalized boundary condition

• Fractional Hamiltonian Dynamics
  • Introduction and overview of results
  • Fractional Hamiltonian analysis for discrete and continuous systems
    • A direct method within Riemann-Liouville fractional derivatives
    • A direct method with Caputo fractional derivatives
    • A direct method within Riesz-Caputo fractional derivatives
  • Fractional Hamiltonian formulation for constrained systems
    • Fractional Hessian matrix
    • The reduced phase space
    • Fractional Ostrogradski’s approach
  • Applications
    • Discrete fractional constrained systems
    • Fractional Hamiltonian formulation in fractional time
    • Fractional Nambu mechanics
    • A fractional supersymmetric model
    • Fractional optimal control formulation
    • The fractional optimal control approach with delay
    • Fractional multi time Hamiltonian equations
    • Hamilton-Jacobi formulation with Caputo fractional derivative
    • Fractional dynamics on extended phase space

The following sections are included:

• Introduction

• The Definition of Continuous-Time Random Walks

• Fractional Diffusion and Limit Theorems

The following sections are included:

• Introduction

• Models of Price Fluctuations in Financial Markets

• Simulation

• Option Pricing

• Other Applications

The following sections are included:

• Appendix A Source Codes
  • The Adams-Bashforth-Moulton Method
  • Lubich’s Fractional Backward Differentiation Formulas
  • Time-fractional Diffusion Equations
  • Computation of the Mittag-Leffler Function
  • Monte Carlo simulation of CTRW

• Bibliography

• Index