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Preface

Acknowledgments

Special Functions of Preface

Acknowledgements

Special Functions of the Fractional Calculuslt

Gamma Function

Mittag-Leffler Function

Wright Function

Fractional Derivatives and Integralslt

The Name of the Game

Gruuml;nwald-Letnikov Fractional Derivatives

Riemann-Liouville Fractional Derivatives

Some Other Approaches

Sequential Fractional Derivatives

Left and Right Fractional Derivatives

Properties of Fractional Derivatives

Laplace Transforms of Fractional Derivatives

Fourier Transforms of Fractional Derivatives

Mellin Transforms of Fractional Derivatives

Existence and Uniqueness Theoremslt

Linear Fractional Differential Equations

Fractional Differential Equation of a General Form

Existence and Uniqueness Theorem as a Method of Solution

Dependence of a Solution on Initial Conditions

The Laplace Transform Methodlt

Standard Fractional Differential Equations

Sequential Fractional Differential Equations

Fractional Green's Functionlt

Definition and Some Properties

Term Equation / One：

Term Equation. / Two：

Calculation of Heat Load Intensity Change in Blast Furnace Walls / Four：

Finite-Part Integrals and Fractional Derivatives

General Case: nlt;/i>-term Equation

Other Methods for the Solution of Fractional-order Equationslt

The Mellin Transform Method

Power Series Method

Babenko's Symbolic Calculus Method

Method of Orthogonal Polynomials

Numerical Evaluation of Fractional Derivativeslt

Approximation of Fractional Derivatives

The "Short-Memory" Principle

Order of Approximation

Computation of Coefficients

Higher-order Approximations

Numerical Solution of Fractional Differential Equationslt

Initial Conditions: Which Problem to Solve? Numerical Solution

Examples of Numerical Solutions

The "Short-Memory" Principle in Initial Value Problems for Fractional Differential Equations

Fractional-Order Systems and Controllers

Fractional-Order Systems and Fractional-Order Controllerslt

Example

on Viscoelasticity

Bode's Analysis of Feedback Amplifiers

Fractional Capacitor Theory

Electrical Circuits

Electroanalytical Chemistry

Electrode-Electrolyte Interface

Fractional Multipoles

Biology

Fractional Diffusion Equations

Control Theory

Fitting of Experimental Data

The "Fractional-Order" Physics? Bibliography

Tables of Fractional Derivativeslt

Index

Preface

Acknowledgments

Special Functions of Preface

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