# Fractional Differential Equations

**An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications**

**By**

- Igor Podlubny, Technical University of Kosice, Slovak Republic

This book is a landmark title in the continuous move from integer to non-integer in mathematics: from integer numbers to real numbers, from factorials to the gamma function, from integer-order models to models of an arbitrary order. For historical reasons, the word 'fractional' is used instead of the word 'arbitrary'.This book is written for readers who are new to the fields of fractional derivatives and fractional-order mathematical models, and feel that they need them for developing more adequate mathematical models.In this book, not only applied scientists, but also pure mathematicians will find fresh motivation for developing new methods and approaches in their fields of research.A reader will find in this book everything necessary for the initial study and immediate application of fractional derivatives fractional differential equations, including several necessary special functions, basic theory of fractional differentiation, uniqueness and existence theorems, analytical numerical methods of solution of fractional differential equations, and many inspiring examples of applications.

View full description### Audience

Researchers in math analysis; engineers; mathematicians; applied scientists wishing to use fractional-order models for modeling and studying processes in a particular field; teachers and students of differential and integral calculus, and mathematical and theoretical physics.

### Book information

- Published: October 1998
- Imprint: ACADEMIC PRESS
- ISBN: 978-0-12-558840-9

### Reviews

This is by no means the first (or the last) book on the subject of fractional calculus, but indeed it is one that would undoubtedly attract the attention (and successfully serve the needs) of mathematical, physical, and engineering scientists looking for applications of fractional calculus. I, therefore, recommend this well-written book to all users of fractional calculus."

--H. M. Srivastava, Zentralblatt MATH

### Table of Contents

Preface. Acknowledgments. **Special Functions Of Preface. Acknowledgements. Special Functions of the Fractional Calculus**. Gamma Function. Mittag-Leffler Function. Wright Function. **Fractional Derivatives and Integrals**. The Name of the Game. Grünwald-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 Theorems**. 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 Method**. Standard Fractional Differential Equations. Sequential Fractional Differential Equations. **Fractional Green's Function**. Definition and Some Properties. One-Term Equation. Two-Term Equation. Three-Term Equation. Four-Term Equation. Calculation of Heat Load Intensity Change in Blast Furnace Walls. Finite-Part Integrals and Fractional Derivatives. General Case: *n*-term Equation. **Other Methods for the Solution of Fractional-order Equations**. The Mellin Transform Method. Power Series Method. Babenko's Symbolic Calculus Method. Method of Orthogonal Polynomials. **Numerical Evaluation of Fractional Derivatives**. Approximation of Fractional Derivatives. The "Short-Memory" Principle. Order of Approximation. Computation of Coefficients. Higher-order Approximations. **Numerical Solution of Fractional Differential Equations**. 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 Controllers**. 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 Derivatives**. **Index**.