Books in Fractional Calculus

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The Fractional Calculus; Theory and Applications of Differentiation and Integration to Arbitrary Order (Mathematics in Science and Engineering, V)
By Keith B. Oldham, Jerome Spanier



  • Publisher: Academic Pr
  • Number Of Pages: 248
  • Publication Date: 1974-11
  • ISBN-10 / ASIN: 0125255500
  • ISBN-13 / EAN: 9780125255509
  • Binding: Hardcover


Product Description:
Not only does this text explain the theory underlying the properties of the generalized operator, but it also illustrates the wide variety of fields to which these ideas may be applied. Topics include integer order, simple and complex functions, semiderivatives and semiintegrals, and transcendental functions. 1974 edition.

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Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering
By J. Sabatier, O. P. Agrawal, J. A. Tenreiro Machado



  • Publisher: Springer
  • Number Of Pages: 552
  • Publication Date: 2007-08-24
  • ISBN-10 / ASIN: 1402060416
  • ISBN-13 / EAN: 9781402060410
  • Binding: Hardcover


Product Description:
In the last two decades, fractional (or non integer) differentiation has played a very important role in various fields such as mechanics, electricity, chemistry, biology, economics, control theory and signal and image processing. For example, in the last three fields, some important considerations such as modelling, curve fitting, filtering, pattern recognition, edge detection, identification, stability, controllability, observability and robustness are now linked to long-range dependence phenomena. Similar progress has been made in other fields listed here. The scope of the book is thus to present the state of the art in the study of fractional systems and the application of fractional differentiation.

As this volume covers recent applications of fractional calculus, it will be of interest to engineers, scientists, and applied mathematicians.​


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Functional Fractional Calculus for System Identification and Controls
By Shantanu Das



  • Publisher: Springer
  • Number Of Pages: 240
  • Publication Date: 2007-11-20
  • ISBN-10 / ASIN: 3540727027
  • ISBN-13 / EAN: 9783540727026
  • Binding: Hardcover


Product Description:
When a new extraordinary and outstanding theory is stated, it has to face criticism and skepticism, because it is beyond the usual concept. The fractional calculus though not new, was not discussed or developed for a long time, particularly for lack of its applications to real life problems. It is extraordinary because it does not deal with ‘ordinary’ differential calculus. It is outstanding because it can now be applied to situations where existing theories fail to give satisfactory results. In this book not only mathematical abstractions are discussed in a lucid manner, but also several practical applications are given particularly for system identification, description and then efficient controls.



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Applications of Fractional Calculus in Physics
By Rudolf Hilfer



  • Publisher: World Scientific Publishing Company
  • Number Of Pages: 463
  • Publication Date: 2000-06
  • ISBN-10 / ASIN: 9810234570
  • ISBN-13 / EAN: 9789810234577
  • Binding: Hardcover


Product Description:
Fractional calculus is a collection of relatively little-known mathematical results concerning generalizations of differentiation and integration to noninteger orders. While these results have been accumulated over centuries in various branches of the mathematics, until recently they have found little appreciation or application in physics and other mathematically oriented sciences. This situation is beginning to change, and there are now a growing number of research areas in physics which employ fractional calculus. This volume provides an introduction to fractional calculus for physicists, and collects easily accessible review articles surveying those areas of physics in which applications of fractional calculus have recently become prominent.​



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Fractional Calculus and Its Applications: Proceedings of the International Conference held at the University of New Haven, June 1974 (Lecture Notes in Mathematics)
By B. Ross



  • Publisher: Springer
  • Number Of Pages: 381
  • Publication Date: 1975-06-17
  • ISBN-10 / ASIN: 354007161X
  • ISBN-13 / EAN: 9783540071617
  • Binding: Paperback

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An Introduction to the Fractional Calculus and Fractional Differential Equations
By Kenneth S. Miller, Bertram Ross



  • Publisher: John Wiley & Sons
  • Number Of Pages: 384
  • Publication Date: 1993-05-19
  • ISBN-10 / ASIN: 0471588849
  • ISBN-13 / EAN: 9780471588849
  • Binding: Hardcover


Product Description:
This study aims to make the solution of certain integral equations simpler to visualize. It presents material in a methodical manner and includes a bibliography covering references on the development of the subject from 1695 to modern times.​



Summary: Fractional calculus explained
Rating: 5
This is a complex topic that the authors do an excellent job in explaining. It is not a book for the layman, but then none of the books on this topic I have read are. They cover the spectrum on what will be done to apply fractional calculus. I really enjoyed the sections for differential equations and the laplace transform and inverse transform.​


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Theory and Applications of Fractional Differential Equations, Volume 204 (North-Holland Mathematics Studies)
By A.A. Kilbas, H.M. Srivastava, J.J. Trujillo



  • Publisher: Elsevier Science
  • Number Of Pages: 540
  • Publication Date: 2006-02-16
  • ISBN-10 / ASIN: 0444518320
  • ISBN-13 / EAN: 9780444518323
  • Binding: Hardcover


Product Description:
This monograph provides the most recent and up-to-date developments on fractional differential and fractional integro-differential equations involving many different potentially useful operators of fractional calculus.

The subject of fractional calculus and its applications (that is, calculus of integrals and derivatives of any arbitrary real or complex order) has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.

Some of the areas of present-day applications of fractional models include Fluid Flow, Solute Transport or Dynamical Processes in Self-Similar and Porous Structures, Diffusive Transport akin to Diffusion, Material Viscoelastic Theory, Electromagnetic Theory, Dynamics of Earthquakes, Control Theory of Dynamical Systems, Optics and Signal Processing, Bio-Sciences, Economics, Geology, Astrophysics, Probability and Statistics, Chemical Physics, and so on.

In the above-mentioned areas, there are phenomena with estrange kinetics which have a microscopic complex behaviour, and their macroscopic dynamics can not be characterized by classical derivative models.

The fractional modelling is an emergent tool which use fractional differential equations including derivatives of fractional order, that is, we can speak about a derivative of order 1/3, or square root of 2, and so on. Some of such fractional models can have solutions which are non-differentiable but continuous functions, such as Weierstrass type functions. Such kinds of properties are, obviously, impossible for the ordinary models.

What are the useful properties of these fractional operators which help in the modelling of so many anomalous processes? From the point of view of the authors and from known experimental results, most of the processes associated with complex systems have non-local dynamics involving long-memory in time, and the fractional integral and fractional derivative operators do have some of those characteristics.

This book is written primarily for the graduate students and researchers in many different disciplines in the mathematical, physical, engineering and so many others sciences, who are interested not only in learning about the various mathematical tools and techniques used in the theory and widespread applications of fractional differential equations, but also in further investigations which emerge naturally from (or which are motivated substantially by) the physical situations modelled mathematically in the book.

This monograph consists of a total of eight chapters and a very extensive bibliography. The main objective of it is to complement the contents of the other books dedicated to the study and the applications of fractional differential equations. The aim of the book is to present, in a systematic manner, results including the existence and uniqueness of solutions for the Cauchy type problems involving nonlinear ordinary fractional differential equations, explicit solutions of linear differential equations and of the corresponding initial-value problems through different methods, closed-form solutions of ordinary and partial differential equations, and a theory of the so-called sequential linear fractional differential equations including a generalization of the classical Frobenius method, and also to include an interesting set of applications of the developed theory.

Key features:

- It is mainly application oriented.
- It contains a complete theory of Fractional Differential Equations.
- It can be used as a postgraduate-level textbook in many different disciplines within science and engineering.
- It contains an up-to-date bibliography.
- It provides problems and directions for further investigations.
- Fractional Modelling is an emergent tool with demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering.
- It contains many examples.
- and so on!​

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Fractional Differential Equations (Mathematics in Science and Engineering) (Mathematics in Science and Engineering)
By Igor Podlubny



  • Publisher: Academic Press
  • Number Of Pages: 340
  • Publication Date: 1999-01-15
  • ISBN-10 / ASIN: 0125588402
  • ISBN-13 / EAN: 9780125588409
  • Binding: Hardcover


Product Description:
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.

Key Features
* A unique survey of many applications of fractional calculus
* Presents basic theory
* Includes a unified presentation of selected classical results, which are important for applications
* Provides many examples
* Contains a separate chapter of fractional order control systems, which opens new perspectives in control theory
* The first systematic consideration of Caputo's fractional derivative in comparison with other selected approaches
* Includes tables of fractional derivatives, which can be used for evaluation of all considered types of fractional derivatives​



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Stochastic Calculus for Fractional Brownian Motion and Applications (Probability and its Applications)
By Francesca Biagini, Yaozhong Hu, Bernt Øksendal, Tusheng Zhang



  • Publisher: Springer
  • Number Of Pages: 332
  • Publication Date: 2008-02-28
  • ISBN-10 / ASIN: 1852339969
  • ISBN-13 / EAN: 9781852339968
  • Binding: Hardcover


Product Description:
Fractional Brownian motion (fBm) has been widely used to model a number of phenomena in diverse fields from biology to finance. This huge range of potential applications makes fBm an interesting object of study.

fBm represents a natural one-parameter extension of classical Brownian motion therefore it is natural to ask if a stochastic calculus for fBm can be developed. This is not obvious, since fBm is neither a semimartingale (except when H = ½), nor a Markov process so the classical mathematical machineries for stochastic calculus are not available in the fBm case.

Several approaches have been used to develop the concept of stochastic calculus for fBm. The purpose of this book is to present a comprehensive account of the different definitions of stochastic integration for fBm, and to give applications of the resulting theory. Particular emphasis is placed on studying the relations between the different approaches.

Readers are assumed to be familiar with probability theory and stochastic analysis, although the mathematical techniques used in the book are thoroughly exposed and some of the necessary prerequisites, such as classical white noise theory and fractional calculus, are recalled in the appendices.

This book will be a valuable reference for graduate students and researchers in mathematics, biology, meteorology, physics, engineering and finance. Aspects of the book will also be useful in other fields where fBm can be used as a model for applications.​


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Special Functions for Applied Scientists
By A.M. Mathai, H.J. Haubold



  • Publisher: Springer
  • Number Of Pages: 470
  • Publication Date: 2008-03-19
  • ISBN-10 / ASIN: 0387758933
  • ISBN-13 / EAN: 9780387758930
  • Binding: Hardcover


Product Description:
Chapter 1 introduces elementary classical special functions. Gamma, beta, psi, zeta functions, hypergeometric functions and the associated special functions, generalizations to Meijer's G and Fox's H-functions are examined here. Discussion is confined to basic properties and selected applications. Introduction to statistical distribution theory is provided. Some recent extensions of Dirichlet integrals and Dirichlet densities are discussed. A glimpse into multivariable special functions such as Appell's functions and Lauricella functions is part of Chapter 1. Special functions as solutions of differential equations are examined. Chapter 2 is devoted to fractional calculus. Fractional integrals and fractional derivatives are discussed. Their applications to reaction-diffusion problems in physics, input-output analysis, and Mittag-Leffler stochastic processes are developed. Chapter 3 deals with q-hyper-geometric or basic hypergeometric functions. Chapter 4 covers basic hypergeometric functions and Ramanujan's work on elliptic and theta functions. Chapter 5 examines the topic of special functions and Lie groups. Chapters 6 to 9 are devoted to applications of special functions. Applications to stochastic processes, geometric infinite divisibility of random variables, Mittag-Leffler processes, alpha-Laplace processes, density estimation, order statistics and astrophysics problems, are dealt with in Chapters 6 to 9. Chapter 10 is devoted to wavelet analysis. An introduction to wavelet analysis is given. Chapter 11 deals with the Jacobians of matrix transformations. Various types of matrix transformations and the associated Jacobians are provided. Chapter 12 is devoted to the discussion of functions of matrix argument in the real case. Functions of matrix argument and the pathway models along with their applications are discussed.


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Integral Transforms and Their Applications, Second Edition
By Lokenath Debnath, Dambaru Bhatta



  • Publisher: Chapman & Hall/CRC
  • Number Of Pages: 728
  • Publication Date: 2006-10-11
  • ISBN-10 / ASIN: 1584885750
  • ISBN-13 / EAN: 9781584885757
  • Binding: Hardcover


Product Description:
Keeping the style, content, and focus that made the first edition a bestseller, Integral Transforms and their Applications, Second Edition stresses the development of analytical skills rather than the importance of more abstract formulation. The authors provide a working knowledge of the analytical methods required in pure and applied mathematics, physics, and engineering. The second edition includes many new applications, exercises, comments, and observations with some sections entirely rewritten. It contains more than 500 worked examples and exercises with answers as well as hints to selected exercises. The most significant changes in the second edition include: · New chapters on fractional calculus and its applications to ordinary and partial differential equations, wavelets and wavelet transformations, and Radon transform · Revised chapter on Fourier transforms, including new sections on Fourier transforms of generalized functions, Poissons summation formula, Gibbs phenomenon, and Heisenbergs uncertainty principle · A wide variety of applications has been selected from areas of ordinary and partial differential equations, integral equations, fluid mechanics and elasticity, mathematical statistics, fractional ordinary and partial differential equations, and special functions · A broad spectrum of exercises at the end of each chapter further develops analytical skills in the theory and applications of transform methods and a deeper insight into the subject A systematic mathematical treatment of the theory and method of integral transforms, the book provides a clear understanding of the subject and its varied applications in mathematics, applied mathematics, physical sciences, and engineering.​


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Mathematical Methods in Science and Engineering
By Selcuk Bayin



  • Publisher: Wiley-Interscience
  • Number Of Pages: 712
  • Publication Date: 2006-07-18
  • ISBN-10 / ASIN: 0470041420
  • ISBN-13 / EAN: 9780470041420
  • Binding: Hardcover


Product Description:
An innovative treatment of mathematical methods for a multidisciplinary audience

Clearly and elegantly presented, Mathematical Methods in Science and Engineering provides a coherent treatment of mathematical methods, bringing advanced mathematical tools to a multidisciplinary audience. The growing interest in interdisciplinary studies has brought scientists from many disciplines such as physics, mathematics, chemistry, biology, economics, and finance together, which has increased the demand for courses in upper-level mathematical techniques. This book succeeds in not only being tuned in to the existing practical needs of this multidisciplinary audience, but also plays a role in the development of new interdisciplinary science by introducing new techniques to students and researchers.

Mathematical Methods in Science and Engineering's modular structure affords instructors enough flexibility to use this book for several different advanced undergraduate and graduate level courses. Each chapter serves as a review of its subject and can be read independently, thus it also serves as a valuable reference and refresher for scientists and beginning researchers.

There are a growing number of research areas in applied sciences, such as earthquakes, rupture, financial markets, and crashes, that employ the techniques of fractional calculus and path integrals. The book's two unique chapters on these subjects, written in a style that makes these advanced techniques accessible to a multidisciplinary audience, are an indispensable tool for researchers and instructors who want to add something new to their compulsory courses.

Mathematical Methods in Science and Engineering includes:
* Comprehensive chapters on coordinates and tensors and on continuous groups and their representations
* An emphasis on physical motivation and the multidisciplinary nature of the methods discussed
* A coherent treatment of carefully selected topics in a style that makes advanced mathematical tools accessible to a multidisciplinary audience
* Exercises at the end of every chapter and plentiful examples throughout the book

Mathematical Methods in Science and Engineering is not only appropriate as a text for advanced undergraduate and graduate physics programs, but is also appropriate for engineering science and mechanical engineering departments due to its unique chapter coverage and easily accessible style. Readers are expected to be familiar with topics typically covered in the first three years of science and engineering undergraduate programs. Thoroughly class-tested, this book has been used in classes by more than 1,000 students over the past eighteen years.​


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Analogue Realization of Fractional Order Controllers


by Petras I., Podlubny I., O'Leary P., Dorcak L., Vinagre B.:
FBERG, Technical University of Kosice,
Kosice, 2002,
84 pages,
ISBN 8070996277.

Table of Contents


  • Preface
  • 1 Introduction
  • 2 Fundamentals of Fractional Calculus
    • 2.1 Introduction to fractional calculus
    • 2.2 Riemann-Liouville fractional integrals and derivatives
      • 2.2.1 Definitions
      • 2.2.2 Properties
    • 2.3 Laplace transform method
      • 2.3.1 Basic facts on the Laplace transform
      • 2.3.2 Laplace transform of fractional integrals
      • 2.3.3 Laplace transform of fractional derivatives
    • 2.4 Fourier transform method
      • 2.4.1 Basic facts on the Fourier transform
      • 2.4.2 Fourier transform of fractional integrals
      • 2.4.3 Fourier transform of fractional derivatives
    • 2.5 Fractional differential equations and transfer functions
  • 3 Fractional-Order Controllers
    • 3.1 Definition of fractional PIλDμ controller
    • 3.2 Properties and characteristics of controller
      • 3.2.1 General considerations
      • 3.2.2 Bode ideal loop transfer function
      • 3.2.3 Illustrative example
    • 3.3 Design of controller parameters
  • 4 Continued Fraction Expansion (CFE)
    • 4.1 CFE: General introduction
    • 4.2 CFE and approximations of functions
    • 4.3 CFE and stability of linear systems
    • 4.4 CFE and nested multiple-loop control systems
    • 4.5 CFE and rational and irrational numbers
  • 5 Rational Approximation of Fractional Operators
    • 5.1 Preliminary considerations
    • 5.2 General CFE method
    • 5.3 Carlson's method
    • 5.4 Matsuda's method
    • 5.5 Oustaloup's method
    • 5.6 Charef's method
    • 5.7 Other approaches
      • 5.7.1 Roy's method
      • 5.7.2 Wan's method
      • 5.7.3 Jones' method
  • 6 General Approach to Fractance Devices
    • 6.1 Introduction to fractance devices
      • 6.1.1 What is fractance?
      • 6.1.2 Negative impedance converters
    • 6.2 Domino ladder circuit
    • 6.3 Transmission lines circuit
    • 6.4 Tree structure circuit
  • 7 Realization of Fractional-Order Controllers
    • 7.1 Introduction to analogue realization
    • 7.2 Realization of fractional Iλ controller
      • 7.2.1 Description of circuit
      • 7.2.2 Experimental results
    • 7.3 Realization of fractional PIλ controller
      • 7.3.1 Description of circuit
      • 7.3.2 Experimental results
  • 8 Conclusion
  • Bibliography
  • Index
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بارك الله فيك يا دكتور عمر و مشكور كتير على هذه المجموعة من الكتب
و بصراحة عندي بس استفسار
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