# Volume 204. Theory and Applications of Fractional Differential Equations

## 1st Edition

Editors:

- Print ISBN 9780444518323
- Electronic ISBN 9780080462073

Editors:

- Print ISBN 9780444518323
- Electronic ISBN 9780080462073

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 s

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 s

Teachers, researchers and graduate students of mathematics and applied fields (especially Physics, Bio-Sciences, Environmental Sciences, Signal and Optical Theory, Solute Transport through Porous Structures, Mechanical Properties of Polymeric Materials, Economics, Chemical Physics).

1. Preliminaries.
2. Fractional Integrals and Fractional Derivatives.
3. Ordinary Fractional Differential Equations. Existence and Uniqueness Theorems.
4. Methods for Explicitly solving Fractional Differential Equations.
5. Integral Transform Methods for Explicit Solutions to Fractional Differential Equations.
6. Partial Fractional Differential Equations.
7. Sequential Linear Differential Equations of Fractional Order.
8. Further Applications of Fractional Models.
Bibliography
Subject Index

- No. of pages:
- 540

- Language:
- English

- Copyright:
- © 2006

- Published:
- 12th January 2006

- Imprint:
- Elsevier Science

- Print ISBN:
- 9780444518323

- Electronic ISBN:
- 9780080462073

Belarusian State University, Minsk, Belarus

University of Victoria, Victoria, British Columbia, Canada
University of Victoria, BC, Canada

University of La Laguna, Tenerife, Spain

"This book presents a nice and systematic treatment of the theory and applications of fractional differential equations."

-ZENTRALBLATT MATH DATABASE 1931-2007

"This book is a valuable resource for any worker in electronic structure theory, both for its insight into the utility of a variety of relativistic methods, and for its assessment of the contribution of relativity to a wide range of experimental properties,"

-THEOR CHEM ACC (2007)

"For obvious reasons, the book is rather technical, but its main goal is the ultimate applications. These are explicitly or implicitly present during the whole text but they are only treated in their mathematical formulation. I.e., the electrotechnical, biological, optical, or whatever exotic context it could have been embedded in is avoided here. The book is thus certainly interesting for the (applied) mathematician, but, also for researchers who are working in one of the quite diverse applied areas where fractional models are more and more used these days."--Bulletin of the Belgian Mathematical Society