By
A.A. Kilbas, Belarusian State University, Minsk, Belarus
H.M. Srivastava, University of Victoria, BC, Canada
J.J. Trujillo, University of La Laguna, Tenerife, Spain
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!
Included in series
North-Holland Mathematics Studies
Audience:
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).
