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Fractional Equations and Models

Trifce Sandev and Živorad Tomovski
Publisher: 
Springer
Publication Date: 
2019
Number of Pages: 
345
Format: 
Hardcover
Series: 
Developments in Mathematics
Price: 
139.99
ISBN: 
978-3-030-29613-1
Category: 
Monograph
[Reviewed by
Bill Satzer
, on
03/1/2020
]
 
This book is a very specialized text that considers fractional calculus and fractional differential equations used in the study of fractional stochastic and kinetic models. The fractional calculus offers ways to define derivatives and integrals of non-integer order. Although the subject has become more prominent in the literature in recent years, it has long historical roots. The first appearance of a fractional derivative was a question about a derivative of order one-half in a letter from l’Hôpital to Leibnitz in 1695. Abel used an elegant early version of the fractional calculus to solve the tautochrone problem of finding the planar curve along which the time of descent of a bead under gravity is independent of the starting point along the curve. Later Oliver Heaviside developed his own unorthodox version of the fractional calculus for solving differential and integral equations relating to electrical transmission.
 
There are now a great many different non-equivalent definitions of fractional derivatives and integrals, and many of them are probably best thought of as operators on function spaces. As far as I can tell, no unified treatment exists. The current book uses several different versions of the fractional derivative and integral. The version that is used depends on the character of the special application that the authors consider.
 
Two very simple examples might help make the concept more concrete for readers who have not encountered it before. One definition of a fractional integral generalizes the Cauchy for repeated integration
 
\( (J^{n} f) (x) = \frac{1}{(n-1)!} \int_{0}^{x} (x-t)^{n-1} f(t) dt \)
 
to the \( a \)th order fractional integral
 
\( (J^{a} f) (x) = \frac{1}{ \Gamma (a)} \int_{0}^{x} (x-t)^{a-1} f(t) dt \)
 
 
where \( \Gamma \) is the gamma function and \( a \) is a real number.   This is a version of what is sometimes called the
Riemann-Liouville fractional integral.
 
One definition of a fractional derivative (for functions f having a Fourier transform) says that if \( g(a) \) is the Fourier transform of \( f (x) \), then the derivative of f of real order \( v \) is the inverse Fourier transform of \( (−ia)^{v} g(a) \). This is sometimes called the Riesz fractional derivative and generalizes the usual relationship between derivatives of integral order and Fourier transforms.  However, these relatively simple examples of the fractional calculus can be very misleading, Many of the usual properties of derivatives and integrals
do not hold in general for fractional derivatives and integrals. The Riemann-Liouville fractional derivative, for example, is non-local because its value is obtained by integrating over a whole range of values. This nonlocality is one of the principal drivers of interest in applications of fractional calculus.  Many interesting physical phenomena have memory effects so that their state does not depend solely on the current time and position but also on previous states. Systems with memory are very difficult to model with conventional differential and integral equations. Fractional calculus offers useful tools for these applications.
 
One subject that gets special attention in the book is the derivation of fractional diffusion and Fokker-Planck equations within continuous time random walk theory as well as their solutions and applications. The authors demonstrate that all the known fractional diffusion equations are specialized cases of one set of generalized diffusion equations with kernels that embody memory effects. The book also includes an analysis of stochastic processes governed by the generalized Langevin equations. A large variety of diffusion behaviors are described by these equations.
 
The current book is essentially a monograph that explores applications using several variations of the fractional calculus. It does not offer easy entry for those new to the subject. A more accessible text is An Introduction to the Fractional Calculus and Fractional Differential Equations by Miller and Ross.

 

Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical systems and celestial mechanics.