Applications of fractional calculus in physics. (English) Zbl 0998.26002

Singapore: World Scientific. vii, 463 p. (2000).

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Fractional calculus deals with investigation of integrals and derivatives of non-integer order. This subject has gained importance and popularity during the past three decades or so due to its applications in various branches of sciences and engineering.
The main features of this monograph are as follows. This monograph provides a systematic treatment of the theory and applications of fractional calculus for physicists. It contains nine review articles surveying those areas in which fractional calculus has become important. All the chapters are self-contained.
Chapter I (pp. 1-85 by P. L. Butzer and U. Westphal) deals with an introductory note on the origin of fractional calculus and provides various definitions given from time to time by various authors. Applications to special functions such as Euler, Bernoulli, and Stirling functions etc. are demonstrated.
Chapter II (pp. 87-130 by R. Hilfer) is devoted to fractional evolution equations and their emergence from coarse graining. Explicit solutions of generalized fractional relaxation equation and generalized fractional diffusion equation are developed in terms of the Mittag-Leffler function, which is an extension of the exponential function.
Chapter III (pp. 131-170 by U. Westphal) covers the theory of fractional powers of infinitesimal generators of semigroups from the functional analytic point of view.
A description of long-time memory, fractional differences and derivatives, stochastic difference equations and fractional diffusion equations is given in Chapter IV (pp. 171-202 by B. J. West and P. Grigolini).
A brief review of the problems of fractional kinetics of Hamiltonian chaotic systems is offered in Chapter V (pp. 203-240 by G. M. Zaslavsky).
Chapter VI (pp. 241-330 by J. F. Douglas) gives a path-integral description of the interacting flexible polymer models which are converted into equivalent integral equation representation expressible in terms of fractional differential equations. These equations are solved by employing the semigroup properties of the fractional order operators and eigenfunction expansion techniques.
Rouse and rheological constitutive models form the basis of Chapter VII (pp. 331-376 by H. Schiessel, Chr. Friedrich and A. Blumen).
Chapter VIII (pp. 377-428 by T. F. Nonnenmacher and R. Metzler) deals with applications of fractional calculus techniques to problems in biophysics.
In the concluding Chapter IX (429 ff. by R. Hilfer), fractional calculus is employed in generalizing the Ehrenfest classification of phase transitions in equilibrium thermodynamics.
It is a very well written monograph and the presentation is commendable. In conclusion, it is a useful monograph for research workers in the areas of fractional calculus, integral and differential equations, physics, chemistry, biology, statistics and engineering.
Reviewer’s remark: The solution of fractional order differential and integral equations discussed in this monograph involves the Mittag-Leffler function, which is a special csse of the \(H\)-function of Fox. A detailed account of the \(H\)-function is available from the monograph written by A. M. Mathai and R. K. Saxena [“The \(H\)-function with applications in statistics and other disciplines” (1978; Zbl 0382.33001)]. A systematic (and historical) account of the investigations carried out by various authors in the field of fractional calculus and its applications up to 1995 is given by H. M. Scrivastava and R. K. Saxena [Appl. Math. Comput. 118, No. 1, 1-52 (2001; Zbl 1022.26012)].

Reviewer: Ram Kishore Saxena (Jodhpur)

MSC:

26-02	Research exposition (monographs, survey articles) pertaining to real functions
26A33	Fractional derivatives and integrals
00B15	Collections of articles of miscellaneous specific interest
26-06	Proceedings, conferences, collections, etc. pertaining to real functions
81-02	Research exposition (monographs, survey articles) pertaining to quantum theory

Keywords:

fractional calculus in physics; Riemann-Liouville operator; Laplace transform; Fourier transform; rheology; biophysics; fractals; fractional calculus; fractional relaxation equation; fractional diffusion equation; Mittag-Leffler function; phase transitions; equilibrium thermodynamics; fractional order differential and integral equations; \(H\)-function

Citations:

Zbl 0382.33001; Zbl 1022.26012