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Generalized fractional calculus and applications. (English) Zbl 0882.26003
Pitman Research Notes in Mathematics Series. 301. Harlow: Longman Scientific & Technical. New York: John Wiley & Sons. x, 388 p. £39.00 (1994).

Fractional calculus deals with the theory of operators of integration and differentiation of arbitrary order and their applications [K. Nishimoto: “Fractional calculus”, Vol. I (1984; Zbl 0605.26006), Vol. II (1987; Zbl 0702.26011), Vol. III (1989; Zbl 0798.26005), and Vol. IV (1991; Zbl 0798.26006); S. G. Samko, A. A. Kilbas and O. I. Marichev: “Integrals and derivatives of fractional order and some of their applications” (Russian: 1987; Zbl 0617.26004; English translation: 1993; Zbl 0818.26003)]. The concept of differintegral of complex order δ, which is a generalization of the ordinary nth derivative and n-times integral, can be introduced in several ways. One of the simple definition of an integral of an arbitrary order is based on an integral transform, called the Riemann-Liouville operator of fractional integration:

R δ f(x)=D -δ f(x)=1 Γ(δ) 0 x (x-t) δ-1 f(t)dt;Re(δ)>0·

The so-called Weyl fractional integral is defined as:

W δ f(x)=1 Γ(δ) x (t-x) δ-1 f(t)dt,Re(δ)>0·

There are several modifications and generalizations of these operators, but the most widely used in applications are the Erdélyi-Kober operators.

This book is devoted to a systematic and unified development of a new generalized fractional calculus. Generalized operators of integration and differentiation of arbitrary multiorder δ (δ 1 0,,δ m 0), m1, are introduced by means of kernels being G m,m m,0 - and H m,m m,0 -functions. Due to this special choice of Meijer’s G-function (and Fox’s H-function) in the single integral representations of the operators considered here, a decomposition into commuting Erdélyi-Kober fractional operators holds under suitable conditions. The author has developed a full chain of operational rules, mapping properties and convolutional structure of the generalized (m-tuple) fractional integrals and the corresponding derivatives.

Historical background and the theme of the book is contained in the Introduction. Chapters 1 and 2 treat the basic concepts and properties of the Erdélyi-Kober fractional integrals. Chapter 3 is devoted to the class of so-called hyper-Bessel integral and differential properties, Poisson-Sonine-Dimovski transmutations and Obrechkoff transform. Some new integral and differintegral formulas for the generalized hypergeometric functions p F q are considered in Chapter 4. Some other applications of the generalized fractional calculus: Abel’s integral equation, theory of univalent functions and generalized Laplace type transforms are treated in the Chapter 5. Fractional integration operators involving Fox’s H m,m m,0 -function are studied here in different functional spaces. To make the book self-contained, the author has given an Appendix dealing with definition and main properties of the Meijer’s G-function, Fox’s H-function, Hyper Bessel, D- and n-Bessel functions, etc. The references include 519 titles and a Citation Index is provided, showing the articles referred to in the Sections.

This book is an exposition of a self-contained new theory of generalized operators of differintegrals. This monograph is very useful for graduate students, lecturers and researchers in Applied Mathematical Analysis and related Mathematical Sciences. This book is a good addition to the existing literature on the subject, and it will stimulate more research in this new exciting field of fractional calculus.


MSC:
26A33Fractional derivatives and integrals (real functions)
26-02Research monographs (real functions)
33-02Research monographs (special functions)
34B30Special ODE (Mathieu, Hill, Bessel, etc.)
44A10Laplace transform
30C45Special classes of univalent and multivalent functions
45E10Integral equations of the convolution type