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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. (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 \(\delta\), which is a generalization of the ordinary \(n\)th 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: \[ R^\delta f(x)= D^{-\delta}f(x)= {1\over\Gamma(\delta)} \int^x_0 (x-t)^{\delta- 1}f(t)dt;\quad\text{Re}(\delta)>0. \] The so-called Weyl fractional integral is defined as: \[ W^\delta f(x)= {1\over\Gamma(\delta)} \int^\infty_x (t-x)^{\delta- 1}f(t)dt,\quad\text{Re}(\delta)>0. \] 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 \(\delta\) \((\delta_1\geq 0,\dots,\delta_m\geq 0)\), \(m\geq 1\), are introduced by means of kernels being \(G^{m,0}_{m,m}\)- and \(H^{m,0}_{m,m}\)-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 \({_pF_q}\) 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 \(H^{m,0}_{m,m}\)-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.
Reviewer: S.L.Kalla (Kuwait)

MSC:
26A33 Fractional derivatives and integrals
26-02 Research exposition (monographs, survey articles) pertaining to real functions
33-02 Research exposition (monographs, survey articles) pertaining to special functions
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
44A10 Laplace transform
30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
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