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Functional fractional calculus for system identification and controls. (English) Zbl 1154.26007
Berlin: Springer (ISBN 978-3-540-72702-6/hbk). xvii, 239 p. (2008).
The book under review, entitled “Functional fractional calculus for system identification and controls”, which is organized into ten chapters, appears to be fairly inspiring for those who are engaged in varied applications of the fractional calculus. The chapters comprising the book are, according to the author’s intention, made as application oriented from various science and engineering fields. Chapter one is introductory, which deals with fundamentals of fractional calculus. Chapter two describes the functions those are used in fractional calculus. Chapters three and four are an overview of the application of fractional calculus such as application of fractional calculus in electrical, thermal, control system, among others. The concept of fractional divergence and curl operator is explained with applications in nuclear reactor and electromagnetism (which is contained in Chapter four). “Insight of fractional integration fractional differentiation and fractional differintegral” (this line is verbatim of the author) is described in terms of physical and geometric meaning.
Reviewer’s remark:
The quoted sentence is not precise, owing to the fact that the fractional differintegrals means fractional differentiation and integration: therefore, either one writes fractional integration and fractional differentiation or it may be chosen to write fractional differintegration.
Chapter six attempts to generalize the concept of initialization function, whereas of the decomposition properties of the fractional differintegration is introduced, where as this chapter also deals with fractional differential equations. Chapter seven deals with Laplace transform theory and the concept of the \(w\)-plane, on which the fractional control system properties are studied is also described. Application of fractional calculus in electrical and electronic circuits are given in Chapter eight and some similar examples of applications of the fractional calculus in other field of science and engineering are given in Chapter nine. Chapter ten deals with fractional order systems, continuous order distribution system, generalized conventional control system, among other relevant applications. The book concludes with a bibliography consisting of 158 references.
Sumptuous work of quality done by eminent researchers is not included in this book, namely A. Erdelyi [Q. J. Math., Oxf. Ser. 11, 293–303 (1940; Zbl 0025.18602 and JFM 66.0522.01)] and H. Kober [Q. J. Math., Oxf. Ser. 11, 193–211 (1940; Zbl 0025.18502 and JFM 66.0520.02)], A. C. McBride [SIAM J. Math. Anal. 6, 583–599 (1975; Zbl 0302.46026), Fractional calculus. (Papers presented at the Workshop held at Ross Priory, University of Strathclyde, England, August 1984). ed. with G. F. Roach, Pitman Advanced Publishing Programme, London, Res. Notes. Math. 138 (1985; Zbl 0595.00008)].
Although plenty of work and contributions were made by C. F. Lorenzo, T. T. Hartely yet they cannot be named as the only authors to have popularized the subject (this is mentioned by the author in Acknowledgement, page x of this book).
The Article/Section 1.5.1, page 9 of the book is Fractional Integration Riemann-Liouville (RL), which sounds mathematically incorrect (i.e., the term commonly spoken by mathematicians). It could have been written as (which is the true form) Riemann-Liouville Fractional Integral Operator. A similar phrase should also be used for Article/Section 1.5.2 and 1.5.3.
On page 10, the block diagram representation of the fractional integration process by convolution is worth to be appreciated (very rarely one comes across such exclusive explanation). On page 14, in Article/Section 1.5.5, the author has very properly quoted the properties of fractional derivatives and integrals.
On page 14, lines 15–19 from above, what is mentioned by the author, is known as Index Formula and these were first described by E. R. Love [J. Aust. Math. Soc. 14, 385–410 (1972; Zbl 0249.26009), Res. Notes. Math. 138, 63–74 (1985; Zbl 0615.26006)].
However, plenty of typographical errors are observed in the bibliography including couple of incomplete references, one such example is, in Reference 36, p.234, the initials of the author Benson are not mentioned: Reference 57, similarly no initials are written for the author Ross, which should be Ross, B. The Reference 153, H. Bateman is not the author of Higher Transcendental Functions, this book was written by A. Erdelyi, W. Magnus, F. Oberhettinger and F. G. Tricomi and known as Bateman Project [Zbl 0064.06302; Zbl 0052.29502; Zbl 0051.30303].

26A33 Fractional derivatives and integrals
00A06 Mathematics for nonmathematicians (engineering, social sciences, etc.)
26-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to real functions