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Kieth Oldham, Jerome Spanier, "The Fractional Calculus : Theory and Applications of Differentiation and Integration to Arbitrary Order"

Not only does this text explain the theory underlying the properties of the generalized operator, but it also illustrates the wide variety of fields to which these ideas may be applied. Topics include integer order, simple and complex functions, semiderivatives and semiintegrals, and transcendental functions. 1974 edition.

Review

This was the first textbook devoted entirely to Fractional Calculus, and remains perhaps the best introduction to the field. It is a graduate level text for mathematicians, engineers and scientists, written in a clear pedagogical style, with excellent references and bibliography.

A brief historical survey traces contributions as far back as Leibnitz. Basic results of the traditional integer-order differential and integral calculus are summarized, and properties of some special functions are examined, as a foundation for what follows. Later in the book, many advanced special functions are described in terms of the hypergeometric function.

The notions of differentiation and integration are unified as differintegration. Grünwald's definition is presented which extends differintegration to non-integer orders, applicable to functions which satisfy certain conditions. Other definitions used historically are shown to be equivalent to, or special cases of the Grünwald definition. General properties of differintegration are examined, such as linearity and homogeneity. Results corresponding to those of traditional calculus are derived, such as Leibnitz' rule. Significant differences from traditional calculus are also shown, such as the lack of a useful chain rule.

The preceding results are illustrated by example, using elementary algebraic, trigonometric, and logarithmic functions, and some special functions, such as hypergeometric, Bessel, and Heaviside. The semiderivative and semiintegral are accorded additional scrutiny, due to their importance in applications. Tables of formulae for semiderivative and semiintegral are provided for elementary functions and common special functions, as well as formulae valid for arbitrary functions. The LaPlace transform or tables of Riemann-Liouville transforms may be used to extend these tables.

Practical techniques are presented both for numerical differintegration, suitable for computer implementation, and analog differentegration by means of electrical circuits. Some extraordinary fractional differential equations and semidifferential equations are solved analytically. Techniques of the fractional calculus are applied to a number of well-known problems in traditional calculus, including Abel's equation and Bessel's equation.

The diffusion equation and its relatives are treated in a short chapter on practical applications of the fractional calculus. The usual formulations for planar and spherical geometries are examined, including incorporation of sources and sinks, and restriction to finite media.

However, the most recent innovations in fractional calculus are not described, such as generalization to complex order. Also, although algorithms for evaluation of differintegrals may be easily derived from the definitions through the book, a set of exemplary pseudocode fragments would be most welcome. These minor flaws are understandable, given the 1974 vintage of the book. A little more material could have been given on Heaviside's Operational Calculus and its relationship to the fractional calculus - it exists only in the references. Similarly, although the link to Weyl calculus is noted, it is not elaborated. Finally, a few examples from rheology would enrich the applications section.

COMPARISON : This book covers much the same area as Kenneth Miller and Bertram Ross in "An Introduction to the Fractional Calculus and Fractional Differential Equations", Wiley, 1993.

Whereas Oldham and Spanier base their treatise largely on the Grünwald definition, Miller and Ross prefer to use the equivalent Riemann-Liouville definition. Both books contain comparable historical surveys, but Oldham and Spannier provide a more extensive bibliography, and employ a better pedagogical style. Miller and Ross provide results in forms better suited to computer implementation, and rely more on matrix techniques. Also, Miller and Ross provide a chapter on Weyl Calculus, and their practical examples include one from hydrology, but omit diffusion problems. Other than this, the two books share most of their strengths and have only a few weaknesses. (Reader on Amazon.com)

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Rating:
2.5 out of 5 by Book123