- Paperback: 256 pages
- Publisher: Dover Publications Inc. (28 April 2006)
- Language: English
- ISBN-10: 0486450015
- ISBN-13: 978-0486450018
- Product Dimensions: 14 x 1.4 x 21.6 cm
- Average Customer Review: 1 customer review
- Amazon Bestsellers Rank: #47,922 in Books (See Top 100 in Books)
The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order (Dover Books on Mathematics) Paperback – 28 Apr 2006
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Fractional calculus (FC) is a misnomer, because irrationals can also be integral indices in this field. This book covers all the major pieces, as well as good history up to 1975. Since then, most of the advances have come in numerical methods and extended applications in fractional conservation of mass, porous media flows, bioacoustics, polymer dampening, etc. Since one of these authors is a chemist, many of these directions were anticipated in the diffuse flow discussions.
Today, some people consider FC an historic field since every FC calculation can now be transformed to an integral with numeric methods on a computer (which ARE covered sans the processor here). That line of thought relegates FC to a notation anomaly with some value in the algorithmic sense of process efficiency. In reality, the METHODS explained here are still used, as their foundations were already in place in the 60's-- vanishing Fouriers with functions on the unit circle, gamma functions, the newer Mellin technique hearkening back to Fourier integral transforms, Cauchy's repeated integration, etc.
A nice feature is that, in current literature (in journals, not texts), FC won't take the time to explain WHY gamma functions are used, whereas this little volume goes to great length to introduce the reader to them and the characteristics that give them value in FC simplification and reduction.
Normal integrals and derivatives need very close proximity, and FCs don't have that naturally, so boundary conditions are key. That means partial derivatives and many other analysis level techniques are employed-- requiring grad level math to fully appreciate.
One strange omission is that the authors don't mention M. Caputo's Fractional Derivative, published in 1967 (the sweet spot of this book), which simplifies Riemann-Liouville. Wonder what that was about???
Highly recommended for coverage, thoroughness and price. Really one of the only ones left at a reasonable price. The "bible" of this field (Samko-- over 1,000 pages) is out of print and has sold for over $1,000 at auctions!
Other than that, this field has "dispersed" itself into its applications, and you'll find chapters on it in those texts, but nothing like this book. Amazing since there are so many fields to which it applies, notation or not!
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Two of these aspects included the hypergeometric p-function used by D.B. Thomas in his paper: `Echo Correlation Analysis and the Acoustic Evidence in the JFK Assassination Revisited' (in Science and Justice, Vol. 41, p. 21) and Dartmouth Computer scientist Hany Farid's 2009 claim that he'd proven the Oswald backyard photos were real, not fakes - as most conspiracy proponents claim.
In extending Thomas' echogram significance analysis, I found the section on Hypergeometric functions (pp. 99-102)to be most useful, and to deal with Farid's claims, 'Properties of the Gamma Function' (pp. 16-24) were well-suited.
In the latter case, when scrutinizing Farid's pixel analysis of the Oswald LIFE photograph, one is - technically speaking- considering how a (1963) emulsion of silver iodide tends to displace or migrate over time (as anyone with old photos can attest) so one actually needs to deal properly with diffusion factors D(x) and D(y) using fractional calculus for computations, such that:
D_t(x^b) = G(b + 1) / G[b - t + 1]^x(b - t)
D_t(y^b) = G(b + 1) / G[b - t + 1]^y(b - t)
Where the G -functions are Euler Gamma functions, t is the time increment, and x^b, y^b are measurable shifts in pixels. The use of D_t(x^b) and D_t(y^b) make the reasonable assumption that any transformation, e.g. from specular dots on silver iodide -based emulsions will undergo some drift in rectangular coordinates (x,y) such that: d'(x) = w_H×(d(x)^bH) and d'(y) = w_V×(d(y)^bV). The preceding measure fractal deviations in pixel density, e.g. from one Oswald photo to the other, while the diffusion factors, D_t(x^b) and D_t(y^b) can discriminate image tampering from natural (silver iodide) migration.
To make a long story short, the use of the fractional calculus enabled me to clearly show slight errors in D.B. Thomas' echogram correlations, while also showing Hany Farid's claims don't hold up.
But if Oldham's and Spanier's book was merely responsible for allowing more extensive analysis of data, materials in a historical event, that would be more than enough. The fact is, the book provides a solid basis for so much more, including engineering applications (as illustrated on pp. 149, 151 and 152- these are in Chapter 8, on 'Techniques in Fractional Calculus').
Meanwhile, the synthesis of Bessel and Struve functions, p. 177, and the differintegrals of the Bessel functions (pp. 97-98) have set me to reworking an arcade model for 2-ribbon solar flares that I laid out in my book, 'Selected Analyses in Solar Flare Plasma Dynamics' (Ch. 3). In that original model, I employed the ordinary Bessel functions J_o(x) and J_1(x) but now find there is scope to refine the model (to perhaps enhance prediction for these flares) by using more refined techniques.
This excellent book on fractional calculus also prompted me to re-examine a cavity resonator model I developed for loop solar flares, especially when I beheld their Fig. 11.5.1 on page 211 for a resistive -capacitative transmission line. As is well known in solar physics models, one often employs the ansatz of "equivalent circuits" - usually incorporating some defined emf source, along with an inductance, L and resistive load, R to simply model a a loop flare, say.
In my cavity resonator model, I attempted to refine this by showing how resonance conditions would be triggered at certain frequencies.
Oldham's and Spanier's figure (and discussion) made me realize there is much scope to refine the nature of solar coronal perturbations, say triggered by magneto-hydrodynamic waves or loop oscillations- which can then be transmitted down the loop feet to the photosphere.
Why am I raving about this book? Because it is a cornucopia in terms of stimulating novel concepts for modeling across a spectrum of disciplines and applications, or improving approaches in others. In almost every chapter, one encounters material, presentations, and examples which prompt the reader toward new ideas, new ways of looking at the same models.....that originally might have been seen as exhausted in terms of utility and novelty.
For that reason alone, I'd recommend it to any research professional. And because fractional calculus can be so widely employed, I would recommend it across disciplines (e.g. physics, engineering, chemistry, biology, bio-physics) - something I very seldom do even for statistical monographs!
The real shame is that most students at university level will never be exposed to the beauty of these fractional calc methods. It is left then for the diligent teacher to make them aware (at least) of this fine and useful book, or for them to pursue self- study.
I am just thankful I found this book when I did.
looked at on this subject, this is by far the best written and most
understandable without requiring knowledge of abstract mathematics.
If you can find one, I would suggest purchasing
an original hardcover edition of this text as the quality of the Dover reprint edition is not
all that great (e.g., small font size)