Fractional Trigonometry: with Applications to Fractional Differential Equations and Science by Lorenzo Carl F.; Hartley Tom T.; & Tom T. Hartley

Author:Lorenzo, Carl F.; Hartley, Tom T.; & Tom T. Hartley
Language: eng
Format: epub
Publisher: John Wiley & Sons, Incorporated
Published: 2016-11-01T00:00:00+00:00

Introduction to Applications

The primary and by far the most important application of the fractional exponential and fractional trigonometric functions is the solution of linear fractional differential equations. In Chapter 12, these functions were used for the solution of linear commensurate-order fractional differential equations with constant coefficients. Analytical tools for such differential equations with unrepeated roots, repeated real roots, and repeated complex roots were presented.

Spiral behavior occurs in many natural phenomena. Galactic spirals, sea shells, whirlpools, and weather patterns are only a few such occurrences seen in the sciences. We have seen numerous examples of fractional spirals as the result of our development of the fractional trigonometries. It is natural to ask if and how the fractional spirals and associated fractional differential equations might apply to these scientific areas. Our goal is to show the application potential to such areas and not necessarily the solution to detailed problems.

Specific areas to be considered in Chapters 13–19 are fractional oscillators, morphology of sea shells, in particular the Nautilus pompilius, hurricanes, and low-pressure system cloud morphology, morphological classification of spiral and ring galaxies, tornado profiles, and whirlpool morphology. While in most cases these potential application areas provide spatial definition of the phenomena, it is expected that the development of compatible cylindrical fractional field equations, considered briefly in Chapter 20, will allow full definition and possible temporal solution. There is much to be done in the development of the fractional trigonometry; Chapter 20 presents some related challenges.

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