Abstract
In the next few lectures we provide a brief overview of Fourier analysis and how it has been used to model lin- ear physical phenomena, particularly the reversible propagation of scalar waves in homogeneous media and the irreversible diffusion of one molecular species within another. The purpose of this review is to orient the reader so that the significance of wave propagation in fractal media will be apparent as will anom- alous diffusion. These latter topics have emerged in the last two decades as the natural successors of the phenomena examined in the 19th and early 20th centuries.
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Bibliography
M. Bologna, Derivata ad Indice Reale, Ets Editrice, Italy (1990).
A. L. Cauchy, Theorie de la propagation des ondes (Prix d’analyses mathe- matique), Concours de 1815 et de 1816, 140–2, 281–2.
P. A. M. Dirac, The Principles of Quantum Mechanics, Oxford University Press, Oxford (1953).
U. Frisch and G. Parisi, in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, M. Ghil, R. Benzi and G. Parisi, eds., North-Holland, Amsterdam (1985).
D. V. Giri, Dirac Delta Functions, unpublished report (1979).
H. Goldstein, Advanced Classical Mechanics, first edition, John Wiley, New York (1955).
I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, corrected and enlarged edition, Academic, New York (1980).
M. Hermite, (Faculte des Sciences de Paris), 5 edition, 155, Paris (1891).
O. Heaviside, Electromagnetic Theory, vol. II, 54–55 and 92, First issue 1893 and reissued London 1922.
D. D. Joseph and L. Preziosi, Rev. Mod. Phys. 61, 41 (1989).
G. R. Kirchoff and S. B. Kon, Adad. Wiss, Berlin Vom. 22, 641, juni (1992)
G. R. Kirchoff and S. B. Kon, Wied. Ann XVIII, 663 (1883).
M. J. Lighthill, Introduction to Fourier Analysis and Generalized Functions, Cambridge University Press, Cambridge (1964).
K. Lindenberg and B.J. West, The first, the biggest, and other such con- siderations, J. Stat. Phys. 42, 201 (1986).
K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley, New York (1993).
S. D. Poisson, Memoire sur la theorie des ondes (lu le 2 octobre et le 18 decembre 1815), 85–86.
S. G. Sainko, A. A. Kilbas and 0. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach, New York (1993).
J. A. Scales and R. Snieder, What is a wave?, Nature 401, 739 (1999).
W. Thomson, Trans. Roy. Soc. of Edinburgh, XXI, (May) (1854).
A. Zygmund, Trigonometric Series, vols. I and II combined, 2 edition, Cambridge University Press, Cambridge, MA (1977).
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West, B.J., Bologna, M., Grigolini, P. (2003). Fractional Fourier Transforms. In: Physics of Fractal Operators. Institute for Nonlinear Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-21746-8_4
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DOI: https://doi.org/10.1007/978-0-387-21746-8_4
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