Fractional Calculus and the Evolution of Fractal Phenomena

Abstract

It is argued that the evolution of complex phenomena ought to be described by fractional, differential, stochastic equations whose solutions have scaling properties and are therefore random, fractal functions. To support this argument we demonstrate that the fractional derivative (integral) of a generalized Weierstrass function (GWF) is another fractal function with a greater (lesser) fractal dimension. We also determine that the GWF is a solution to such a fractional differential stochastic equation of motion.

7 pages

References in corpus (1)

• Holder exponents of irregular signals and local fractional derivatives

Cited by in corpus (5)

• A physically based connection between fractional calculus and fractal geometry
• Memory Effects and Macroscopic Manifestation of Randomness
• Complexity Analysis of Chaos and Other Fluctuating Phenomena
• Fractal index, central charge and fractons
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