The Relationship Between Fractional Calculus and Fractals

The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).