Self-similarity and response of fractional differential equations under white noise input

Abstract

Self-similarity, fractal behaviour and long-range dependence are observed in various branches of physical, biological, geological, socioeconomics and mechanical systems. Self-similarity, also termed self-affinity, is a concept that links the properties of a phenomenon at a certain scale with the same properties at different time scales as it happens in fractal geometry. The fractional Brownian motion (fBm), i.e. the Riemann–Liouville fractional integral of the Gaussian white noise, is self-similar; in fact by changing the temporal scale t→at (a>0), the statistics in the new time axis (at) remain proportional to those calculated in the previous axis (t). The proportionality coefficient is a2H being H>0 the Hurst index. In the practical applications, the phenomena are usually ruled by fractional differential equations involving more terms. In this paper it is shown that the response of a multi-term fractional differential equation is a linear combination of self-similar processes with increasing order of Hurst exponent. The consequences of self-similarity are discussed in detail, closed forms of correlation and variance are presented for the general case and particularized for the cases of engineering interest.