Duality of fractional systems

Abstract In this paper we reveal a common property of generalized fractional equations used for the description of many physical systems related to non-exponential relaxation and anomalous diffusion. It follows from conjugate pairs of Bernstein functions being Laplace exponents of random processes that participate at subordination of simple processes such as ordinary exponential relaxation or Brownian motion. The reciprocals of such pair after the inverse Laplace transform obey the Sonine equation in which the Sonine pair consists of two interconnected memory functions being kernels of two generalized fractional equations. Their solutions are transformed into each other. This property is considered as duality in which each generalized fractional system has a partner.

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