A discussion on nonlocality: From fractional derivative model to peridynamic model

Abstract

Characterization of nonlocality is an open problem in physics and engineering. This paper conducts a detailed investigation on two nonlocal models, namely, the fractional derivative model and the peridynamic model for anomalous diffusion. A generalized nonlocal model combining the advantages of the fractional derivative model and the peridynamic model, is introduced. In this paper, analytical solutions and the mean squared displacements of the two models are provided and discussed. In addition, their intrinsic relations and notable differences are investigated. Preliminary applications indicate that the peridynamic model can well capture an unremarkable transition from normal-diffusion to super-diffusion, while the fractional derivative model presents super-diffusion behaviors in the whole process. At last, a generalized nonlocal operator is proposed as a more general strategy to solve nonlocal problems.