In this paper, we first apply cosine radial basis function neural networks to solve the fractional differential equations with initial value problems or boundary value problems. In the examples, we successfully obtained the numerical solutions for the fractional Riccati equations and fractional Langevin equations. The computer graphics and numerical solutions show that this method is very effective.

The Generalized Elastic Model is a linear stochastic model which accounts for thebehaviour of many physical systems in nature, ranging from polymeric chains to single-filesystems. If an external perturbation is exerted only on a single pointx⋆ (taggedprobe), it propagates throughout the entire system. Within the fractionalLangevin equation framework, we study the effect of such a perturbation, in cases of aconstant force applied. We report most of the results arising from our previous analysisand, in the present work, we show that the Fox H-functions formalismprovides a compact, elegant and useful tool for the study of the scaling properties of anyobservable. In particular we show how the generalized Kubo fluctuation relations can beexpressed in terms of H-functions.