Coupled systems of fractional equations related to sound propagation: analysis and discussion

Abstract

In this note we analyse the propagation of a small density perturbation in a one-dimensional compressible fluid by means of fractional calculus modelling, replacing thus the ordinary time derivative with the Caputo fractional derivative in the constitutive equations. By doing so, we embrace a vast phenomenology, including subdiffusive, superdiffusive and also memoryless processes like classical diffusions. From a mathematical point of view, we study systems of coupled fractional equations, leading to fractional diffusion equations or to equations with sequential fractional derivatives. In this framework we also propose a method to solve partial differential equations with sequential fractional derivatives by analysing the corresponding coupled system of equations.