On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems

Abstract

The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Three physical examples illustrate this point.