Infinitesimal Prolongation for Fractional Derivative \(\varPsi \)-Caputo Variable Order and Applications

In this work, we are concerned with the well-known Lie group theory to find symmetries of differential equations with fractional derivative \(\varPsi \) $\Psi$ -Caputo variable order. In this sense, we discuss the Leibniz-type rule and also the chain-type rule for the fractional derivative of this operator. In the end, we apply the results obtained in the fractional Harry Dym-type equation to find its symmetries, and we present a solution for a fractional Harry Dym-type equation with constant order.