The $$\psi $$-Hilfer fractional calculus of variable order and its applications

Abstract

In this paper, we present the $$ \psi $$ -Hilfer fractional derivatives of variable order (FDVO) of 3 types I, II and III, versions A and B, as well as their combinations. In addition, we propose approximations and relations between both derivatives, i.e., $$ \psi $$ -Hilfer FDVO and $$ \psi $$ -Caputo FDVO. With regard to the $$ \psi $$ -Hilfer FDVO type II, we discuss the stability of the FVO nonlinear systems 7 solutions by means of one-parameter Mittag-Leffler functions of variable order. Examples 8 involving the FDVO Lu and Chen systems are also presented.