ABSTRACT
In this chapter, we introduce the variable-order fractional calculus with respect to another function variable-order fractional calculus with respect to another function , first proposed by Yang and Machado in 2017 [114], and the variable-order general fractional calculus involving the Mittag-Leffler function with respect to another function, proposed by Sousa and Oliveira in 2018 for the first time [324]. The variable-order general fractional derivatives containing the kernel of the special functions, such as the Mittag-Leffler function and weak power-law function are presented. The definitions of the variable-order fractional derivatives and the variable-order general fractional integrals with respect to another function variable-order general fractional integrals with respect to another function are introduced. The Hilfer type fractional derivatives of variable order with respect to another function and the Riesz type fractional derivatives of variable order with respect to another function are also proposed. The mathematical models for the relaxation and rheological models are discussed in detail.
