(Non)Commutativity and associativity of general fractional derivatives with different Sonin kernels

Abstract

The usual properties and rules of integer-order derivatives and integrals are violated for fractional derivatives and integrals of non-integer order. For example, the product rule, the chain rule, and the additivity property of the integer-order derivatives are violated for fractional derivatives of non-integer order. These unusual properties allow us to describe important unusual properties of various processes and systems with memory and non-locality. These properties of fractional derivatives and integrals also lead to difficulties and errors when these operators are not used accurately. In this paper, we consider commutativity, associativity and semigroup properties of general fractional derivatives (GFDs) with different type of Sonin kernels. The fulfillment of the semigroup property for fractional derivatives imposes strong restrictions on the existence of the commutativity property. The semigroup property is not a necessary condition for the commutativity and associativity of GFDs. In this paper, we prove that the commutativity and associativity properties of the GFDs can be satisfied in the general case when the semigroup property is violated. We prove that the semigroup property is generally not satisfied for GFDs of the Caputo type, but the commutativity and associativity properties of these GFDs with different Sonin kernels are satisfied. This paper also proves that the GFDs of the Riemann–Liouville type with different Sonin kernels are non-commutative associative operators. Exact equations that describe the violation of the commutativity and the semigroup properties are derived. These equations allow us to derive conditions under which the semigroup and commutativity properties are satisfied for the GFDs.