Caputo–Fabrizio operator in terms of integer derivatives: memory or distributed lag?

Abstract

In this paper, we prove that linear and nonlinear equations with the Caputo–Fabrizio operators can be represented as systems of differential equations with derivatives of integer orders. The order of these equations is not more than one with respect to the integer part of the highest order of the Caputo–Fabrizio operators. We state that the Caputo–Fabrizio operators with exponential kernel cannot describe nonlocality and memory (temporal nonlocality) in processes and systems. Using the principle of nonlocality for fractional derivatives of noninteger orders (“No nonlocality. No fractional derivative”), we can state that the Caputo–Fabrizio operators cannot be considered as a fractional derivative. A general physical and economic interpretation (meaning) of the Caputo–Fabrizio operator is proposed. We state that physical and economic meaning of the Caputo–Fabrizio operators is continuously (exponentially) distributed lags.