Fractional and integer derivatives with continuously distributed lag

Abstract

We propose new type of derivatives and integrals of non-integer orders. The fractional derivatives and integrals with continuously distributed lag (time delay) are defined as a composition of the translation operator with lag, which is continuously distributed with a weighting function, and the fractional derivatives of non-integer order such as the Riemann–Liouville, Liouville, Caputo, and other fractional derivatives. The weighting function satisfies the condition of non-negativity and the normalization condition. These properties of the weighting function allow us to interpret it as the density of a probability distribution of the lag. This gives a possibility to apply different probability distributions, which are defined on the positive semiaxis, to describe continuously distributions of lags. The properties of the suggested operators with non-integer orders and distributed lag are described. The solution of equations that contain the fractional derivatives with gamma distributed lag and it special cases such as exponential and Erlang distributions of lag are obtained. We use the Laplace method and the method that is based of representation by equations with fractional derivatives without lag. The fractional derivatives and integrals with continuously distributed scaling (dilation) are considered on the basis of the Kober and Erdelyi–Kober fractional operators as a generalization of the approach proposed for continuously distributed lag. The suggested generalization of the fractional integrals and derivatives, which includes the translation and dilation operators, can be used to describe the continuously distributed lag and scale effects in physical, economic, and other processes with memory.