On new types of fractional operators and applications

We introduce two kinds of fractional integral operators; the one is defined via the exponential-integral function $$E_1(x)=\int_x^\infty \frac{e^{-t}}{t}\,dt,\quad x>0,$$ and the other is defined via the special function $$\mathcal{S}(x)=e^{-x} \int_0^\infty \frac{x^{s-1}}{\Gamma(s)}\,ds,\quad x>0.$$ We establish different properties of these operators, and we study the relationship between the fractional integrals of first kind and the fractional integrals of second kind. Next, we introduce a new concept of fractional derivative of order $\alpha>0$, which is defined via the fractional integral of first kind. Using an approximate identity argument, we show that the introduced fractional derivative converges to the standard derivative in $L^1$ space, as $\alpha\to 0^+$. Several other properties are studied, like fractional integration by parts, the relationship between this fractional derivative and the fractional integral of second kind, etc. As an application, we consider a new fractional model of the relaxation equation, we establish an existence and uniqueness result for this model, and provide an iterative algorithm that converges to the solution.