Fractional-Type Laplace Transforms

Since the Laplace transform plays a central role in the solution of differential equations, it seems natural to extend it in the field of fractional calculus, since many applications of this topic have been proposed, and are becoming more and more important. In this paper we extend the classical Laplace Transform by replacing the usual kernel with a suitable one, both in the classical and Laguerre-type case, obtained by constructing the reciprocal of some exponential-type functions with respect to an appropriate differential operator. Some examples are shown, derived using the computer algebra system Mathematica.