Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative

We propose a (new) definition of a fractional Laplace's transform, or Laplace's transform of fractional order, which applies to functions which are fractional differentiable but are not differentiable, in such a manner that they cannot be analyzed by using the Djrbashian fractional derivative. After a short survey on fractional analysis based on the modified Riemann-Liouville derivative, we define the fractional Laplace's transform. Evidence for the main properties of this fractal transformation is given, and we obtain a fractional Laplace inversion theorem. (C) 2009 Elsevier Ltd. All rights reserved.