  We describe a general operational method that can be used in the analysis of
fractional initial and boundary value problems with additional analytic
conditions. As an example, we derive analytic solutions of some fractional
generalisation of differential equations of mathematical physics. Fractionality
is obtained by substituting the ordinary integer-order derivative with the
Caputo fractional derivative. Furthermore, operational relations between
ordinary and fractional differentiation are shown and discussed in detail.
Finally, a last example concerns the application of the method to the study of
a fractional Poisson process.
