  Some fractional and anomalous diffusions are driven by equations involving
fractional derivatives in both time and space. Such diffusions are processes
with randomly varying times. In representing the solutions to those diffusions,
the explicit laws of certain stable processes turn out to be fundamental. This
paper directs one's efforts towards the explicit representation of solutions to
fractional and anomalous diffusions related to Sturm-Liouville problems of
fractional order associated to fractional power function spaces. Furthermore,
we study a new version of the Bochner's subordination rule and we establish
some connections between subordination and space-fractional operator
