The work in this paper is four-fold. Firstly, we introduce an alternative
approach to solve fractional ordinary differential equations as an expected
value of a random time process. Using the latter, we present an interesting
numerical approach based on Monte Carlo integration to simulate solutions of
fractional ordinary and partial differential equations. Thirdly, we show that
this approach allows us to find the fundamental solutions for fractional
partial differential equations (PDEs), in which the fractional derivative in
time is in the Caputo sense and the fractional in space one is in the
Riesz-Feller sense. Lastly, using Riccati equation, we study families of
fractional PDEs with variable coefficients which allow explicit solutions.
Those solutions connect Lie symmetries to fractional PDEs.Comment: 23 pages, 5 figure