Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations

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