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 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.

Automated summary

The work in this paper has four aspects. Firstly, an alternative approach is introduced to solve fractional ordinary differential equations by representing them as the expected value of a random time process. Secondly, based on Monte Carlo integration, an interesting numerical method is presented to simulate solutions of fractional ordinary and partial differential equations. Thirdly, it is shown that this approach can find the fundamental solutions for fractional partial differential equations, where the fractional derivative in time is in the Caputo sense and the fractional derivative in space is in the Riesz-Feller sense. Lastly, using the Riccati equation, families of fractional partial differential equations with variable coefficients that have explicit solutions are studied, and these solutions connect Lie symmetries to fractional partial differential equations.