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

TL;DR

This paper introduces a probabilistic Monte Carlo method for solving fractional differential equations, providing new numerical techniques, fundamental solutions, and explicit solutions for variable coefficient fractional PDEs, linking Lie symmetries to these equations.

Contribution

It presents an alternative probabilistic approach to solve fractional differential equations and derives fundamental solutions and explicit solutions for variable coefficient cases.

Findings

Monte Carlo integration effectively simulates fractional PDE solutions

Fundamental solutions for fractional PDEs are derived using the probabilistic approach

Explicit solutions for variable coefficient fractional PDEs are obtained via Riccati equations

Abstract

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.