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

Authors

Tamer Oraby, The University of Texas Rio Grande Valley
Erwin Suazo, The University of Texas Rio Grande Valley
Harrinson Arrubla, The University of Texas Rio Grande Valley

Document Type

Article

Publication Date

1-2023

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.

Comments

Original published version available at https://doi.org/10.1016/j.chaos.2022.112901

Recommended Citation

Oraby, Tamer, E. Suazo, and H. Arrubla. "Probabilistic solutions of fractional differential and partial differential equations and their Monte Carlo simulations." Chaos, Solitons & Fractals 166 (2023): 112901. https://doi.org/10.1016/j.chaos.2022.112901

Creative Commons License

This work is licensed under a Creative Commons Attribution-NonCommercial-No Derivative Works 4.0 International License.

Publication Title

Chaos, Solitons & Fractals

DOI

https://doi.org/10.1016/j.chaos.2022.112901