A stochastic method for solving time-fractional differential equations

We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resortin...

Full description

Saved in:
Bibliographic Details
Published in:Computers & mathematics with applications (1987) 2024-04, Vol.159, p.240-253
Main Authors: Guidotti, Nicolas L., Acebrón, Juan A., Monteiro, José
Format: Article
Language:English
Subjects:
Anomalous diffusion
High performance computing
Matrix functions
Mittag-Leffler function
Monte Carlo method
Parallel algorithms
Time-fractional differential equations
Citations: Items that this one cites
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We present a stochastic method for efficiently computing the solution of time-fractional partial differential equations (fPDEs) that model anomalous diffusion problems of the subdiffusive type. After discretizing the fPDE in space, the ensuing system of fractional linear equations is solved resorting to a Monte Carlo evaluation of the corresponding Mittag-Leffler matrix function. This is accomplished through the approximation of the expected value of a suitable multiplicative functional of a stochastic process, which consists of a Markov chain whose sojourn times in every state are Mittag-Leffler distributed. The resulting algorithm is able to calculate the solution at conveniently chosen points in the domain with high efficiency. In addition, we present how to generalize this algorithm in order to compute the complete solution. For several large-scale numerical problems, our method showed remarkable performance in both shared-memory and distributed-memory systems, achieving nearly perfect scalability up to 16,384 CPU cores.
ISSN:0898-1221
1873-7668
DOI:10.1016/j.camwa.2024.02.020