Supervised learning and meshless methods for two-dimensional fractional PDEs on irregular domains

Recently, several numerical methods have been developed for solving time-fractional differential equations not only on rectangular computational domains but also on convex and non-convex non-rectangular computational geometries. On the other hand, due to the existence of integrals in the definition of space-fractional operators, there are few numerical schemes for solving space-fractional differential equations on irregular regions. In this paper, we develop a novel numerical solution based on the machine learning technique and a generalized moving least squares approximation for two-dimensional fractional PDEs on irregular domains. The scheme is constructed on the monomials, and this is the strength of this technique. Moreover, it will be used to approximate the space derivatives on convex and non-convex non-rectangular computational domains. The numerical results are extended to solve the fractional Bloch-Torrey equation, fractional Gray-Scott equation, and fractional Fitzhugh-Nagumo equation. (c) 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

Automated summary

In this paper, a novel numerical solution based on machine learning technique and a generalized moving least squares approximation is developed for solving two-dimensional fractional partial differential equations on irregular domains. The method approximates spatial derivatives on convex and non-convex non-rectangular computational domains and is validated on various specific problems.