A modified EM method and its fast implementation for multi-term Riemann-Liouville stochastic fractional differential equations

TL;DR

This paper introduces a modified Euler-Maruyama method for multi-term Riemann-Liouville stochastic fractional differential equations, proving its convergence and developing a fast implementation that enhances computational efficiency with numerical validation.

Contribution

The paper develops a new modified EM method with proven convergence for multi-term stochastic fractional equations and introduces a fast implementation using sum-of-exponentials approximation.

Findings

The modified EM method has a strong convergence order of min{1-α_m, 0.5}.

The fast EM method significantly reduces computational time.

Numerical examples confirm the theoretical convergence and efficiency improvements.

Abstract

In this paper, a modified Euler-Maruyama (EM) method is constructed for a kind of multi-term Riemann-Liouville stochastic fractional differential equations and the strong convergence order min{1-{\alpha}_m, 0.5} of the proposed method is proved with Riemann-Liouville fractional derivatives' orders 0<{\alpha}_1<{\alpha}_2<...<{\alpha}_m <1. Then, based on the sum-of-exponentials approximation, a fast implementation of the modified EM method which is called a fast EM method is derived to greatly improve the computational efficiency. Finally, some numerical examples are carried out to support the theoretical results and show the powerful computational performance of the fast EM method.