Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions

TL;DR

This paper introduces a new second-order numerical method based on a modified Gr"{u}nwald--Letnikov formula for solving multi-term fractional differential equations, achieving high accuracy for both smooth and non-smooth solutions.

Contribution

A novel modified WSGL formula with correction terms is developed, enabling efficient and accurate solutions without explicit knowledge of analytical solutions.

Findings

Achieves second-order convergence for smooth and non-smooth solutions

Requires only a few correction terms for high accuracy

Numerical results outperform existing methods

Abstract

Starting with the asymptotic expansion of the error equation of the shifted Gr\"{u}nwald--Letnikov formula, we derive a new modified weighted shifted Gr\"{u}nwald--Letnikov (WSGL) formula by introducing appropriate correction terms. We then apply one special case of the modified WSGL formula to solve multi-term fractional ordinary and partial differential equations, and we prove the linear stability and second-order convergence for both smooth and non-smooth solutions. We show theoretically and numerically that numerical solutions up to certain accuracy can be obtained with only a few correction terms. Moreover, the correction terms can be tuned according to the fractional derivative orders without explicitly knowing the analytical solutions. Numerical simulations verify the theoretical results and demonstrate that the new formula leads to better performance compared to other known numerical approximations with similar resolution.