Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions
This work provides a more accurate and stable numerical method for solving multi-term fractional differential equations, which are important in modeling complex systems with memory effects.
The authors developed a modified weighted shifted Grünwald-Letnikov formula with correction terms that achieves second-order convergence for multi-term fractional differential equations, even for non-smooth solutions, and demonstrated improved performance over existing methods through numerical simulations.
Starting with the asymptotic expansion of the error equation of the shifted Grünwald--Letnikov formula, we derive a new modified weighted shifted Grünwald--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.