Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions☆
Second-order numerical methods for multi-term fractional differential equations: Smooth and non-smooth solutions☆
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DOI：
10.1016/j.cma.2017.08.029
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发表时间：
2017-01
影响因子：
7.2
通讯作者：
Fanhai Zeng;Zhongqiang Zhang;G. Karniadakis
中科院分区：
文献类型：
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作者：
Fanhai Zeng;Zhongqiang Zhang;G. Karniadakis
从位移Grünwald-Letnikov公式误差方程的渐近展开出发，通过引入适当的修正项，得到了一种新的修正加权Grünwald-Letnikov(WSGL)公式。然后，我们利用修正的WSGL公式的一个特例来求解多项分数阶常微分方程组和偏微分方程组，并在解的正则性已知的情况下，证明了光滑解和非光滑解的线性稳定性和二阶收敛。从理论和数值上证明，只需少量的修正项即可得到精度较高的数值解。此外，修正项可以根据分数导数阶进行调整，而不需要明确知道解析解。数值模拟验证了理论结果，并证明了新的WSGL公式比其他已知的类似分辨率的数值近似具有更好的性能。
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 when the regularity of the solutions is known. We show theoretically and numerically that numerical solutions with good 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 WSGL formula leads to better performance compared to other known numerical approximations with similar resolution.