A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications
A new fractional numerical differentiation formula to approximate the Caputo fractional derivative and its applications
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一种新的逼近Caputo分数阶导数的分数数值微分公式及其应用
DOI：
10.1016/j.jcp.2013.11.017
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发表时间：
2014-02-15
影响因子：
4.1
通讯作者：
Zhang, Hong-wei
中科院分区：
文献类型：
--
作者：
Gao, Guang-hua;Sun, Zhi-zhong;Zhang, Hong-wei
本文首先建立了一个新的分数阶数值微分公式（称为L1-2公式）来近似alpha （0 < alpha < 1）阶的Caputo分数阶导数。对于被积函数f(t)在每个小区间[t(j), t(j)] （j >= 2）上的被积函数f(t)，采用三个点(t(j-2), f(t（j)), f(t(j))）和(t(j), f（t(j)）的二次插值近似建立，而在第一个小区间[t(0), t(1)]上应用线性插值近似。因此，新公式可以形式上看作是对经典L1公式的修正，经典L1公式是通过对f (t)分段线性逼近得到的。新公式的计算效率和数值精度均优于L1公式。详细讨论了该公式的系数和截断误差。两个算例验证了L1-2公式的数值精度。其次，利用新公式，分别构造了在有界空间域和无界空间域上求解时间分数次扩散方程的两种高阶时间精度的改进有限差分格式。此外，还介绍了新公式在求解分数阶常微分方程中的应用。计算了几个数值算例。通过与L1公式的有限差分法计算结果的比较，表明新的L1-2公式在数值求解时间分数阶微分方程时比L1公式更有效、更精确。(C) 2013爱思唯尔公司版权所有。
In the present work, first, a new fractional numerical differentiation formula (called the L1-2 formula) to approximate the Caputo fractional derivative of order alpha (0 < alpha < 1) is developed. It is established by means of the quadratic interpolation approximation using three points (t(j-2), f(t(j-2))), (t(j-1), f(t(j-1))) and (t(j), f(t(j))) for the integrand f(t) on each small interval [t(j-1), t(j)] (j >= 2), while the linear interpolation approximation is applied on the first small interval [t(0), t(1)]. As a result, the new formula can be formally viewed as a modification of the classical L1 formula, which is obtained by the piecewise linear approximation for f (t). Both the computational efficiency and numerical accuracy of the new formula are superior to that of the L1 formula. The coefficients and truncation errors of this formula are discussed in detail. Two test examples show the numerical accuracy of L1-2 formula. Second, by the new formula, two improved finite difference schemes with high order accuracy in time for solving the time-fractional sub-diffusion equations on a bounded spatial domain and on an unbounded spatial domain are constructed, respectively. In addition, the application of the new formula into solving fractional ordinary differential equations is also presented. Several numerical examples are computed. The comparison with the corresponding results of finite difference methods by the L1 formula demonstrates that the new L1-2 formula is much more effective and more accurate than the L1 formula when solving time-fractional differential equations numerically. (C) 2013 Elsevier Inc. All rights reserved.