Abstract In order to approximate the Caputo fractional derivative of order α, 0 α 1 , we construct here a new class of formulae on the basis of B-spline interpolation. These new formulae called S1, S2 and S3 have 2 − α , 3 − α and 4 − α order of convergence, respectively. The proposed formulae are as simple as the well-known L1 formula and the main advantage of them lies in the fact that their accuracy is fixed in the whole interval of integration while the previous formulae such as L1-2 have lower accuracy at the beginning of the interval. Hence in comparison with the previous formulae, new ones have better accuracy and their computational costs are comparable. We then modify S2 and S3 formulae for approximating the Caputo fractional derivative of order α, 1 α 2 . Some numerical examples as well as two applications in solving fractional ordinary and partial differential equations (PDEs) are provided to demonstrate the applicability and accuracy of the new formulae.

中文翻译：

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逼近 Caputo 分数阶导数的一些高阶公式

摘要 为了逼近 α, 0 α 1 阶的 Caputo 分数阶导数，我们在 B 样条插值的基础上构造了一类新的公式。这些称为 S1、S2 和 S3 的新公式分别具有 2 − α 、3 − α 和 4 − α 级收敛。所提出的公式与众所周知的 L1 公式一样简单，它们的主要优点在于它们的精度在整个积分区间内是固定的，而之前的公式如 L1-2 在开始时的精度较低间隔。因此，与之前的公式相比，新的具有更好的准确性，并且它们的计算成本是可比的。然后我们修改 S2 和 S3 公式以逼近 α, 1 α 2 阶的 Caputo 分数阶导数。