In this paper, we propose a new sixth-order finite difference weighted essentially non-oscillatory (WENO) scheme for solving the fractional differential equations which may contain non-smooth solutions at a later time, even if the initial solution is smooth enough. After splitting the Caputo fractional derivative of order \(\alpha \) \((1<\alpha \le 2)\) into a weakly singular integral and a classical second derivative, the classical Gauss–Jacobi quadrature is used to solve the weakly singular integral and a new spatial WENO-type reconstruction methodology is proposed to approximate the second derivative. There are two advantages of the new WENO scheme: the first is that the linear weights can be any positive numbers on condition that their summation equals one, and the second is its simplicity in the engineering applications. The new WENO reconstruction is a convex combination of a quartic polynomial with two linear polynomials defined on three unequal-sized spatial stencils in a traditional WENO fashion. This new sixth-order WENO scheme uses smaller number of cell average information than that of the same order classical WENO schemes and could eliminate non-physical oscillations near strong discontinuities when solving the fractional differential equations. Some benchmark examples are given to demonstrate the efficiency, robustness, and good performance of this new finite difference WENO scheme.

在本文中，我们提出了一种新的六阶有限差分加权基本非振动（WENO）方案，用于求解分数阶微分方程，该分数阶微分方程可能在以后的某个时间包含非光滑解，即使初始解足够平滑。拆分阶\（\ alpha \） \（（1 <\ alpha \ le 2）\）的Caputo分数导数后分解为弱奇异积分和经典二阶导数，使用经典高斯-雅各比积分求解弱奇异积分，并提出了一种新的空间WENO型重构方法来近似二阶导数。新的WENO方案有两个优点：首先是线性权重的总和等于1时可以是任何正数，其次是在工程应用中的简单性。新的WENO重建是四次多项式与两个线性多项式的凸组合，以传统的WENO方式在三个不等大小的空间模具上定义了两个线性多项式。这种新的六阶WENO方案比相同阶数的经典WENO方案使用的单元平均信息数量更少，并且在求解分数阶微分方程时可以消除强不连续点附近的非物理振荡。给出了一些基准示例，以证明这种新的有限差分WENO方案的效率，鲁棒性和良好的性能。