Abstract Fractional differential equations have become an important modeling technique in describing various natural phenomena. A variety of numerical methods for solving fractional differential equations has been developed over the last decades. Among them, finite difference methods are most popular owing to relative easiness for implementation. In this paper, we show that the finite difference method with the Riemann-Liouville (RL) fractional derivative yields inconsistent and oscillatory numerical solutions to fractional differential equations of discontinuous problems for the fractional order α, 0 α 1 . Such an inconsistency affects even smooth problems since the numerical solution can be regarded as a discontinuous function over grids. We show that the inconsistency inherited in discontinuous problems causes the numerical solution for smooth problems to be oscillatory under certain conditions although the magnitude of the oscillations decreases as the number of grids, N, increases. That is, although the truncation error is decaying as N → ∞ and the method is consistent for smooth problems, the numerical solution can be oscillatory for any value of N. To illustrate the inconsistency and the oscillation phenomenon with the RL method, we also consider the finite difference methods with the Caputo and Grunwald-Letnikov fractional derivatives and compare the results with by with the RL method. We also show that the integral approach for the RL method can resolve the issues of the inconsistency and the oscillation phenomenon. Numerical results are presented to support the statements.

中文翻译：

---

关于有限差分近似与 0 < α < 1 的 Riemann-Liouville 分数阶导数的一致性

摘要 分数阶微分方程已成为描述各种自然现象的重要建模技术。在过去的几十年里，已经开发了多种求解分数阶微分方程的数值方法。其中，有限差分法由于相对易于实现而最受欢迎。在本文中，我们表明，对于分数阶 α, 0 α 1 的不连续问题的分数微分方程，使用 Riemann-Liouville (RL) 分数阶导数的有限差分方法会产生不一致和振荡的数值解。这种不一致甚至会影响平滑问题，因为数值解可以被视为网格上的不连续函数。我们表明，在不连续问题中继承的不一致性导致平滑问题的数值解在某些条件下是振荡的，尽管振荡的幅度随着网格数量 N 的增加而减小。也就是说，虽然截断误差随着 N → ∞ 而衰减，并且该方法对于光滑问题是一致的，但是对于任何 N 值，数值解都可以是振荡的。 为了说明 RL 方法的不一致性和振荡现象，我们还考虑使用 Caputo 和 Grunwald-Letnikov 分数阶导数的有限差分方法，并将结果与​​ RL 方法进行比较。我们还表明，RL 方法的积分方法可以解决不一致和振荡现象的问题。提供了数值结果以支持这些陈述。