The Grünwald–Letnikov approximation is a well-known discretization to approximate a Riemann–Liouville derivative of order $\alpha>0$. This approximation has been proved to be a consistent approximation, of order $1 $, when the domain is the real line, using Fourier transforms. However, in recent years, this approximation has been applied frequently to solve fractional differential equations in bounded domains and the result proved for the real line has been assumed to be true in general. In this work we show that when assuming a bounded domain, the Grünwald–Letnikov approximation can be inconsistent, for a large number of cases, and when it is consistent we have mostly an order of $n-\alpha $, for $n-1<\alpha <n$.

中文翻译：

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有界域中 Grünwald-Letnikov 近似的一致性分析

Grünwald-Letnikov 逼近是一种众所周知的离散化方法，用于逼近 $\alpha>0$ 阶的 Riemann-Liouville 导数。使用傅里叶变换，当域是实线时，该近似已被证明是一致的近似，阶数为 $1 $。然而，近年来，这种近似已被频繁地应用于求解有界域中的分数阶微分方程，并且通常假设实线证明的结果是正确的。在这项工作中，我们表明当假设一个有界域时，Grünwald-Letnikov 近似在大量情况下可能是不一致的，当它一致时，我们主要有一个 $n-\alpha $ 的阶数，对于 $n- 1＜α＜n$。