In this paper, the initial condition independence property of Grünwald–Letnikov fractional difference is revealed for the first time. For example, the solution x(k) of equation aG∇kαx(k) = f(x(k)),  k > a + 1, cannot be calculated with initial condition x(a). First, the initial condition independence property is carefully investigated in both time domain and frequency domain. Afterwards, some possible schemes are formulated to make the considered system connect to initial condition. Armed with this information, the concerned property is examined on three modified Grünwald–Letnikov definitions. Finally, results from illustrative examples demonstrate that the developed schemes are sharp.

中文翻译：

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如何用可用的初始条件赋予 Grünwald-Letnikov 分数差分方程？

本文首次揭示了 Grünwald-Letnikov 分数差的初始条件独立性。例如，方程 aG∇kαx(k) = f(x(k)) 的解 x(k)，k > a + 1，不能用初始条件 x(a) 计算。首先，在时域和频域中仔细研究了初始条件独立性。然后，制定了一些可能的方案，使所考虑的系统连接到初始条件。有了这些信息，相关属性将根据三个修改后的 Grünwald-Letnikov 定义进行检查。最后，说明性示例的结果表明，所开发的方案是敏锐的。