In this work the Monte Carlo method, introduced recently by the authors for orders of differentiation between zero and one, is further extended to differentiation of orders higher than one. Two approaches have been developed on this way. The first approach is based on interpreting the coefficients of the Grünwald–Letnikov fractional differences as so called signed probabilities, which in the case of orders higher than one can be negative or positive. We demonstrate how this situation can be processed and used for computations. The second approach uses the Monte Carlo method for orders between zero and one and the semi-group property of fractional-order differences. Both methods have been implemented in MATLAB and illustrated by several examples of fractional-order differentiation of several functions that typically appear in applications. Computational results of both methods were in mutual agreement and conform with the exact fractional-order derivatives of the functions used in the examples.

在这项工作中，作者最近针对零和一之间的微分阶数引入的蒙特卡洛方法进一步扩展到高于一阶的微分。以这种方式开发了两种方法。第一种方法是基于将 Grünwald-Letnikov 分数差的系数解释为所谓的有符号概率，在高于 1 的阶数的情况下，它可以是负数或正数。我们演示了如何处理这种情况并将其用于计算。第二种方法使用 Monte Carlo 方法来计算 0 和 1 之间的阶数以及分数阶差的半群性质。这两种方法都已在 MATLAB 中实现，并通过几个通常出现在应用程序中的函数的分数阶微分示例进行了说明。