基于广义模糊泰勒展开的模糊分数阶微分方程的模糊解法
A fuzzy method for solving fuzzy fractional differential equations based on the generalized fuzzy Taylor expansion
摘要：
在许多数学类型的研究中，为了求解模糊分数阶微分方程，必须将其转化为清晰的对应问题，通过求解这些对应问题，才能得到其近似解。本文的目的是提出一种新的直接法来求解模糊分数阶微分方程而不需要这种变换。利用模糊Caputo分数阶导数的意义，给出了模糊值函数的模糊广义Taylor展开式。将模糊广义欧拉方法应用于求解模糊分数阶微分方程。为了证明该方法的准确性和有效性，确定了局部和全局截断误差。并详细证明了广义欧拉方法的一致性、收敛性和稳定性。最后，数值例子，特别是在切换点的情况下，显示了灵活性和所提出的方法的能力。
英文摘要：
In many mathematical types of research, in order to solve the fuzzy fractional differential equations, we should transform these problems into crisp corresponding problems and by solving them the approximate solution can be obtained. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler's method is applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, the convergence and the stability of the generalized Euler's method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.
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