The use of constant order differential equations to describe the evolution of complex systems is often unable to describe some of the changing characteristics of the systems accurately. Variable order fractional derivatives provide us with new tools to solve such problems. In this paper, the accumulation and derivative orders of the classic grey model are expanded from constants to functions, and a variable order fractional grey model is established to describe the evolution process of complex systems. Firstly, this paper defines the variable order fractional accumulation generation sequence. On the basis of this sequence, a variable order fractional derivative grey model is established, the parameters of the model are estimated using the least square method, and the quantum particle swarm optimization algorithm is used to solve the order of fractional derivative and accumulation. Sadik transform and Laplace transform are adopted to obtain the analytical solution of the new model. Lastly, the effectiveness of the new model is verified through four cases. Compared with other models, the variable order fractional model can describe the development process of complex systems more accurately.

使用常数阶微分方程来描述复杂系统的演化通常无法准确描述系统的某些变化特征。可变阶分数导数为我们提供了解决此类问题的新工具。本文将经典灰色模型的累积和导数阶数从常数扩展为函数，并建立了一个可变阶分数阶灰色模型来描述复杂系统的演化过程。首先，本文定义了可变阶分数累积生成序列。在此序列的基础上，建立了可变阶分数阶导数灰色模型，并使用最小二乘法估算了模型的参数，并采用量子粒子群优化算法求解分数阶导数和累积的阶数。采用萨迪克变换和拉普拉斯变换获得新模型的解析解。最后，通过四个案例验证了新模型的有效性。与其他模型相比，可变阶分数模型可以更准确地描述复杂系统的开发过程。