Variable order fractional grey model and its application

The use of constant order differential equations to describe the evolution of complex sys-tems is often unable to describe some of the changing characteristics of the systems accu-rately. Variable order fractional derivatives provide us with new tools to solve such prob-lems. 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 es-tablished to describe the evolution process of complex systems. Firstly, this paper defines the variable order fractional accumulation generation sequence. On the basis of this se-quence, 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 accumula-tion. 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 devel-opment process of complex systems more accurately. (c) 2021 Elsevier Inc. All rights reserved.