Variable order fractional grey model and its application

• Published in: Applied Mathematical Modelling Vol. 97; pp. 619 - 635
• Main Authors: Yuxiao, Kang, Shuhua, Mao, Yonghong, Zhang
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.09.2021; Elsevier BV
• Subjects: Accumulation; Algorithms; Complex systems; Derivatives; Differential equations; Evolution; Exact solutions; Fractional derivative; Laplace transforms; Parameter estimation; Particle swarm optimization; Quantum particle swarm optimization; Variable order accumulation generation; Variable order grey model; Quantum particle swarm optimization; Variable order grey model; Variable order accumulation generation; Fractional derivative
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2021.03.059

•A variable order fractional accumulation generating operator is proposed.•A variable order derivative is introduced into a grey model.•The effect of the new model is better than that of similar competitive models.•The new model can be used to predict problems with variable order characteristics. 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.