变阶分数阶微分方程的数值解法综述

近年来,分数阶微积分作为一种工具已经被广泛应用于工程中的各个领域.较常阶分数阶微积分算子而言,变阶分数阶微积分算子能够更加准确地描述复杂系统的物理特性,变阶分数阶微积分建模作为一个强大的数学工具,为工程建模提供了便利.在前人优秀研究成果的基础上,结合近几年的国内外相关学者的研究成果对变分数阶微积分方程的研究作全面的综述.以变阶分数阶微分方程、变阶时间分数阶对流-扩散方程、变阶分数阶反应-扩散方程、变阶分数阶积分-微分方程和时滞变阶分数阶微分方程为主要研究对象,从变分数阶微积分算子的相关定义、模型、数值解及在工程中的相关应用等几个方面进行介绍.研究发现,近年来的算法多集中在多项式算法的基础上,通过构建不同的运算矩阵来实现变阶微分方程到代数方程组的转换.该综述可为相关领域的研究学者提供参考.

A review of numerical solutions of variable-order fractional differential equations

In recent years, fractional calculus as a tool has been widely used in various fields of engineering. Compared with the constant order fractional calculus operator, the variable order fractional calculus operator can describe the physical properties of complex systems more accurately. As a powerful mathematical tool, the variable order fractional calculus modeling provides convenience for engineering modeling. On the basis of previous excellent research results, combined with the research results of domestic and foreign scholars in recent years, this paper makes a comprehensive review of the research of variable fractional calculus equations. The variable-order fractional differential equations(VO-FDEs), the variable-order time fractional convection-diffusion equations(VOTFC-DEs), the variable-order fractional reaction-diffusion equations(VOTFR-DEs), the variable-order fractional integral-differential equations(VO-FIDEs), the variable-order delay fractional differential equations(VO-DFDEs) and the variable-order fractional optimal control problems(V-FOCPs) are taken as the main research objects, which are introduced from several aspects of variable fractional calculus operator related definitions, models, numerical solutions and applications in engineering. It is found that most of the algorithms in recent years are based on polynomial algorithms, which can realize the transformation of variable order differential equations to algebraic equations by constructing different operation matrices. This review can provide reference for researchers in related fields.