摘要
分数阶微分方程通过非局部性质精准捕捉系统的记忆性和历史依赖特性,在力学、生物化学及电气工程等领域突破传统整数阶微分方程框架局限,这使其在许多工程和科学领域越来越有吸引力。谱方法凭借其指数收敛精度与对分数阶导数非局部特性的天然适配性,在求解分数阶微分方程时,成为一种高效的数值求解工具。分数阶松弛方程作为分数阶微分方程的重要应用范例,常见于黏弹性阻尼材料、复杂介质以及多物理场耦合系统之中,用于描述具有长程记忆效应或多尺度动力行为的物理过程。本文针对分数阶松弛方程,提出一种具有创新性的基于多重区间划分的Legendre配置法。该方法的核心思路在于运用先进的数学理论与方法,将分数阶松弛方程转化为与之严格等价的Volterra积分方程形式,在此基础上,构造出用于高效近似求解分数阶松弛方程的数值解法。
Fractional-order differential equations(FDEs)uniquely capture the long-range memory effects and history-dependent dynamics of systems through their nonlocal operators,thereby addressing the limitations of classical integer-order differential equations in fields such as mechanics,biochemistry,and electrical engineering.This has established FDEs as a robust tool for modeling complex phenomena across engineering and scientific domains.Spectral methods,celebrated for their exponential convergence rates and natural alignment with the nonlocal nature of fractional derivatives,serve as highly efficient numerical solvers for FDEs.The fractional relaxation equation—a canonical model of FDEs in viscoelastic materials,heterogeneous media,and multiphysics-coupled systems—describes physical processes characterized by long-term memory or multiscale temporal behaviors.This paper presents a Legendre spectral collocation method based on multi-subinterval partitioning for the efficient numerical solution of fractional relaxation equations.Initially,the fractional relaxation equation is reformulated into an equivalent Volterra integral equation.Subsequently,by utilizing this transformed integral equation,an efficient numerical algorithm is developed to approximate the solution of the original equation.
作者
孙桂磊
SUN Guilei(Suzhou Institute of Technology,Jiangsu University of Science and Technology,Suzhou,Jiangsu 215600,China)
出处
《中国科技论文在线精品论文》
2025年第2期181-183,共3页
Highlights of Sciencepaper Online
关键词
计算数学
分数阶松弛微分方程
Legendre配置法
数值解
初值问题
computational mathematics
fractional relaxation differential equation
Legendre collocation method
numerical solution
Initial value problem
