线性分数阶两点边值问题的奇点分离Chebyshev配置方法  

CHEBYSHEV COLLOCATION METHOD WITH SINGULARITY SEPARATION FOR LINEAR FRACTIONAL TWO-POINT BOUNDARY VALUE PROBLEMS

作　　者：娄汝馨 廉欢 王同科[1] Lou Ruxin;Lian Huan;Wang Tongke(School of Mathematical Sciences,Tianjin Normal University,Tianjin 300387)

机构地区：[1]天津师范大学数学科学学院,天津300387

出　　处：《高等学校计算数学学报》2024年第1期23-54,共32页Numerical Mathematics A Journal of Chinese Universities

基　　金：国家自然科学基金项目(11971241);天津市高等学校创新团队培养计划(TD13-5078)项目。

摘　　要：1引言随着科学技术的发展,许多工程计算研究表明利用非整数阶(分数阶)微分方程可以更准确地描述一些实际问题[1],因此越来越多的学者对分数阶微分方程(FDE)展开了系统的研究.本文主要研究分数阶两点边值问题解的奇异展开与高精度配置方法.分数阶导数主要包括Riemann-Liouville(RL)分数阶导数和Caputo分数阶导数.The fractional two-point boundary value problem is transformed into a Volterra-Fredholm integral equation.The truncated fractional series expansion with a parameter for the solution at the singularity is obtained by Picard iteration,from which the singular behavior of the solution is accurately described.Based on the series expansion,the interval is splitted into a singular subinterval and a regular subinterval.On the singular subinterval,the series expansion is used to approximate the solution,and on the regular subinterval an augmented Chebyshev collocation method is designed to simultaneously obtain the numerical solution and the undetermined parameter in the series expansion.The optimal convergence order of the collocation solution with respect to maximum norm is strictly proved.Numerical examples confirm the correctness of the series expansion and the high accuracy of the Chebyshev collocation method.

关 键 词：CAPUTO分数阶导数 两点边值问题 奇点 分数阶微分方程 配置方法 LIOUVILLE RIEMANN 

分 类 号：O248.1[理学—计算数学]