两项线性分数阶微分方程的奇点分离分段配置法  

Singularity separation piecewise collocation method for two-term linear fractional differential equations

作　　者：李玉玉 廉欢 王同科[1] LI Yuyu;LIAN Huan;WANG Tongke(College of Mathematical Science,Tianjin Normal University,Tianjin 300387,China)

机构地区：[1]天津师范大学数学科学学院,天津300387

出　　处：《天津师范大学学报（自然科学版）》2022年第3期1-10,共10页Journal of Tianjin Normal University：Natural Science Edition

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

摘　　要：分数阶微分方程初值问题的典型特征是解在初始点非充分光滑,进而影响标准数值方法的计算精度.针对此问题,本文设计了一种高精度求解两项线性分数阶微分方程的奇点分离分段配置方法.首先,将方程转化为等价的两项Volterra积分方程,通过Picard迭代及符号运算求出解在零点的渐近展开式,其在零点附近具有很高的精度.其次,在包含零点的一个小区间上利用该级数展开式代替方程的解.在剩余区间上,利用Lagrange插值设计分段配置法求配置点处的数值解.对配置格式进行收敛性分析,得到了误差阶估计.最后,通过2个数值算例说明所提方法在全区间上具有最优逼近精度.The typical feature of the initial value problem for fractional differential equation is that the solution is not sufficiently smooth at the origin,which makes the standard numerical methods have very low accuracy.Aiming at this problem,a high-precision method is designed to solve two-term linear fractional differential equation by piecewise collocation method via singularity separation.Firstly,the equation is transformed into an equivalent two-term Volterra integral equation.The asymptotic expansion of the solution at the origin is obtained through Picard iteration and symbolic operation,which has high accuracy near the origin.Secondly,the solution is approximated by the obtained asymptotic expansion in a small subinterval including the origin.In the remaining interval,the equation is discretized using piecewise collocation method based on the Lagrange interpolation at the collocation points.The error analysis of the collocation scheme is conducted,and the convergence order is obtained.Finally,two numerical examples are provided to demonstrate that the proposed method has optimal approximate accuracy on the whole interval.

关 键 词：两项线性分数阶微分方程 VOLTERRA积分方程 渐近展开式 奇点分离 分段配置法 误差估计 

分 类 号：O241.81[理学—计算数学]