分数阶弱奇异积分微分方程的多项式数值解法  

Polynomial method for solving the numerical solution of fractional order integro-differential equation with weakly singular kernel

作　　者：李志文[1] 尹建华[1] 耿万海[2] 

机构地区：[1]河北民族师范学院数学与计算机系,河北承德067000 [2]燕山大学理学院,河北秦皇岛066004

出　　处：《西北师范大学学报（自然科学版）》2016年第2期13-17,共5页Journal of Northwest Normal University(Natural Science)

基　　金：国家自然科学基金资助项目(61176089)

摘　　要：为了求分数阶变系数且带有弱奇异积分核Volterra-Fredholm积分微分方程的数值解,本文提出了Legendre多项式算子矩阵法,利用Legendre多项式的定义及其性质给出了分数阶微分算子矩阵,同时也给出了任意阶弱奇异积分的近似求积公式.通过简化所求分数阶积分微分方程,并离散化简后的方程,可将原问题转换为求代数方程组的解.收敛性分析证明了本文方法是收敛的,数值算例验证了该方法的有效性.In order to obtain the numerical solution of fractional order variable coefficients VolterraFredholm integro-differential equation with weakly singular kernels,an operational matrix method is presented in this paper.An approximate formula which solves solution of arbitrary order weakly singular integral is given by using the definition of Legendre polynomial and some properties.And an operational matrix of fractional derivatives of Legendre polynomial is also obtained.Then the original problem of the equation is changed into a system of algebraic equation through simplifying and descreting the fractional integro-differential equation.The convergence analysis proves that the method is convergent.The numerical examples show that the approach is effective.

关 键 词：分数阶微分 弱奇异 积分微分方程 LEGENDRE多项式 算子矩阵 数值解 

分 类 号：O241[理学—计算数学]