再生核移位勒让德基函数法求解分数阶微分方程

摘要/Abstract

图/表 5

参考文献 30

Metrics

摘要：

该文以再生核理论为基础，用移位Legendre多项式作为基函数构造了一个新的再生核空间，并给出了该空间下的再生核函数.与经典的再生核函数有所不同的是该空间下的再生核函数不再是分段函数，因此可以减小分数阶算子作用在核函数上时的计算量，使近似解更为精确.数值算例表明该方法的有效性.

关键词: 分数阶微分方程, 移位Legendre多项式, 再生核方法

Abstract:

Based on the theory of the reproducing kernel, this paper constructed a new reproducing kernel space and the reproducing kernel function of the space is given by using the shifted Legendre polynomials as the basis function. It is different from former that this function is no longer a piecewise function, so when the fractional operator is used on the kernel function, the computation is reduced. Thus the approximate solution is more accurate. Finally, the numerical examples illustrate the validity of the method.

Key words: Fractional differential equations, Shifted Legendre polynomials, Reproducing kernel method

中图分类号:

O241.8

巩全壹,么焕民. 再生核移位勒让德基函数法求解分数阶微分方程[J]. 数学物理学报, 2020, 40(2): 441-451.

Quanyi Gong,Huanmin Yao. Reproducing Kernel Shifted Legendre Basis Function Method for Solving the Fractional Differential Equations[J]. Acta mathematica scientia,Series A, 2020, 40(2): 441-451.

表 1

方程(7.1)中 $\alpha$ 取不同值的绝对误差"

	本文所取节点数	文献[5]中所取节点数
$n$	5	8
$\alpha$	本文中的误差	文献[5]中的误差
1.9	5.67888E-07	2.18E-03
1.7	9.49771E-08	2.17E-03
1.5	9.51231E-08	2.16E-03
> 1.2	5.07967E-07	2.16E-03
1.1	3.50476E-04	2.16E-03

表 1

表 2

求解方程(7.2), $n$ 取不同值的绝对误差"

x_n₁	$n_{1} = 6$	$x_{n_{2}}$	$n_{2} = 7$
$\frac{1}{6}$	3.56401E-08	$\frac{1}{7}$	1.44916E-08
$\frac{1}{3}$	3.99251E-08	$\frac{2}{7}$	1.57484E-08
$\frac{1}{2}$	4.35407E-08	$\frac{3}{7}$	1.68102E-08
$\frac{2}{3}$	4.64885E-08	$\frac{4}{7}$	1.76847E-08
$\frac{5}{6}$	4.87702E-08	$\frac{5}{7}$	1.83794E-08

表 2

表 3

方程(7.2)的绝对误差与文献[5]的比较结果"

		$\frac{6}{7}$	1.89021E-08
	本文所取节点数		文献[5]中所取节点数
$n$	6	7	16
绝对误差	4.31271E-08	1.70707E-08	3.07E-04

表 3

图 1

绝对误差 $\mid u_{5}(x)-u(x)\mid$ , $\mid u_{5}'(x)-u'(x)\mid$ "

图 1

图 2

绝对误差 $\mid u_{10}(x)-u(x)\mid$ , $\mid u_{10}'(x)-u'(x)\mid$ "

图 2

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