再生核移位勒让德基函数法求解分数阶微分方程

Abstract

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

CLC Number:

O241.8

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.

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Figures/Tables 5

References 30

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	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

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

		$\frac{6}{7}$	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

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[2]	Marko Kostić, Li Chenggang, Li Miao. Abstract Multi-Term Fractional Differential Equations with Riemann-Liouville Derivatives [J]. Acta mathematica scientia,Series A, 2016, 36(4): 601-622.
[3]	SHU Xiao-Bao, DAI Bin-Xiang. S-asymptotically ω-periodic Solutions of Semi-linear Neutral Fractional Differential Equations [J]. Acta mathematica scientia,Series A, 2014, 34(1): 16-26.

Reproducing Kernel Shifted Legendre Basis Function Method for Solving the Fractional Differential Equations

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