解带有扰动数据的第一类Volterra积分方程的谱正则化方法

Abstract

Abstract:

The main purpose of this work is to solve the Volterra integral equation of the first kind with noise data by using both a Legendre-collocation method and a regularization strategy. We provide a rigorous convergence analysis for the proposed method. Some numerical tests are illustrated to demonstrate the validity and effectiveness of the proposed method.

Key words: Spectral method, Regularization strategy, Volterra integral equation, Convergence analysis

CLC Number:

O241.83

Lixin Feng,Xiaoxu Yang. Spectral Regularization Method for Volterra Integral Equation of the First Kind with Noise Data[J].Acta mathematica scientia,Series A, 2020, 40(3): 650-661.

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References 16

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$N$	6	8	10	12	14	16
${\rm RMSE}$	$ 1.51\times 10^{-14}$	$ 2.53\times 10^{-14}$	$ 1.85\times 10^{-13}$	$ 4.73\times 10^{-13}$	$ 2.72\times 10^{-13}$	$ 7.91\times 10^{-12}$

$N$	4	6	8	10	12	14	16
${\rm RMSE}$	$ 0.0264$	$ 0.0065$	$0.0597$	$ 0.0823$	$ 0.0897$	$ 0.0930$	$ 0.0961$

$\alpha$	0.1	0.01	0.001	0.0001	0.00001	0
${ N=4}$	$ 0.2094$	$ 0.0413$	$ 0.0032$	$0.0004$	$0.0012$	$0.0014$
${ N=6}$	$ 0.2715$	$ 0.0679$	$ 0.0090$	$0.0032$	$0.0007$	$0.0013$
$ N=8$	$ 0.2927$	$ 0.1036$	$ 0.0078$	$0.0055$	$0.0024$	$0.0031$

$N$	4	6	8	10	12	14	16
${\rm RMSE}$	$ 10.245 $	$ 1.349 $	$ 0.099$	$ 0.005 $	$ 1.42\times 10^{-4}$	$ 3.25\times 10^{-6}$	$ 5.64\times 10^{-8}$

$\alpha$	0.1	0.01	0.001	0.0001	0.00001	0
${ N=8}$	$ 6.302$	$ 0.744$	$ 0.160$	$0.100$	$0.053$	$0.101$
${ N=10}$	$ 6.264$	$ 0.667$	$ 0.065$	$0.013$	$0.007$	$0.002$
${ N=12}$	$ 6.262$	$ 0.658$	$ 0.066$	$0.006$	$0.004$	$0.003$

Spectral Regularization Method for Volterra Integral Equation of the First Kind with Noise Data

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