解带有扰动数据的第一类Volterra积分方程的谱正则化方法

数学物理学报 ›› 2020, Vol. 40 ›› Issue (3): 650-661.

解带有扰动数据的第一类Volterra积分方程的谱正则化方法

冯立新^1,^2,*(),杨晓旭²

¹ 黑龙江大学数学科学学院哈尔滨 150080
² 黑龙江省复杂系统理论与计算重点实验室哈尔滨 150080

收稿日期:2019-01-15 出版日期:2020-06-26 发布日期:2020-07-15
通讯作者: 冯立新 E-mail:fenglixin@hlju.edu.cn
基金资助:
国家自然科学基金(11871198);黑龙江省高校基础研究基金黑龙江大学专项基金项目(RCYJTD201804)

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

Lixin Feng^1,^2,*(),Xiaoxu Yang²

¹ Department of Mathematics, Heilongjiang University, Harbin 150080
² Heilongjiang Provincial Key Laboratory of the Theory and Computation of Complex Systems, Harbin 150080

Received:2019-01-15 Online:2020-06-26 Published:2020-07-15
Contact: Lixin Feng E-mail:fenglixin@hlju.edu.cn
Supported by:
NSFC(11871198);the Fundamental Research Funds for the Universities of Heilongjiang Province Heilongjiang University Special Fund Project(RCYJTD201804)

摘要/Abstract

摘要：

该文的主要目的是通过使用Legendre配置方法和正则化策略来求解带有噪声数据的第一类Volterra积分方程，并给出该方法收敛性分析的严格数学证明.数值实验表明了该方法的有效性.

关键词: 谱方法, 正则化策略, Volterra积分方程, 收敛分析

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

中图分类号:

O241.83

冯立新,杨晓旭. 解带有扰动数据的第一类Volterra积分方程的谱正则化方法[J]. 数学物理学报, 2020, 40(3): 650-661.

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.

图/表 5

表 1

表 2

表 3

表 4

表 5

参考文献 16

1	Brunner H . A note on collocation methods for Volterra integral equations of the first kind. Computing, 1979, 23, 179- 187 doi: 10.1007/BF02252096
2	Brunner H . Collocation Methods for Volterra Integral and Related Functional Quations Methods. Cambridge: Cambridge University Press, 2004
3	Brunner H , Kauthen J P . The numerical solution of two-dimensional Volterra integral equation. IMA J Numer Anal, 1989, 9, 45- 59
4	Brunner H , Tang T . Polynomial spline collocation methods for the nonlinear Basset equation. Comput Math, 1989, 18, 449- 457
5	Canuto C , Hussaini M Y , Quarteroni A , Zang T A . Spectral Methods:Fundamentals in Single Domains. New York: Springer-Verlag, 2006
6	Delves L M , Mohanmed J L . Computational Methods for Integral Equations. Cambridge: Cambridge University Press, 1985
7	Elnagar C N , Kazemi M . Chebyshev spectral solution of nonlinear Volterra-Hammerstein integral equations. J Comput, 1996, 76, 147- 158
8	Frank D H , Richard W . High methods for Volterra integral equations of the first kind. Siam J Numer Anal, 1973, 10, 647- 664 doi: 10.1137/0710057
9	Guo B , Wang L . Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J Approx Theory, 2004, 128, 1- 41 doi: 10.1016/j.jat.2004.03.008
10	Holyhead P A W , Mckee S , Taylor P J . Multistep methods for solving linear Voltrra integral equations of the first kind. Siam J Numer Anal, 1975, 12, 698- 711 doi: 10.1137/0712052
11	Kauthen J P , Brunner H . Continuous collocation approximations to solutions of first kind Volterra equations. Mathematics of Computation, 1997, 66, 1441- 1459 doi: 10.1090/S0025-5718-97-00905-8
12	Mckee S , Tang T , Diogo T . An Euler-type method for two-dimensional Volterra integral equations of the first kind. IMA J Numer Anal, 2000, 20, 423- 440 doi: 10.1093/imanum/20.3.423
13	Qu C K , Wong R . Szego's conjecture on Lebesgue constants for Legendre series. Pacific J Math, 1998, 135, 157- 188
14	Shen J , Tang T . Spectral and High-Order Methods with Applications. Beijing: Science Press, 2006
15	Tian H C. Spectral Method for Volterra Integral Equation[D]. Vancouver: Simon Fraser University, 1995
16	Tang T , Xu X , Cheng J . On spectral methods for Volterra integral equations and the convergence analysis. J Compu Math, 2008, 26, 825- 837

$N$	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$	$0.006$	$0.004$	$0.003$