分片光滑边值问题的再生核方法

数学物理学报 ›› 2020, Vol. 40 ›› Issue (5): 1333-1340.

分片光滑边值问题的再生核方法

赵志红*(),林迎珍

^{北京理工大学珠海学院广东珠海 519088}

收稿日期:2018-11-21 出版日期:2020-10-26 发布日期:2020-11-04
通讯作者: 赵志红 E-mail:zhaozh39@163.com
基金资助:
广东省普通高校青年创新人才项目(2018KQNCX338);广东省普通高校特色创新项目(2019KTSCX217)

Reproducing Kernel Method for Piecewise Smooth Boundary Value Problems

Zhihong Zhao*(),Yingzhen Lin

^{School of Science, Zhuhai Campus, Beijing Institute of Technology, Guangdong Zhuhai 519088}

Received:2018-11-21 Online:2020-10-26 Published:2020-11-04
Contact: Zhihong Zhao E-mail:zhaozh39@163.com
Supported by:
the Young Innovative Talents Program in Universities and Colleges of Guangdong Province(2018KQNCX338);the Characteristic Innovative Scientific Research Project of Guangdong Province(2019KTSCX217)

摘要/Abstract

摘要：

一种高效的再生核算法（RKM）被提出用于解决分片光滑边值问题（BVP）.通过定义算子${\cal L}：{W_{2}^{3}[0, 1]}\rightarrow{L_{2}[0, 1]}$，应用再生核算法，解决了较为复杂的分片光滑边值问题.对定理的证明保证了算法理论的正确性，进而得到近似解$u_{n}（x）$以$O（h^2）$收敛于精确解.即：在范数$\left\|\displaystyle.\right\|_{W_2^{3}}$意义下$u_{n}（x）$有不低于二阶的收敛性.数值算例表明算法正确、简便、有效.

关键词: 再生核方法, 分片光滑, 收敛阶

Abstract:

In this paper, an efficient reproducing kernel method(RKM) is proposed for solving the piecewise smooth BVP. By defining an operator $ {\cal L}:{W_{2}^{3}[0, 1]}\rightarrow {L_{2}[0, 1]}$ and applying the reproducing kernel property, we have skillfully solved the complex piecewise smooth BVP. The theorems show the accuracy of the algorithm. Furthermore, we get that the approximate solution $u_{n}(x)$ convergence to the exact one with order $O(h^2)$. That is, $u_{n}(x)$ has second order convergence in the sense of norm $\left\|\displaystyle. \right\|_{W_2^{3}}$. Finally, some numerical experiments are given to illustrate the algorithm is accuracy, simple and efficient.

Key words: Reproducing kernel method, Piecewise smooth, Convergence order

中图分类号:

O241

赵志红,林迎珍. 分片光滑边值问题的再生核方法[J]. 数学物理学报, 2020, 40(5): 1333-1340.

Zhihong Zhao,Yingzhen Lin. Reproducing Kernel Method for Piecewise Smooth Boundary Value Problems[J]. Acta mathematica scientia,Series A, 2020, 40(5): 1333-1340.

图/表 3

参考文献 23

1	Mohammadi M , Mokhtari R . Solving the generalized regularized long wave equation on the basis of a reproducing kernel space. Journal of Computational and Applied Mathematics, 2011, 235 (14): 4003- 4014 doi: 10.1016/j.cam.2011.02.012
2	Doha E H , Abd-Elhameed W M , Youssri Y H . New algorithms for solving third- and fifth-order two point boundary value problems based on nonsymmetric generalized Jacobi method. Journal of Advanced Research, 2015, 6 (5): 673- 686
3	Dai D Q , Huang Y . Nonlocal boundary problems for a third-order one-dimensional nonlinear pseudoparabolic equation. Nonlinear Analysis, 2007, 66 (1): 179- 191
4	Lin Y Z , Lin J . A numerical algorithm for solving a class of linear nonlocal boundary value problems. Applied Mathematics Letters, 2010, 23 (9): 997- 1002 doi: 10.1016/j.aml.2010.04.025
5	Chen Z , Lin Y Z . The exact solution of a linear integral equation with weakly singular kernel. Journal of Mathematical Analysis and Applications, 2010, 344 (2): 726- 734
6	Niu J , Lin Y Z , Zhang C P . Approximate solution of nonlinear multi-point boundary value problem on the half-line. Mathematical Modelling and Analysis, 2012, 17 (2): 190- 202
7	Xu M Q , Lin Y Z . Simplified reproducing kernel method for fractional differential equations with delay. Appl Math Lett, 2016, 52, 156- 161 doi: 10.1016/j.aml.2015.09.004
8	Niu J , Sun L , Xu M Q , Hou J . A reproducing kernel method for solving heat conduction equations with delay. Appl Math Lett, 2020, 100, 106036 doi: 10.1016/j.aml.2019.106036
9	Zhou Y F , Cui M G , Lin Y Z . A computational method for nonlinear 2m-th order boundary value problems. Mathematical Modelling and Analysis, 2010, 15 (4): 571- 586
10	Niu J , Li P . Numerical algorithm for the third-order partial differential equation with three-point boundary value problem. Abstract and Applied Analysis, 2014, 2014, 1- 7
11	Yao H M , Lin Y Z . Solving singular boundary-value problems of higher even-order. Journal of Computational and Applied Mathematics, 2009, 223 (2): 703- 713
12	Mohammadi M , Mokhtari R . A new algorithm for solving nonlinear Schrodinger equation in the reproducing kernel space. Iranian Journal of Science and Technology, Transaction A, 2013, 37, 513- 526
13	Mokhtari R , Isfahani T , Mohammadi M . Solving a class of nonlinear differential-difference equations in the reproducing kernel space. Abstract and Applied Analysis, 2012, 2012, 1- 8
14	Andrzejczak G . Spline reproducing kernels on R and error bounds for piecewise smooth LBV problems. Applied Mathematics and Computation, 2018, 320 (C): 27- 44
15	Geng F Z . Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method. Applied Mathematics and Computation, 2009, 215 (6): 2095- 2102 doi: 10.1016/j.amc.2009.08.002
16	Geng F Z . A novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method. Applied Mathematics and Computation, 2011, 218 (8): 4211- 4215 doi: 10.1016/j.amc.2011.09.052
17	Geng F Z , Cui M G . A reproducing kernel method for solving nonlocal fractional boundary value problems. Applied Mathematics Letters, 2012, 25 (5): 818- 823 doi: 10.1016/j.aml.2011.10.025
18	Lin Y Z , Niu J , Cui M G . A numerical solution to nonlinear second order three-point boundary value problems in the reproducing kernel space. Applied Mathematics and Computation, 2012, 218 (14): 7362- 7368 doi: 10.1016/j.amc.2011.11.009
19	Zhang R M , Lin Y Z . A novel method for nonlinear boundary value problems. Journal of Computational and Applied Mathematics, 2015, 282, 77- 82 doi: 10.1016/j.cam.2014.12.035
20	赵志红, 徐敏强, 林迎珍. 基于再生核三次样条基底的构造及其应用. 数学的实践与认识, 2017, 47 (6): 274- 278
	Zhao Z H , Xu M Q , Lin Y Z . Structure and application of cubic spline basement based on reproducing kernel. Mathematics in Practice and Theory, 2017, 47 (6): 274- 278
21	Zhao Z H , Lin Y Z , Niu J . Convergence order of the reproducing kernel method for solving boundary value problems. Mathematical Modelling and Analysis, 2016, 21 (4): 466- 477
22	Xu M Q , Zhao Z H , Niu J , et al. A simplified reproducing kernel method for 1-D elliptic type interface problems. Journal of Computational and Applied Mathematics, 2019, 351, 29- 40 doi: 10.1016/j.cam.2018.10.027
23	吴勃英, 林迎珍. 应用型再生核空间. 北京: 科学出版社, 2012
	Wu B Y , Lin Y Z . Applied Reproducing Kernel Space. Beijing: Science Press, 2012

$ n$	$\|u_{n}(x)-u(x)\|_{\max}^{[14]}$	$\|u_{n}(x)-u(x)\|_{\max}$	$\|u'_{n}(x)-u'(x)\|_{\max}$	$\|u''_{n}(x)-u''(x)\|_{\max}$
$ 10$	$4.0\times10^{-4}$	$5.877\times10^{-5}$	$ 2.867\times10^{-4}$	$1.263\times10^{-5}$
$ 100$	$4.0\times10^{-6}$	$5.894\times10^{-7}$	$ 2.868\times10^{-6}$	$1.260\times10^{-7}$
$ 1000$	$/$	$5.893\times10^{-9}$	$ 2.867\times10^{-8}$	$1.263\times10^{-9}$
阶	$/$	$2.000$	$ 2.000$	$2.000$

$ n$	$\|u_{n}(x)-u(x)\|_{\max}$	$\|u'_{n}(x)-u'(x)\|_{\max}$	$\|u''_{n}(x)-u''(x)\|_{\max}$
$ 100$	$1.868\times10^{-6}$	$ 1.051\times10^{-5}$	$1.839\times10^{-5}$
$ 1000$	$1.868\times10^{-8}$	$ 1.051\times10^{-7}$	$1.839\times10^{-7}$
阶	$2.000$	$ 2.000$	$2.000$

[1]	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.
[2]	韩传峰, 张超 . 用复参数迭代法解不适定自共轭紧算子方程[J]. 数学物理学报, 2011, 31(4): 887-891.
[3]	袁学刚王敏. 关于S.N.Bernstein问题的新研究[J]. 数学物理学报, 2000, 20(2): 256-260.