Sturm-Liouville问题的特征值与特征函数的渐近公式

doi:10.6040/j.issn.1671-9352.0.2019.626

《山东大学学报(理学版)》 ›› 2020, Vol. 55 ›› Issue (2): 57-62.doi: 10.6040/j.issn.1671-9352.0.2019.626

• • 上一篇

Sturm-Liouville问题的特征值与特征函数的渐近公式

冉茂军,高承华^*

西北师范大学数学与统计学院, 甘肃兰州 730070

发布日期:2020-02-14
作者简介:冉茂军(1994— ),男,硕士研究生,研究方向为常微分方程及其应用. E-mail:rmj2521122@163.com*通信作者简介:高承华(1983— ),男,教授,研究方向为常微分方程及其应用. E-mail:gaokuguo@163.com
基金资助:
国家自然科学基金资助项目(11961060);甘肃省自然科学基金资助项目(18JR3RA084)

Asymptotic formula for eigenvalues and eigenfunctions of Sturm-Liouville question

RAN Mao-jun, GAO Cheng-hua^*

College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, Gansu, China

Published:2020-02-14

摘要/Abstract

摘要： 根据经典的分析方法,研究了边界条件依赖非线性特征参数的四阶Sturm-Liouville问题的特征值与特征函数的渐近公式。通过常微分方程的常数变易法及特征函数零点渐近估计,获得了边界条件含有非线性特征参数的特征值与特征函数的渐近公式。

关键词: Sturm-Liouville问题, 边界条件依赖特征参数, 渐近公式

Abstract: According to the classical analysis method, the asymptotic formula of eigenvalues and eigenfunctions of Sturm-Liouville question with boundary conditions dependent on nonlinear eigenparameters are studied. By means of constant variation method of ordinary differential equation and asymptotic estimation of eigenfunction zero, the asymptotic formula of eigenvalues and eigenfunctions of boundary conditions with nonlinear eigenparameters are obtained.

Key words: Sturm-Liouville question, eigenparameter-dependent boundary condition, asymptotic formula

中图分类号:

O175.8

冉茂军,高承华. Sturm-Liouville问题的特征值与特征函数的渐近公式[J]. 《山东大学学报(理学版)》, 2020, 55(2): 57-62.

RAN Mao-jun, GAO Cheng-hua. Asymptotic formula for eigenvalues and eigenfunctions of Sturm-Liouville question[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2020, 55(2): 57-62.

参考文献

[1] FULTON C, PRUESS S. Numerical methods for a singular eigenvalue problem with eigenparameter in the boundary conditions[J]. Journal of Mathematical Analysis and Applications, 1979, 71(2):431-462.
[2] KAPUSTIN N. On the basis property of the system of eigenfunctions of a problem with squared spectral parameter in a boundary condition[J]. Differential Equations, 2015, 51(10):1274-1279.
[3] BINDING P, PATRICK B, SEDDIGHI K. Sturm-Liouville problems with eigenparameter dependent boundary conditions[J]. Proceedings of the Edinburgh Mathematical Society, 1993, 37(1):57-72.
[4] 牡丹, 孙炯, 姚斯琴. 具有转移条件的两个区间Sturm-Liouville算子的Green函数和渐近展开[J]. 应用数学, 2014, 27(3):658-672. MU Dan, SUN Jiong, YAO Siqin. Asymptotic behaviors and Green's function of two-interval Sturm-Liouville problems with transmission conditions[J]. Mathematica Applicata, 2014, 27(3):658-672.
[5] 杨秋霞, 王万义. 一类正则Sturm-Liouville问题特征的渐近分析[J]. 应用泛函分析学报, 2009, 11(4):301-308. YANG Qiuxia, WANG Wanyi. Asymptotic behavior of a regular Sturm-Liouville problem[J]. Acta Analysis Functionalis Applicata, 2009, 11(4):301-308.
[6] 张益宁, 孙炯, 李昆,等. 一类具有转移边界条件的Sturm-Liouville问题的特征值的渐近估计[J]. 数学的实践与认识, 2018, 48(6):263-270. ZHANG Yining, SUN Jiong, LI Kun, et al. Asymptotic formulars of eigenvalues for Sturm-Liouville problem with transmission conditions[J]. Mathematics in Practice and Theory, 2018, 48(6):263-270.
[7] BANKS D, KUROWSKI G. A Prüfer transformation for the equation of a vibrating beam subject to axial forces[J]. Journal of Differential Equations, 1977, 24(1):57-74.
[8] KERIMOV N, ALIYEV Z. On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in a boundary condition[J]. Differential Equations, 2007, 43(7):905-915.
[9] AMARA J. Fourth-order spectral problem with eigenvalue in the boundary conditions[J]. Functional Analysis and Its Applications, 2004, 197(2):49-58.
[10] ALIYEV Z, KERIMOV N. Spectral properties of the differential operators of the fourth-order with eigenvalue parameter dependent boundary condition[J]. International Journal of Mathematics and Mathematical Sciences, 2012, 7(2):1-28.
[11] ALIYEV Z. Basis properties in L_P of systems of root functions of a spectral problem with spectral parameter in a boundary condition[J]. Differential Equations, 2011, 47(6):766-777.
[12] GAO Chenghua, LI Xiaolong, MA Ruyun. Eigenvalues of a linear fourth-order differential operator with squared spectral parameter in a boundary condition[J]. Mediterranean Journal of Mathematics, 2018, 15(1):1-14.
[13] 曹之江.常微分算子[M]. 上海:上海科学技术出版社, 1987. CAO Zhijiang. Ordinary differential operator[M]. Shanghai: Shanghai Scientific & Technical Publishers, 1987.
[14] 刘景麟.常微分算子谱论[M]. 北京:科学出版社, 2009. LIU Jinglin. Spectral theory of ordinary differential operators[M]. Beijing: Science Press, 2009.
[15] NAIMARK M. Linear differential operators[M]. Moscow: Nauka, 1969.

多维度评价

Viewed

Full text

Abstract

Cited

Shared

Discussed

Sturm-Liouville问题的特征值与特征函数的渐近公式

Asymptotic formula for eigenvalues and eigenfunctions of Sturm-Liouville question

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 5

多维度评价

本文评价

推荐阅读 0

[1]	郭汝廷,王明强. 一个求和公式中的几个常数[J]. 山东大学学报（理学版）, 2017, 52(4): 68-71.
[2]	赵娜. 时标上Sturm-Liouville问题的有限谱[J]. J4, 2013, 48(09): 96-102.
[3]	赵月英,宋晓秋*. 指数有界C余弦算子函数的概率型表示[J]. J4, 2010, 45(4): 77-81.
[4]	王晓瑛1，张瑾1,2. 一类算数函数的均值估计[J]. J4, 2009, 44(12): 14-16.
[5]	张德瑜,翟文广 . 关于整数n的k次补数[J]. J4, 2006, 41(5): 4-07 .