[1]林府标,马丽荣.KdV型粘性分数阶方程的数值解[J].南京师范大学学报(自然科学版),2020,43(03):28-33.[doi:10.3969/j.issn.1001-4616.2020.03.006]
　Lin Fubiao,Ma Lirong.Numerical Methods for KdV Type Fractional Order Equationwith a Nonlocal Viscous Term[J].Journal of Nanjing Normal University(Natural Science Edition),2020,43(03):28-33.[doi:10.3969/j.issn.1001-4616.2020.03.006]
点击复制

KdV型粘性分数阶方程的数值解()
分享到：

《南京师范大学学报》（自然科学版）[ISSN:1001-4616/CN:32-1239/N]

卷:
第43卷
期数:
2020年03期
页码:
28-33
栏目:
·数学·
出版日期:
2020-09-30

文章信息/Info

Title:
Numerical Methods for KdV Type Fractional Order Equationwith a Nonlocal Viscous Term
文章编号:
1001-4616(2020)03-0028-06
作者:
林府标1; 马丽荣2
(1.贵州财经大学数统学院,贵州 贵阳 550025)(2.石家庄邮电职业技术学院会计系,河北 石家庄 050000)
Author(s):
Lin Fubiao1; Ma Lirong2
(1.School of Mathematics and Statistics,Guizhou University of Finance and Economics,Guiyang 550025,China)(2.Accounting Department,Shijiazhuang Vocational and Technical College of Posts and Telecommunications,Shijiazhuang 050000,China)
关键词:
分数阶方程; 稳定性; 谱方法; 衰减率
Keywords:
fractional equation; stability; spectral method; decay rates
分类号:
O156.5
DOI:
10.3969/j.issn.1001-4616.2020.03.006
文献标志码:
A
摘要:
构造了一种求解KdV型粘性分数阶方程的数值格式,分析了格式的稳定性,证明了格式都是无条件稳定的. 数值结果验证了方程中存在分数阶项时,时间方向是1.5阶,不存在分数阶项时,时间方向是2阶. 最后用数值例子讨论这两类方程解的长时间衰减率,并讨论了不同参数对解的衰减率的影响. 数值例子表明,这方程的衰减率是:L2范数接近-0.25; L∞范数接近-0.5,这与已知的理论结果是吻合的.
Abstract:
We turn to study the numerical solution of the Fractional order equation with a nonlocal viscous term. We propose a numerical scheme to solve this equation. A detailed analysis is carried out for this scheme,and we prove that the scheme is unconditionally stable. The numerical results verify that the fractional order equation with a nonlocal viscous term is of order 1.5,when a nonlocal viscous term does not exist,the scheme is of order 2. At last,we use the proposed methods to investigate the asymptotical decay rate of the solutions to fractional order equation with a nonlocal viscous term. We equally discuss the role of the diffusion terms,the geometric dispersion and the nonlinearity respectively. The performed numerical experiment confirms that the decay rates in L2-norm,L∞-norm,and are very close to -0.25,and -0.5 respectively. These numerical results are consistent with the known theoretical prediction.

参考文献/References:

[1] KAKUTANI T,MATSUUCHI K.Effect of viscosity on long gravity waves[J]. Journal of the physical society of Japan,1975,39(39):237-246.
[2]LIU P L F,ORFILA A.Viscous effects on transient long-wave propagation[J]. Journal of fluid mechanics,2004,520(1):83-92.
[3]SAUT J C,BONA J L,CHEN M.Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I:derivation and linear theory[J]. Journal of nonlinear science,2002,12(4):283-318.
[4]DUTYKH D,DIAS F.Viscous potential free-surface flows in fluid layer of finite depth[J]. Comptes rendus mathematique,2007,345(2):113-118.
[5]DUTYKH D.Visco-potential free-surface flows and long wave modelling[J]. European journal of mechanics-B/fluids,2009,28(3):430-443.
[6]CHEN M,DUMONT S,DUPAIGNE L,et al. Decay of solutions to a water wave model with a nonlocal viscous dispersive term[J]. Discrete and continuous dynamical systems,2010,27(4):1473-1492.
[7]CHEN M.Numerical investigation of a two-dimensional Boussinesq system[J]. Discrete and continuous dynamical systems,2009,28(4):1169-1190.
[8]GOUBET O,WARNAULT G.Decay of solutions to a linear viscous asymptotic model for water waves[J]. Chinese annals of mathematics-series B,2010,31(6):841-85.
[9]DUMONT S,DUVAL J B.Numerical investigation of the decay rate of solutions to models for water waves with nonlocal viscosity[J]. International journal of numerical analysis and modeling,2012,10(2):333-349.
[10]ZHANG J,XU C. Finite difference/spectral approximations to a water wave model with a nonlocal viscous term[J]. Applied mathematical modelling,2014,38(19/20):4912-4925.
[11]LIN Y M,XU C J.Finite difference spectral approximations for the time-fractional diffusion equation[J]. Journal of computational physics,2007,225(2):1533-1552.

相似文献/References:

[1]幸玲,刘宣亮.具饱和传染率的一类捕食者-食饵模型的分析[J].南京师范大学学报(自然科学版),2011,34(03):25.
　Xing Ling,Liu Xuanliang.Analysis of a Predatro-Prey Model With a Saturated Infection Rate[J].Journal of Nanjing Normal University(Natural Science Edition),2011,34(03):25.
[2]方玲玲,齐龙兴.一类SEIRS模型稳定性分析(英文)[J].南京师范大学学报(自然科学版),2013,36(03):21.
　Fang Lingling,Qi Longxing.The Stability Analysis of an SEIRS Model[J].Journal of Nanjing Normal University(Natural Science Edition),2013,36(03):21.
[3]崔文文,王 成,王小亮,等.平面BnBe(n=1～7)团簇的结构和稳定性理论研究[J].南京师范大学学报(自然科学版),2013,36(03):61.
　Cui Wenwen,Wang Cheng,Wang Xiaoliang,et al.A Theoretical Study on Structure and Stability of Planar BnBe(n=1～7)[J].Journal of Nanjing Normal University(Natural Science Edition),2013,36(03):61.
[4]王 成,崔文文,邵景玲,等.Cn B4( n = 2 ~ 7) 团簇的结构、稳定性和成键性质[J].南京师范大学学报(自然科学版),2014,37(02):79.
　Wang Cheng,Cui Wenwen,Shao Jingling,et al.Structure,Stability and Bonding Nature of Cn B4( n = 2 ~ 7)Clusters[J].Journal of Nanjing Normal University(Natural Science Edition),2014,37(03):79.
[5]王小亮,杨广丽,崔文文,等.平面BnBe3(n=1～6)团簇的结构和稳定性的理论研究[J].南京师范大学学报(自然科学版),2014,37(03):78.
　Wang Xiaoliang,Yang Guangli,Cui Wenwen,et al.A Theoretical Study on Structure and Stability ofPlanar BnBe3(n=1-6)[J].Journal of Nanjing Normal University(Natural Science Edition),2014,37(03):78.
[6]颜晓光,徐爱民.关于Ding投射范畴的稳定性[J].南京师范大学学报(自然科学版),2015,38(01):25.
　Yan Xiaoguang,Xu Aimin.On Stability of Ding Projective Categories[J].Journal of Nanjing Normal University(Natural Science Edition),2015,38(03):25.
[7]岳瑞英,杨广丽,朱小蕾.平面BnCP(n=1-6)团簇的结构和稳定性的理论研究[J].南京师范大学学报(自然科学版),2016,39(04):0.[doi:10.3969/j.issn.1001-4616.2016.04.021]
　Yue Ruiying,Yang Guangli,Zhu Xiaolei.A Theoretical Study on Structure and Stability of Planar BnCP(n=1-6)[J].Journal of Nanjing Normal University(Natural Science Edition),2016,39(03):0.[doi:10.3969/j.issn.1001-4616.2016.04.021]
[8]白 婵,万 辉.一个传染病模型中的后向分支问题（英文）[J].南京师范大学学报(自然科学版),2017,40(03):5.[doi:10.3969/j.issn.1001-4616.2017.03.002]
　Bai Chan,Wan Hui.Backward Bifurcation in an Epidemic Model[J].Journal of Nanjing Normal University(Natural Science Edition),2017,40(03):5.[doi:10.3969/j.issn.1001-4616.2017.03.002]
[9]王 丽,岳瑞英,朱小蕾.平面BnCPt(n=1-5)团簇的结构、稳定性和成键性质的密度泛函理论研究[J].南京师范大学学报(自然科学版),2017,40(04):74.[doi:10.3969/j.issn.1001-4616.2017.04.012]
　Wang Li,Yue Ruiying,Zhu Xiaolei.Study of Structure,Stability,and Bond Nature of Planar BnCPt(n=1-5)Clusters Based on Density Functional Theory[J].Journal of Nanjing Normal University(Natural Science Edition),2017,40(03):74.[doi:10.3969/j.issn.1001-4616.2017.04.012]
[10]徐 茜,赵 烨,杨玉洁.Lotka-Volterra交错扩散方程组平衡解的局部渐近稳定性[J].南京师范大学学报(自然科学版),2018,41(04):7.[doi:10.3969/j.issn.1001-4616.2018.04.002]
　Xu Qian,Zhao Ye,Yang Yujie.Stability of Steady State Solution for a Lotka-VolterraModel with Cross Diffusion[J].Journal of Nanjing Normal University(Natural Science Edition),2018,41(03):7.[doi:10.3969/j.issn.1001-4616.2018.04.002]

备注/Memo

备注/Memo:
收稿日期:2019-05-21.
基金项目:贵州省科技计划基金项目(黔科合基础[2019]1051)、贵州省教育厅青年科技人才成长项目(黔教合KY字[2017]150)、2018年度贵州财经大学校级科研项目资助(2018XYB04)、贵州财经大学创新探索及学术新苗项目(黔科合平台人才[2017]5736-020).
通讯作者:林府标,博士,副教授,研究方向:计算数学,应用数学. E-mail:linfubiao0851@163.com
更新日期/Last Update: 2020-09-15