一类奇异摄动Fredholm型积分微分方程的二阶自适应移动网格方法

doi:10.13438/j.cnki.jdzk.2024.04.004

吉首大学学报（自然科学版） ›› 2024, Vol. 45 ›› Issue (4): 15-25.DOI: 10.13438/j.cnki.jdzk.2024.04.004

一类奇异摄动Fredholm型积分微分方程的二阶自适应移动网格方法

罗丹,毛志

(1.吉首大学数学与统计学院,湖南吉首 416000;2.铜仁学院大数据学院,贵州铜仁 554300)

出版日期:2024-07-25 发布日期:2024-07-23
作者简介:罗丹(1997—),女,湖北十堰人,吉首大学数学与统计学院硕士生,主要从事微分方程数值解研究
基金资助:
贵州省第六届高层人才创新项目(2022-(2020)-040)

The Second Order Adaptive Moving Mesh Method for a Singularly Perturbed Fredholm Integro-Differential Equation

LUO Dan,MAO Zhi

(1.School of Mathematics and Statistics,Jishou University,Jishou 416100,Hunan China;2.School of Data Science,Tongren University,Tongren 554300,Guizhou China)

Online:2024-07-25 Published:2024-07-23

摘要/Abstract

摘要：利用基函数方法和复合梯形公式构建了指数拟合有限差分格式,用该有限差分格式对一类奇异摄动Fredholm型积分微分方程进行了离散.用推导出的数值解的截断误差估计,并结合边界估计和稳定性分析,证明了自适应移动网格方法关于摄动参数在无穷范数意义下具有二阶一致收敛性.在理论分析的基础上,设计出二阶自适应移动网格生成算法.

关键词： 奇异摄动, Fredholm型积分方程, 自适应移动网格, 一致收敛

Abstract: An exponentially fitted finite difference scheme constructed by the base function method and the compound trapezoidal formula is applied to make discrete a class of singularly perturbed Fredholm integral differential equation.Combining the truncation error estimate,the boundary estimate and the stability analysis,the second order uniform convergence in the infinity norm with respect to the perturbation parameter for the adaptive moving mesh method is proved.Based on the above theoretical analysis,the algorithm to produce the mesh is designed.

Key words: singularly perturbed problem, Fredholm integral equation, adaptive moving mesh, uniform convergence

罗丹, 毛志. 一类奇异摄动Fredholm型积分微分方程的二阶自适应移动网格方法[J]. 吉首大学学报（自然科学版）, 2024, 45(4): 15-25.

LUO Dan, MAO Zhi. The Second Order Adaptive Moving Mesh Method for a Singularly Perturbed Fredholm Integro-Differential Equation[J]. Journal of Jishou University(Natural Sciences Edition), 2024, 45(4): 15-25.

参考文献

[1] SMITS ALEXANDER,MCKEON BEVERLEY,IVAN MARUSIC.High-Reynolds Number Wall Turbulence[J].Annual Review of Fluid Mechanics,2011,43:353-375.
[2] DUKHIN STANISLAV.Electrophoresis at Large Peclet Numbers[J].Advances in Colloid and Interface Science,1991,36:219-248.
[3] HUNT J C R,SHERCLIFF J A.Magnetohydrodynamics at High Hartmann Number[J].Annual Review of Fluid Mechanics,1971,3(1):37-62.
[4] KEIMER ALEXANDER,PFLUG LUKAS.Modeling Infectious Diseases Using Integro-Differential Equations:Optimal Control Strategies for Policy Decisions and Applications in COVID-19[EB/OL].(2020-05-09)[2023-04-14].https:∥www.researchgate.net/publication/341265820.html.
[5] 刘华彦.散射与反散射问题的适定性分析[D].杭州:浙江大学,2019:23.
[6] 郭钊,郭子涛,易玲艳.多裂纹问题计算分析的本征COD边界积分方程方法[J].应用数学和力学,2019,40(2):200-209.
[7] RATHORE AJAY SINGH,SHANTHI VEMBU,RAMOS HIGINIO.A Fitted Numerical Method for a Singularly Perturbed Fredholm Integro-Differential Equation with Discontinuous Source Term[J].Applied Numerical Mathematics,2023,185:88-100.
[8] DURMAZ MUHAMMET ENES,CAKIR MUSA,AMIRALI ILHAME,et al.Numerical Solution of Singularly Perturbed Fredholm Integro-Differential Equations by Homogeneous Second Order Difference Method[J].Journal of Computational and Applied Mathematics,2022,412:114327.DOI:10.1016/j.cam.2022.114327.
[9] KUMAR DEVENDRA,DESWAL KOMAL,SINGH SATPAL.Wavelet-Based Approximation with Nonstandard Finite Difference Scheme for Singularly Perturbed Partial Integrodifferential Equations[J].Computational and Applied Mathematics,2022,41:341.DOI:10.1007/s40314-022-02053-4.
[10] TORSTEN LINSS.Analysis of a System of Singularly Perturbed Convection-Diffusion Equations with Strong Coupling[J].SIAM Journal on Numerical Analysis,2009,47(3):1847-1862.
[11] AMIRALIYEV GABIL,DURMAZ MUHAMMET ENES,KUDU MUSTAFA.Uniform Convergence Results for Singularly Perturbed Fredholm Integro-Differential Equation[J].Journal of Mathematical Analysis,2018,9(6):55-64.
[12] AMIRALIYEV GABIL,DURMAZ MUHAMMET ENES,KUDU MUSTAFA.Fitted Second Order Numerical Method for a Singularly Perturbed Fredholm Integro-Differential Equation[J].Bulletin of the Belgian Mathematical Society-Simon Stevin,2020,27:71-88.
[13] CAKIR MUSA,EKINCI YILMAZ,CIMEN ERKAN.A Numerical Approach for Solving Nonlinear Fredholm Integro-Differential Equation with Boundary Layer[J].Computational and Applied Mathematics,2022,41:259.DOI:10.1007/s40314-022-01933-z.
[14] MAO Zhi,LUO Dan.A Robust Adaptive Grid Method for First-Order Nonlinear Singularly Perturbed Fredholm Integro-Differential Equations[J].Networks Heterogeneous Media,2023,18(3):1006-1023.
[15] DURMAZ MUHAMMET ENES,AMIRALI ILHAME,AMIRALIYEV GABIL.An Efficient Numerical Method for a Singularly Perturbed Fredholm Integro-Differential Equation with Integral Boundary Condition[J].Journal of Applied Mathematics and Computing,2023,69:505-528.
[16] DOOLAN E P,MILLER J J H,SCHIDERS W H A.Uniform Numerical Methods for Problems with Initial and Boundary Layers[M].Dublin:Boole Press,1980:11-14;17.
[17] 梁颖.奇异摄动Volterra积分微分方程（组）的自适应移动网格方法研究[D].南宁:南宁师范大学,2021:37.
[18] NATALIA KOPTEVA,NIALL MADDEN,MARTIN STYNES.Grid Equidistribution for Reaction-Diffusion Problems in One Dimension[J].Numerical Algorithms,2005,40:305-322.

一类奇异摄动Fredholm型积分微分方程的二阶自适应移动网格方法

The Second Order Adaptive Moving Mesh Method for a Singularly Perturbed Fredholm Integro-Differential Equation

PDF (PC)

可视化

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 0

编辑推荐

Metrics

本文评价