第一类柯西奇异积分方程的有效配置求解法
The Efficient Collocation Method for Solving the First Kind of Cauchy Singular Integral Equation
DOI: 10.12677/PM.2023.134102, PDF,    国家自然科学基金支持
作者: 陈 锐：西华师范大学数学与信息学院，四川 南充；陈 冲：西华师范大学公共数学学院，四川 南充
关键词: 柯西奇异积分方程；q-Bessel多项式；高斯–切比雪夫求积公式；误差分析；Cauchy Singular Integral Equation； q-Bessel Polynomial； Gauss-Chebyshev Quadrature Formula； Error Analysis
摘要: 本文提出了求解第一类柯西奇异积分方程的一种有效配置法，即基于q-Bessel多项式并结合第一类、第二类高斯–切比雪夫求积公式的离散配置法将第一类柯西奇异积分方程转化为线性方程组进行近似求解，结合插值理论对该方法进行误差分析。通过数值算例验证了该方法的可行性和有效性。
Abstract: This paper presents an efficient collocation method to solve Cauchy singular integral equations of the first kind. Namely, based on the q-Bessel polynomial and the discrete collocation method of the first and second Gauss-Chebyshev quadrature formulas, the first Cauchy singular integral equation is transformed into a linear system of equations for approximate solution, and the error analysis of the method is carried out by combining the interpolation theory. The effectiveness and feasibility of the method are verified by numerical examples.
文章引用：陈锐, 陈冲. 第一类柯西奇异积分方程的有效配置求解法[J]. 理论数学, 2023, 13(4): 968-975. https://doi.org/10.12677/PM.2023.134102

参考文献

[1] 黄晋. 多维奇异积分方程的高精度算法[M]. 北京: 北京科学出版社, 2007.
[2] Wen, Q. and Du, Q.G. (2020) An Approximate Numerical Method for Solving Cauchy Singular Integral Equations Composed of Multiple Implicit Pa-rameter Functions with Unknown Integral Limits in Contact Mechanics. Journal of Mathematical Analysis and Appli-cations, 482, 1-14. [Google Scholar] [CrossRef] 
[3] 崔智刚, 刘铁军. 压电涂层-功能梯度压电界面层-基底结构的二维接触问题研究[J]. 固体力学学报, 2022, 43(3): 284-295.
[4] 臧炜煜, 刘铁军. 功能梯度压电界面层在球形压头作用下的接触问题[J/OL]. 应用力学学报, 2023: 1-11.
http://kns.cnki.net/kcms/detail/61.1112.O3.20221202.1250.001.html, 2023-04-24
[5] 小巴桑次仁, 多布杰, 德吉玉珍, 等. 基于电磁波反问题的一类非线性积分方程解的唯一性[J]. 高原科学研究, 2022, 6(2): 109-112.
[6] 戴世坤, 陈轻蕊, 凌嘉宣, 李昆. 空间-波数域三维大地电磁场积分方程数值模拟[J]. 地球物理学报, 2022, 65(6): 2294-2310.
[7] Zolotov, N.B. and Pozharskii, D.A. (2022) Periodic Contact Problems for a Half-Space with a Partially Fixed Boundary. Mechanics of Solids, 57, 1758-1765. [Google Scholar] [CrossRef] 
[8] Jin, X., Keer, L.M. and Wang, Q. (2008) A Practical Method for Singular Integral Equations of the Second Kind. Engineering Fracture Mechanics, 75, 1005-1014. [Google Scholar] [CrossRef] 
[9] Moghaddam, B.P., Tenreiro, J.A., Shajari, P.S., and Mostaghim, Z.S. (2020) A Numerical Algorithm for Solving the Cauchy Singular Integral Equation Based on Hermite Polynomials. Hacettepe Journal of Mathematics and Statistics, 49, 974-983.
[10] Karczmarek, P., Pylak, D. and Sheshko, M.A. (2006) Application of Jacobi Polynomials to Approximate Solution of a Singular Integral Equation with Cauchy Kernel. Applied Mathematics and Computation, 181, 697-707. [Google Scholar] [CrossRef] 
[11] Eshkuvatov, Z.K., Nik Long, N.M.A. and Abdulkawi, M. (2009) Approximate Solution of Singular Integral Equations of the First Kind with Cauchy Kernel. Applied Mathematics and Letter, 22, 651-657. [Google Scholar] [CrossRef] 
[12] Setia, A., Sharma, V. and Liu, Y. (2015) Numerical Solution of Cauchy Singular Integral Equation with an Application to a Crack Problem. Neural, Parallel and Scientific Computations, 23, 387-392.
[13] Setia, A. (2014) Numerical Solution of Various Cases of Cauchy Type Singular Integral Equation. Applied Mathematics and Compution, 230, 200-207. [Google Scholar] [CrossRef] 
[14] Seifi, A., Lotfi, T., Allahviranloo, T. and Paripour, M. (2017) An Effective Collocation Technique to Solve the Singular Fredholm Integral Equations with Cauchy Kernel. Advances in Difference Equations, 2017, Article No. 280. [Google Scholar] [CrossRef] 
[15] Seifi, A. (2020) Numerical Solution of Certain Cauchy Singular Integral Equations Using a Collocation Scheme. Advances in Different Equations, 2020, Article No. 537. [Google Scholar] [CrossRef] 
[16] Lifanov, I.K. (1996) Singular Integral Equations and Discrete Vortices. Walter De Gruyter Incorporated, Berlin. [Google Scholar] [CrossRef] 
[17] Mumtaz, R. and Subuhi, K. (2019) A Determinant Approach to q-Bessel Polynomials and Applications. Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, 113, 1571-1583. [Google Scholar] [CrossRef] 
[18] Dezhbord, A., Lotfi, T. and Mahdiani, K. (2016) A New Efficient Method for Cases of the Singular Integral Equation of the First Kind. Journal of Computational and Applied Mathematics, 296, 156-169. [Google Scholar] [CrossRef] 

为你推荐