指标1-型积分代数方程的配置边值方法
A Collocation Boundary Value Method for Integral Algebraic Equations of Index-1
DOI: 10.12677/AAM.2022.115257, PDF,    国家自然科学基金支持
作者: 曾 港*, 任 浩, 李恒杰：贵州大学数学与统计学院，贵州 贵阳
关键词: 指标-1型积分代数方程；配置边值方法；收敛性分析；Integral Algebraic Equations Index-1； Collocation Boundary Value Method； Convergence Analysis
摘要: 通过构造配置边值方法求解指标-1型积分代数方程。利用拉格朗日插值公式，选取未计算的节点作插值节点，将原方程离散为一个线性方程组。由绕圈数以及剩余理论分析了该方法的可解性和收敛性。证明了该方法求解指标-1型积分代数方程具有较高的收敛阶，同时给出数值算例验证了我们的理论结果。
Abstract: The index-1 integral algebraic equation is solved by constructing boundary value method. Using Lagrange interpolation formula, the nodes that not calculated are selected as interpolation nodes, then original equation is discreted into a system of linear equations. And the solvability and con-vergence of the method are analyzed based on the winding number and residual theory. It is proved that this method has higher convergence order for solving index-1 type integral algebraic equations, and numerical examples are given to verify our theoretical results.
文章引用：曾港, 任浩, 李恒杰. 指标1-型积分代数方程的配置边值方法[J]. 应用数学进展, 2022, 11(5): 2441-2453. https://doi.org/10.12677/AAM.2022.115257

参考文献

[1] Gomilko, A.M. (2003) A Dirichlet Problem for the Biharmonic Equation in a Semi-Infinite Strip. Journal of Engineering Mathematics, 46, 253-268. [Google Scholar] [CrossRef] 
[2] Jumarhon, B., Lamb, W., McKee, S., et al. (1996) A Volterra Integral Type Method for Solving a Class of Nonlinear Initial-Boundary Value Problems. Nu-merical Methods for Partial Differential Equations: An International Journal, 12, 265-281. [Google Scholar] [CrossRef] 
[3] Zenchuk, A.I. (2008) Combination of Inverse Spectral Transform Method and Method of Characteristics: Deformed Pohlmeyer Equa-tion. Journal of Nonlinear Mathematical Physics, 15, 437-448. [Google Scholar] [CrossRef] 
[4] Kafarov, V.V., Mayorga, B. and Dallos, C. (1999) Mathematical Method for Analysis of Dynamic Processes in Chemical Reactors. Chemical Engineering Science, 54, 4669-4678. [Google Scholar] [CrossRef] 
[5] Gear, C.W. (1990) Differential Algebraic Equations, Indices, and Integral Algebraic Equations. SIAM Journal on Numerical Analysis, 27, 1527-1534. [Google Scholar] [CrossRef] 
[6] Gear, C.W. (1988) Differential-Algebraic Equation Index Transformations. SIAM Journal on Scientific and Statistical Computing, 9, 39-47. [Google Scholar] [CrossRef] 
[7] Kauthen, J.P. (2001) The Numerical Solution of Integral-Algebraic Equations of Index 1 by Polynomial Spline Collocation Methods. Mathematics of Computation, 70, 1503-1514. [Google Scholar] [CrossRef] 
[8] Hadizadeh, M., Ghoreishi, F. and Pishbin, S. (2011) Jacobi Spectral Solution for Integral Algebraic Equations of Index-2. Applied Numerical Mathematics, 61, 131-148. [Google Scholar] [CrossRef] 
[9] Pishbin, S., Ghoreishi, F. and Hadizadeh, M. (2011) A Posteriori Error Estimation for the Legendre Collocation Method Applied to Inte-gral-Algebraic Volterra Equations. Electronic Transactions on Numerical Analysis, 38, 327-346.
[10] Farahani, M.S. and Hadizadeh, M. (2018) Direct Regularization for System of Integral-Algebraic Equations of Index-1. Inverse Prob-lems in Science and Engineering, 26, 728-743. [Google Scholar] [CrossRef] 
[11] Liang, H. and Brunner, H. (2013) Integral-Algebraic Equations: Theory of Collocation Methods I. SIAM Journal on Numerical Analysis, 51, 2238-2259. [Google Scholar] [CrossRef] 
[12] Liang, H. and Brunner, H. (2016) Inte-gral-Algebraic Equations: Theory of Collocation Methods II. SIAM Journal on Numerical Analysis, 54, 2640-2663. [Google Scholar] [CrossRef] 
[13] Sajjadi, S.A. and Pishbin, S. (2021) Convergence Analysis of the Product Integration Method for Solving the Fourth Kind Integral Equations with Weakly Singular Kernels. Numerical Algorithms, 86, 25-54. [Google Scholar] [CrossRef] 
[14] Pishbin, S. (2015) Optimal Convergence Results of Piecewise Polynomial Collocation Solutions for Integral-Algebraic Equations of Index-3. Journal of Computational and Applied Mathematics, 279, 209-224. [Google Scholar] [CrossRef] 
[15] Sohrabi, S. and Ranjbar, H. (2019) On Sinc Discretization for Systems of Volterra Integral-Algebraic Equations. Applied Mathematics and Computation, 346, 193-204. [Google Scholar] [CrossRef] 
[16] Zhang, T., Liang, H. and Zhang, S. (2020) On the Convergence of Multistep Collocation Methods for Integral-Algebraic Equations of Index 1. Computational and Applied Mathematics, 39, Article No. 294. [Google Scholar] [CrossRef] 
[17] Brugnano, L. and Trigiante, D. (1998) Solving Differential Problems by Multistep Initial and Boundary Value Methods. Gordon and Breach Science Publisher, Amster-dam.
[18] Chen, H. and Zhang, C. (2011) Boundary Value Methods for Volterra Integral and Integro-Differential Equa-tions. Applied Mathematics and Computation, 218, 2619-2630. [Google Scholar] [CrossRef] 
[19] Chen, H. and Zhang, C. (2012) Block Boundary Value Methods for Volterra Integral and Integro-Differential Equations. Journal of Computational and Applied Mathematics, 236, 2822-2837. [Google Scholar] [CrossRef] 
[20] Ma, J. and Xiang, S. (2017) A Collocation Boundary Value Method for Linear Volterra Integral Equations. Journal of Scientific Computing, 71, 1-20. [Google Scholar] [CrossRef] 
[21] Liu, L. and Ma, J. (2021) Block Collocation Boundary Value Solutions of the First-Kind Volterra Integral Equations. Numerical Algorithms, 86, 911-932. [Google Scholar] [CrossRef] 
[22] Bottcher, A. and Grudsky, S. (2005) Spectral Properties of Banded Toeplitz Matrices. Society for Industrial and Applied Mathematics, Philadelphia. [Google Scholar] [CrossRef] 

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