二维 Volterra-Fredholm 积分方程数值算法研究及收敛性分析

doi:10.3969/j.issn.1005-3085.2022.06.014

工程数学学报

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二维 Volterra-Fredholm 积分方程数值算法研究及收敛性分析

解加全^1,2, 刘霄琪¹, 张佳乐¹

1. 太原师范学院数学系，晋中 030619；2. 太原师范学院工程科学计算山西省高等学校重点实验室，晋中 030619

出版日期:2022-12-15 发布日期:2022-12-15
基金资助:
国家自然科学基金 (52005360)；山西省高等学校科技创新项目 (2021L403).

Numerical Algorithm for Two-dimensional Volterra-Fredholm Integral Equations and Its Convergence Analysis

XIE Jiaquan^1,2, LIU Xiaoqi¹, ZHANG Jiale¹

1. Department of Mathematics, Taiyuan Normal University, Jinzhong 030619;
2. Key Laboratory for Engineering & Computational Science, Shanxi Provincial Department of Education, Taiyuan Normal University, Jinzhong 030619

Online:2022-12-15 Published:2022-12-15
Supported by:
The National Natural Science Foundation of China (52005360); the Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi Province (2021L403).

摘要/Abstract

摘要：

针对一类二维非线性Volterra-Fredholm积分方程，提出利用二维Block-Pulse函数为基函数进行数值求解。首先，引入Block-Pulse函数的定义及基函数的向量表示形式；其次，根据二维Block-Pulse函数的不相交性和正交性推导了基向量的积分算子矩阵和乘积算子矩阵；然后，基于该算子矩阵将待求问题转化为一系列向量的乘积形式，利用配点法离散未知变量获得原问题的数值解；最后，通过两个具体的数值算例对所提算法的可行性和收敛性进行了验证。

关键词: Volterra-Fredholm积分方程, 二维Block-Pulse函数, 算子矩阵, 数值解, 收敛性

Abstract: A two-dimensional nonlinear Volterra-Fredholm Hammerstein integral equation is numerically solved by using the two-dimensional block pulse function as the basis function. Firstly, the definition of the block pulse function and the vector representation of the basis function are introduced. Secondly, according to the disjointness and orthogonality of

two-dimensional block pulse functions, the integral operator matrix and product operator matrix of the basis vector are derived. Thirdly, the operator matrix is used to transform the problem to be solved into the product form of a series of vectors, and the unknown variables are discretized by the collocation method to obtain the numerical solution of the original problem, Finally, the feasibility and convergence of the proposed algorithm are verified by two numerical examples.

Key words: Volterra-Fredholm integral equations, two-dimensional Block-Pulse functions, ope-rational matrix, numerical solutions, convergence

中图分类号:

O241.8

解加全, 刘霄琪, 张佳乐. 二维 Volterra-Fredholm 积分方程数值算法研究及收敛性分析[J]. 工程数学学报, doi: 10.3969/j.issn.1005-3085.2022.06.014.

XIE Jiaquan, LIU Xiaoqi, ZHANG Jiale. Numerical Algorithm for Two-dimensional Volterra-Fredholm Integral Equations and Its Convergence Analysis[J]. Chinese Journal of Engineering Mathematics, doi: 10.3969/j.issn.1005-3085.2022.06.014.

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二维 Volterra-Fredholm 积分方程数值算法研究及收敛性分析

Numerical Algorithm for Two-dimensional Volterra-Fredholm Integral Equations and Its Convergence Analysis

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