二维 Volterra-Fredholm 积分方程数值算法研究及收敛性分析

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。

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

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.