一类线性Volterra积分方程组的一类数值解法  

A Numerical Method for System of Linear Volterra Integral Equations

作　　者：张益 陈冲 ZHANG Yi;CHEN Chong(School of Mathematics and Information,China West Normal University,Nanchong 637009,China;School of Mathematics Education,China West Normal University,Nanchong 637009,China)

机构地区：[1]西华师范大学数学与信息学院,四川南充637009 [2]西华师范大学公共数学学院,四川南充637009

出　　处：《西安航空学院学报》2025年第1期65-73,共9页Journal of Xi’an Aeronautical Institute

基　　金：博士启动基金项目(17E083)。

摘　　要：提出了一种具有连续核函数的线性Volterra积分方程组的一类近似求解方法:利用一元多项式逼近方程组的未知函数,再结合Galerkin方法,将该积分方程组转化成矩阵方程组进行求解,得出Volterra积分方程组的近似解。利用正交变换将一元多项式进行正交变换,在此基础上对该数值近似方法的收敛性进行分析。通过数值算例验证算法的有效性和可行性,并与文献报道的结果进行比较。结果表明,在精确解为多项式类型,且多项式近似的最高阶数等于近似解最高阶数时,近似解即为精确解。当不是多项式类型解时,随着多项式近似时最高阶数逐渐增大,近似解与精确解的误差随之减小,当最高阶数大于固定取值时,近似解与精确解绝对误差减小速度变缓。A numerical method for system of linear Volterra integral equations with continuous kernel functions is proposed.The unknown functions of the equation system are approximated by univariate polynomials,and combined with the Galerkin method,the system of Volterra integral equations is transformed into a matrix system,thereby obtaining approximate solutions.The orthogonal transformation of univariate polynomials is carried out by using orthogonal transformation,and based on which the convergence of this numerical approximation method is analyzed.The effectiveness and feasibility of the algorithm are verified through numerical examples with comparisons made against existing results in the existing literature.The results show that when the exact solution is a polynomial function and the highest degree of polynomial approximation matches that of the exact solution,the approximate solution coincides with the exact one.When the solution is not of polynomial type,as the highest degree of polynomial approximation gradually increases,the error between the approximate solution and the exact solution decreases.When the highest degree is greater than a fixed value,the rate of decrease of the absolute error between the approximate solution and the exact solution decreases.

关 键 词：Volterra积分方程组 GALERKIN方法 近似解 收敛性分析 

分 类 号：O241.83[理学—计算数学]