带线性延迟项的Volterra积分方程研究（英文）

郑伟珊

Abstract This paper is concerned about the Volterra integral equation with linear delay. First we transfer the integral interval [0，T] into interval [-1， 1] through the conversion of variables. Then we use the Gauss quadrature formula to get the approximate solutions. After that the Chebyshev spectral-collocation method is proposed to solve the equation. With the help of Gronwall inequality and some other lemmas， a rigorous error analysis is provided for the proposed method， which shows that the numerical error decay exponentially in the innity norm and the Chebyshev weighted Hilbert space norms. In the end， numerical example is given to confirm the theoretical results.

Key words Chebyshev spectral-collocation method； linear delay； Volterra integral equations； error analysis

中圖分类号 O242.2文献标识码 A文章编号 1000-2537（2017）04-0083-06

摘 要 本文主要研究带线性延迟项的Volterra型积分方程收敛情况. 首先通过线性变换， 我们将原先定义在[0，T]区间上带线性延迟项的Volterra型积分方程转换成定义在固定区间[-1，1]上的方程， 然后利用Gauss积分公式求得近似解， 进而再利用Chebyshev谱配置方法分析该方程的收敛性， 最终借助格朗沃不等式及相关引理分析获得方程在L∞和L2ωc 范数意义下呈现指数收敛的结论. 最后给出数值例子， 验证理论证明的结论.

关键词 Chebyshev谱配置方法； 线性延迟项； Volterra型积分方程； 误差分析

Equations of this type arise as models in many fields， such as the Mechanical problems of physics， the movement of celestial bodies problems of astronomy and the problem of biological population original state changes. They are also applied to network reservoir， storage system， material accumulation， different fields of industrial process etc， and solve a lot problems from mathematical models of population statistics， viscoelastic materials and insurance abstracted. The Volterra integral equation with linear delay is one of the important type of Volterra integral equations with great significance in both theory and applications. There are many methods to solve Volterra integral equations， such as Legendre spectral-collocation method[1]， Jacobi spectral-collocation method[2]， spectral Galerkin method[3-4]， Chebyshev spectral-collocation method[5] and so on. In this paper， inspired by[5] and [6]， we use a Chebyshev spectral-collocation method to solve Volterra integral equations with linear delay.

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