鄭偉珊

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

中圖分類號(hào) O242.2文獻(xiàn)標(biāo)識(shí)碼 A文章編號(hào) 1000-2537（2017）04-0083-06

摘 要 本文主要研究帶線性延遲項(xiàng)的Volterra型積分方程收斂情況. 首先通過(guò)線性變換， 我們將原先定義在[0，T]區(qū)間上帶線性延遲項(xiàng)的Volterra型積分方程轉(zhuǎn)換成定義在固定區(qū)間[-1，1]上的方程， 然后利用Gauss積分公式求得近似解， 進(jìn)而再利用Chebyshev譜配置方法分析該方程的收斂性， 最終借助格朗沃不等式及相關(guān)引理分析獲得方程在L∞和L2ωc 范數(shù)意義下呈現(xiàn)指數(shù)收斂的結(jié)論. 最后給出數(shù)值例子， 驗(yàn)證理論證明的結(jié)論.

關(guān)鍵詞 Chebyshev譜配置方法； 線性延遲項(xiàng)； Volterra型積分方程； 誤差分析

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.

References：

[1] TANG T， XU X， CHENG J. On Spectral methods for Volterra integral equation and the convergence analysis[J]. J Comput Math， 2008，26（6）：825-837.

[2] CHEN Y， TANG T. Convergence analysis of the Jacobi spectral-collocation methods for Volterra integral equation with a weakly singular kernel[J]. Math Comput， 2010，79（269）：147-167.

[3] WAN Z， CHEN Y， HUANG Y. Legendre spectral Galerkin method for second-kind Volterra integral equations[J]. Front Math China， 2009，4（1）：181-193.

[4] XIE Z， LI X， TANG T. Convergence analysis of spectral galerkin methods for Volterra type integral equations[J]. J Sci Comput， 2012，53（2）：414-434.

[5] GU Z， CHEN Y. Chebyshev spectral collocation method for Volterra integral equations[J]. Contem Math， 2013，586：163-170.

[6] LI J， ZHENG W， WU J. Volterra integral equations with vanishing delay[J]. Appl Comput Math， 2015，4（3）：152-161.

[7] CANUTO C， HUSSAINI M， QUARTERONI A， et al. Spectral method fundamentals in single domains[M]. New York： Spring-Verlag， 2006.

[8] SHEN J， TANG T. Spectral and high-order methods with applications[M]. Beijing： Science Press， 2006.

[9] MASTROIANNI G， OCCORSIO D. Optional system od nodes for Lagrange interpolation on bounded intervals[J]. J Comput Appl Math， 2001，134（1-2）：325-341.

[10] KUFNER A， PERSSON L. Weighted inequality of Hardys Type[M]. New York： World Scientific， 2003.

[11] NEVAI P. Mean convergence of Lagrange interpolation[J]. Trans Amer Math Soc， 1984，282：669-698.