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      Studies on the Volterra Integral Equation with Linear Delay

      2017-09-03 10:13:32-
      關(guān)鍵詞:強(qiáng)校韓山國家自然科學(xué)基金

      -

      (Colloge of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, China)

      Studies on the Volterra Integral Equation with Linear Delay

      ZHENGWei-shan*

      (Colloge of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, China)

      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.

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

      The Volterra integral equation with linear delay is as follow:

      (1)

      where the unknown functiony(τ) is defined on [0,T],T<+∞ andqis constant with 0

      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.

      1 Chebyshev spectral-collocation method

      (2)

      Then equation (1) becomes

      and the above equation can be rewritten as follows.

      (3)

      Obviously, the above equations hold atximentioned above,xi∈[-1,1]. Thus, we have

      (4)

      Using aN+1-point Legendre quadrature formula (2), corresponding weightwk, we can obtain that

      (5)

      2 Convergence analysis

      2.1ConvergenceanalysisinL∞(-1,1)space

      Theorem1Supposeu(x)istheexactsolutiontoequations(3)anduN(x)istheapproximatesolutionobtainedbyusingtheChebyshevspectralcollocationmethod.ThenforNsufficientlylarge,weget

      (6)

      where

      ProofMinusing(4)by(5)gives

      Notee(x)=u(x)-uN(x), then we have

      (7)

      By Theorem 4.3, 4.7 and 4.10 in [7], we get

      |J1(x)|≤CN-m|K(x,·)|Hm,N(-1,1)‖uN(sx(·))‖L2(-1,1).

      (8)

      u-uN+INu-u=e(x)+INu-u.

      (9)

      Now we are in the position to estimate the above inequality term by term. Firstly the second conclusion in Theorem 1.8.4 in [8] shows that

      (10)

      For the estimation of ‖INJ1‖L∞(-1,1), firstly using the first inequality in (8), we obtain

      CN-mK*(‖e‖L∞(-1,1)+‖u‖L∞(-1,1)).

      Then, due to Theorem 3.3 in [9], we have

      ‖INJ1(x)‖L∞(-1,1)≤‖IN‖L∞(-1,1)‖J1(x)‖L∞(-1,1)≤

      CN-m(logN)K*(‖e‖L∞(-1,1)+‖u‖L∞(-1,1)).

      For ‖J3(x)‖L∞(-1,1), applying (7) and lettingm=1, we get

      From what discussed above, we can see

      Since

      This completes the proof of this theorem.

      Theorem2Supposeu(x)istheexactsolutiontoequation(3)anduN(x)istheapproximatesolutionobtainedbytheChebyshevspectralcollocationmethodwhichdefinedin(5).ThenforNsufficientlylarge,weget

      ProofAsthesameprocedureinthedeductionfrom(7)to(9)inTheorem1,wecanderivethefollowinginequality

      Further more applying the generalized Hardy’s inequality with weights[10], we get

      Now we estimate each item from left to right for the above inequality. ForJ0(x), the first conclusion in Theorem 1.8.4 in [8] shows that

      (11)

      By Theorem 1 in [11] and (8), we get

      Due to the conclusion in Theorem 1, we get

      (12)

      3 Algorithm implementation and numerical experiment

      The corresponding exact solution is given byu(x)=e4x,x∈ [-1,1].

      a.The errors u-uN in L∞ and norms. b.The comparison between approximate solution uN and the exact solution u.Fig.1 Numerical results

      N68101214L∞?error0.0300419.0557×10-42.4424×10-55.6363×10-79.5384×10-9L2ωc?error0.0287919.2524×10-42.4507×10-55.2913×10-78.6576×10-9N1618202224L∞?error1.2083×10-101.1866×10-122.1316×10-141.4211×10-147.1054×10-15L2ωc?error1.0916×10-101.1034×10-128.6652×10-148.3351×10-149.2255×10-15

      [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,etal. 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 Hardy’s Type[M]. New York: World Scientific, 2003.

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

      (編輯 HWJ)

      2016-10-29

      國家自然科學(xué)基金資助項(xiàng)目(11626074);韓山師范學(xué)院創(chuàng)新強(qiáng)校項(xiàng)目(Z16027);中山大學(xué)廣東省計(jì)算科學(xué)重點(diǎn)實(shí)驗(yàn)室開放基金資助項(xiàng)目(2016011);韓山師范學(xué)院扶持項(xiàng)目(201404)

      O242.2

      A

      1000-2537(2017)04-0083-06

      帶線性延遲項(xiàng)的Volterra積分方程研究

      鄭偉珊*

      (韓山師范學(xué)院數(shù)學(xué)與統(tǒng)計(jì)學(xué)院, 中國 潮州 521041)

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

      10.7612/j.issn.1000-2537.2017.04.014

      *通訊作者,E-mail:weishanzheng@yeah.net

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