Studies on the Volterra Integral Equation with Linear Delay

2017-09-03-

湖南师范大学自然科学学报 2017年4期

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

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

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(编辑 HWJ)

2016-10-29

国家自然科学基金资助项目(11626074);韩山师范学院创新强校项目(Z16027);中山大学广东省计算科学重点实验室开放基金资助项目(2016011)；韩山师范学院扶持项目(201404)

O242.2

1000-2537(2017)04-0083-06

带线性延迟项的Volterra积分方程研究

郑伟珊*

(韩山师范学院数学与统计学院, 中国潮州 521041)