中图分类号 O175.29 文献标识码 A 文章编号 2096-5281(2018)02-0082-06

摘 要 本文对一类延迟微积分方程进行勒让德误差分析, 首先通过适当的函数变换和变量变化把方程的定义域化为标准区间, 然后利用勒让德谱配置方法进行分析, 最终获得方程的在L2和L∞模意义下呈现谱收敛的结论.

关键词 延迟微积分方程;勒让德谱方法;误差分析

We consider some integro-differential equations with quadratic delay term in the upper integral. As we know these kinds of equations arise widely in mathematical models of physical and biological phenomena. Extensive studies have been done for these topics[1-6]. Most of them are convergence analysis, owing to the fact that the exact solution is difficult to get. In practice, spectral method has excellent convergence property of exponential convergence rate, so we will provide a Legendre spectral method for the delay integro-differential equation. We get the conclusion that not only the error of the approximate solution but also the error of approximate derivative both decay exponentially in L2 and L∞-norms. The latter is an extension of [6], where we only deal with the integral equation with linear delay. In this article, we focus on the integro-differential equation with nonlinear delay. At the same time we get two kinds of error. Now we introduce the equation:

y′(τ)=a(τ)y(τ)+f(τ)+R(τ,η)y(η)dη,

(1)

where y(0)=y0, a(τ),f(τ)∈Cm(I), R(τ,η)∈Cm(D), for τ∈[0,T],m≥1, D:={(τ,η):0≤η≤qτ2≤τ≤T}. y(τ) is the unknown function and 0<q<1.

For convenience of analysis, we will describe the spectral method on the standard interval [-1,1]. Hence, we employ the transformation

 

and then the problem (1) becomes

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(2)

with u(-1)=y0, x∈[-1,1] and

 

 

1 Spectral method

Set the collocation points as the set of N+1-point Legendre Gauss, or Legendre Gauss-Radau, or Legendre Gauss-Lobatto points, (see, e.g. [7]). Then (2) holds at xi

 

(3)

for 0≤i≤N. Make a linear transformation to transfer the integral interval to [-1,1], we get

 

(4)

and (3) becomes

 

(5)

Consider the initial condition

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u(xi)=u(-1)+u′(t)dt,

‖u-INu‖Hl(I)≤CN2l-1/2-m|u|Hm,N(I),1≤l≤m.

(6)

and we use the similar variable changes as (4) to obtain that

 

(7)

Gauss quadrature formula gives

 

(8)

 

(9)

where uiu(xi), 0≤i≤N. Now we use Lagrange interpolation polynomials to expand u′ and u as for Fk is the k-th Lagrange basis function. The Legendre spectral method is to seek and such that the following equations hold

 

(10)

 

(11)

2 Convergence analysis

This section is devoted to the error deduction. We first introduce five important lemmas, which are essential for the derivation of the main result.

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Since have and |e′(x)|≤C|e′(t)|dt+|H(x)|, for

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(12)

where

 

(13)

 

(14)

Lemma 2[8] For every bounded function v(x), there exists a constant C independent of v such that

 

(15)

Lemma 3[7] Assume that u∈Hm(I) with I=(-1,1) and denote INu its interpolation polynomial associated with the (N+1)-point Gauss points namely,

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(16)

Then the following estimates hold

‖u-INu‖L2(I)≤CN-m|u|Hm,N(I),

(17)

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(18)

Lemma 4 (Gronwall inequality) If a non-negative integrable function u(x) satisfies

u(x)≤Lu(s)ds+v(x),-1<x≤1,

(19)

where v(x) is an integrable function, then there exists a constant C such that

Further more, (31) and (32) can be denoted as

(20)

INu(x)-uN(x)=IN(e′(t)dt).

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‖u(x)‖Lp(I)≤C‖v(x)‖Lp(I), p≥1.

(21)

Lemma 5[9] Assume that Fj(x) is the j-th Lagrange interpolation polynomial associated with the Legendre-Gauss points. Then

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(22)

where B0 is a constant.

Now we carry out the error analysis in L2 and L∞ spaces respectively. Firstly we relabel the numerical scheme of (10)-(11). Then we make subtraction between (6)-(7) and (27)-(28) to get the error functions of approximate solution and approximate derivative. Finally with the help of the above lemmas, we get the desired conclusion.

Theorem 1 Let u be the exact solution of (2). Note that where are given by (10)-(11). If u∈Hm+1(I) with I=(-1,1) for some m≥1, we have

 

(23)

 

(24)

 

(25)

 

(26)

provided that N is sufficiently large, where and C is a constant indepedent of N.

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Proof Firstly, we carry out our convergence analysis in L2 space. Using the symbol of (14), we note that

 

and we can rewrite (10) and (11) as

 

(27)

 

(28)

Lemma 3 implies that

 

(29)

Let e(x)=u(x)-uN(x), and subtract (27) from (5), lead to

 

(30)

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For each side of (30), multiplying Fi(x) and summing up from i=0 to i=N, we get

 

(31)

where Take the same procedure for (29), yields

and

(32)

u(x)≤Cv(s)ds+v(x),-1<x≤1,

 

(33)

 

(34)

 

Combining (33) and (34), we get

 

(35)

where

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Now we deal with the double integrals in (35). Let and By Dirichlet′s formula we have

 

 

 

 

 

 

(36)

Lemma 1[7] Assume that an N+1-point Gauss quadrature formula relative to the Legendre weight is used to integrate the product uψ, where u∈Hm(I) with I=(-1,1) for some m≥1 and ψ∈PN. Then there exists a constant C independent of N such that

According to Lemma 4, we have

 

(37)

Now we estimate Jj, j=1,2…,6. From Lemma 1, we get

 

 

(38)

From Lemma 2, there is

 

Make subtraction between (6) and (28), we get

 

Note that ‖u-INu‖L2(I)≤CN-m|u|Hm,N(I) due to Lemma 3. Let m=1, we have ‖u-INu‖L2(I)≤CN-1×|u|H1,N(I). Conseqnently, we have

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≤CN-1‖e‖L2(I),

 

The above inequalities, combining the estimates of ‖e‖L2(I) and ‖Ji‖L2(I), i=1,2,…,6, yield

 

Which is the conclusion of (24). Furthermore, by (34), we have

 

leading to (23). Next, following the same process as we have done for the estimate in L2 space, we come to estimate the errors in L∞ space. Gronwall inequality indicates By [7], we have for all w∈H1(a,b). This together with Lemma 3 yields

 

After that we can get the estimates of ‖Ji‖L∞(I),i=2,4,…,6. By (38) and Lemma 5 we From (34), we can have the error estimate for ‖e‖L∞(I).

3 Numerical scheme

(1) Denote and UN=(u0,u1,…,uN)T. Then, we can rewrivte (10)-(11) in the following matrix form

where G=(g(x0),g(x1),…,g(xN))T, A=diag(A(x0),A(x1),…,A(xN)).

(2) Select a concrete example, then calculate and UN using step (1).

(3) With the help of step (1), we can get the approximate derivative and the approximate solution

(4) Finally calculate L2 and L∞ error respectively.

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