Weishan ZHENG (鄭偉珊)

College of Mathematics and Statistics，Hanshan Normal University，Chaozhou 521041，China E-mail:weishanzheng@yeah.net

Yanping CHEN (陳艷萍)?

School of Mathematical Sciences，South China Normal University，Guangzhou 510631，China E-mail:yanpingchen@scnu.edu.cn

Abstract In this paper，a Jacobi-collocation spectral method is developed for a Volterraintegro-differential equation with delay，which contains a weakly singular kernel.We use a function transformation and a variable transformation to change the equation into a new Volterra integral equation defined on the standard interval[-1，1]，so that the Jacobi orthogonal polynomial theory can be applied conveniently.In order to obtain high order accuracy for the approximation，the integral term in the resulting equation is approximated by Jacobi spectral quadrature rules.In the end，we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in both the L∞-norm and the weighted L2-norm.

Key words Volterra integro-differential equation;pantograph delay;weakly singular kernel;Jacobi-collocation spectral methods;error analysis;convergence analysis

1 Introduction

Volterra integro-differential equations with delay arise often in mathematical models of physical and biological phenomena.As they are widely encountered and applied，they must be solved successfully with efficient numerical methods.There has been a lot of study on this subject，such as[6，7，17，21，23，24].This topic has also attracted the attention of famous mathematicians，such as Ali，Brunner and Tang[1]，Ishiwata and Muroya[10]，Wei and Chen[18].

As far as we know，very little work has been done on the Volterra delay-integro-differential equations with a weakly singular kernel using spectral approximation.Spectral methods are a class of techniques used in applied mathematics and scientific computing to numerically solve certain Volterra equations[3，4，19，20，22，25]，and they are favoured due to their excellent error properties and their“exponential convergence”being the fastest possible.In this paper，we provide a Jacobi-collocation spectral method for Volterra integro-differential equations with a pantograph delay that contain a weakly singular kernel.In the end，we provide a rigorous error analysis for the proposed method.The spectral rate of convergence for the proposed method is established in the L∞-norm and the weighted L2-norm.

In this paper，we study the pantograph Volterra delay-integro-differential equation with a weakly singular kernel of the form

where 0＜μ＜1，0＜q＜1，the functions a (t)，b (t)，g (t)∈C1(I)，y (t) are the unknown functions and are supposed to be sufficiently smooth，and K (t，t)0 for t∈I:=[0，T].(t-s)-μis a weak kernel and y (qt) is the pantograph delay.

To use the theory of orthogonal polynomials，we make the change of variable

Furthermore，to transfer the integral interval[0，T (1+x)/2]to the interval[-1，x]，we make a linear transformation s=T (1+τ)/2，τ∈[-1，x].Then eq.(1.2) becomes

where

The main purpose of this work is to use a Jacobi-collocation method to numerically solve the Volterra integro-differential equations with a pantograph delay that contain a weakly singular kernel.We will provide a rigorous error analysis which theoretically justifies the spectral rate of convergence.The rest of the paper is organized as follows:in Section 2，we introduce the Jacobi-collocation spectral approach for (1.3).Some useful lemmas are provided in Section 3;These are are important for the convergence analysis.In Section 4 the convergence analysis is outlined，and Section 5 contains numerical results which will be used to a the theoretical results obtained in the former section.Finally，in Section 6，we end with a conclusion and a discussion of future work.

Throughout the paper C will denote a generic positive constant that is independent of N，but dependant on T，the given functions and the index μ.

2 Jacobi-Collocation Methods

Now we introduce the Jacobi polynomialsof indices α，β＞-1 which are the solutions to singular Sturm-Liouville problems

Hereafter，we denote the Jacobi weight function of index (α，β) by ωα，β(x)=(1-x)α(1+x)β(see[2，8，9，16]).We define the“usual”weighted Sobolev spaces as follows:

For a given N≥0，we denote bythe Jacobi Gauss points，and bythe corresponding Jacobi weights.Then，the Jacobi Gauss integration formula is

In particular，we denote that

In order to use the Jacobi-collocation methods naturally，we restate (1.4) as

Letting v (x)=u (qx+q-1)，we have that

First，eqs.(1.3)，(2.7) and (2.8) hold at the collocation pointson[-1，1]，associated with ω-μ，-μ，i.e.，

In order to obtain high order accuracy for the problem (2.9)-(2.11)，the main difficulty is to compute the integral term.In particular，for small values of xi，there is little information available for u (τ).To overcome this difficulty，we transfer the integral interval[-1，xi]to a fixed interval[-1，1]，

by using the following variable change:

Next，using the Jacobi Gauss integration formula，the integration term in (2.12) can be approximated by

We use uiandto approximate the function values u (xi) and u′(xi)(0≤i≤N)，respectively，and use

Remark 2.1Since Fj，j=0，1，···，N are polynomials of a degree not exceeding N，we have that

3 Some Useful Lemmas

In this section，we will give some lemmas which are important for the derivation of the main results of the subsequent section.

Lemma 3.1(see[2]) Let PNdenote the space of all polynomials of a degree not exceeding N.Assume that the Gauss quadrature formula relative to the Jacobi weight is used to integrate the product vφ，where v∈for some m≥1 and φ∈PN.Then there exists a constant C independent of N such that

Then the following estimates hold:

ProofThe inequality (1) can be found in[2].We only prove (2).Let∈PNdenote the interpolant of v at the Chebyshev Gauss points.From (5.5.28) in[2]，the interpolation error estimate in the maximum norm is given by

By using (3.7)，Lemma 3.2 and (3.6)，we obtain that

where J (x) is an integrable function，then

Lemma 3.4(Gronwall inequality) If a non-negative integrable function E (x) satisfies

where q is a constant and 0＜q＜1，we get

ProofUsing the variable change

where 0＜q＜1，qx+q-1=q (x+1)-1＜x+1-1=x，for x∈(-1，1].

This，together with (3.11)，gives us

This leads to the result found in (3.12) and (3.13). □

Lemma 3.5For nonnegative integer r and κ∈(0，1)，there exists a constant Cr，κ＞0 such that for any function v∈Cr，κ([-1，1])，there exists a polynomial function TNv∈PNsuch that

where‖·‖r，κis the standard norm in Cr，κ([-1，1]).Actually，as stated in[14，15]，TNis a linear operator from Cr，κ([-1，1]) into PN.

Lemma 3.6(see[5]) Let κ∈(0，1) and M be defined by

Then，for any function v∈C ([-1，1])，there exists a positive constant C such that

under the assumption that 0＜κ＜1-μ，for any x′，x′′∈[-1，1]and x′x′′.This implies that

Lemma 3.7(see[11]) For all measurable functions f≥0，the generalized Hardy inequality

holds if and only if

for the case 1＜p≤q＜∞.Here，T is an operator of the form (Tf)(x)=，with k (x，t) a given kernel，u，v weight functions，and-∞≤a＜b≤∞.

Lemma 3.8(see[13]) For every bounded function v (x)，there exists a constant C independent of v such that

where Fi(x) is the Lagrange interpolation basis function associated with the Jacobi collocation points.

4 Error Analysis

This section is devoted to providing a convergence analysis for the numerical scheme (2.18)-(2.20).The goal is to show that the rate of convergence is exponential and the spectral accuracy can be obtained for the proposed approximations.First，we carry out the convergence analysis in L∞space.

Theorem 4.1Let u (x) be the exact solution of (1.3) with (1.4)，which is assumed to be sufficiently smooth.Assume thatare obtained by using the spectral collocation scheme (2.18)-(2.20)，together with a polynomial interpolation (2.17).If μ associated with the weakly singular kernel satisfies 0＜μ＜1 and u∈，then

provided that N is sufficiently large，where C is a constant independent of N but which will depend on the bounds of the functionsand the index μ，and

ProofFirst，using the weighted inner product，we note that

and，by using the discrete inner product，we set

Then，the numerical scheme (2.18)-(2.20) can be written as

where (4.6) and (4.7) are obtained by Remark 2.1.By subtracting (4.5) from (2.12)，subtracting (4.6) from (2.10) and subtracting (4.7) from (2.11) and letting eu(x)=u (x)-，eu′(x)=u′(x)-，we obtain that

Using the integration error estimate in Lemma 3.1，we have

Multiplying Fi(x) on both sides of eqs.(4.8) and (4.9) and summing up from i=0 to i=N yields

Due to eqs.(4.14)-(4.15)，and using Dirichlet’s formula，which states that

and provided that the integral exists，we obtain

Denoting D:={(x，s):-1≤s≤x，x∈[-1，1]}，we have

Eq.(4.14) gives

It follows from the Gronwall inequality in Lemma 3.4 that

It follows from (4.15) that

Using Lemma 3.2，and the estimates (4.11) and (4.17)，we have

Due to Lemma 3.3，

By virtue of Lemma 3.3(2) with m=1，we have that

We now estimate the term J5(x).It follows from Lemmas 3.5 and 3.6 that

where in the last step we used Lemma 3.6 under the following assumption:

We now obtain the estimate forby using (4.16):

The above estimate，together with (4.17)，yields that

This completes the proof of the theorem. □

Next we will give the error analysis inspace.

Theorem 4.2If the hypotheses given in Theorem 4.1 hold，then

for any κ∈(0，1-μ)，provided that N is sufficiently large and C is a constant independent of N，where

ProofBy using the Gronwall inequality (Lemma 3.4) and the Hardy inequality (Lemma 3.7)，we obtain that

Now，using Lemma 3.8，we have that

By the convergence result in Theorem 4.1(m=1)，we have that

Due to Lemma 3.3，

By virtue of Lemma 3.3(1) with m=1，

Finally，it follows from Lemmas 3.5 and 3.8 that

where，in the last step，we used Lemma 3.6 for any κ∈(0，1-μ).By the convergence result in Theorem 4.1，we obtain that

for N sufficiently large and for any κ∈(0，1-μ).The desired estimates (4.18) and (4.19) are obtained. □

5 Numerical Example

Writing U′=，U=(u0，u1，···，uN)T，U-1=u-1×(1，1，···，1)Tand V=(v0，v1，···，vN)T，we obtain the following equations of the matrix form from (2.18)-(2.20):

The entries of the matrices are given by

We give a numerical example to con firm our analysis.Consider weakly singular Volterra integro-differential equations with a pantograph delay

Figure 1 Comparison between approximate solution uN and exact solution u (left);Comparison between approximate derivative and exact derivative u′(right)

Figure 2 The errors u-uN(left) andu′-(right) versus the number of collocation pointsin L∞ and norms

Table 1 The errors

Table 1 The errors

Table 2 The errors

Table 2 The errors

6 Conclusion and Future Work

This paper has given a Jacobi-collocation spectral method for Volterra-integro-differential equations with a pantograph delay which contain a weakly singular kernel (t-s)-μ，0＜μ＜1，under the hypothesis that the solution is smooth.The main point of this work is that it has demonstrated rigorously that the errors of spectral approximations decay exponentially in both the L∞-norm and the-norm;This is a desired feature for a spectral method.In a future article we will extend our work to the fractional Volterra-integro-differential equations that contain a pantograph delay.