Applied and Computational Mathematics
Volume 4, Issue 3, June 2015, Pages: 152-161

Volterra Integral Equations with Vanishing Delay

Xiaoxuan Li, Weishan Zheng*, Jiena Wu

Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, China

Email address:

(Weishan Zheng)

To cite this article:

Xiaoxuan Li, Weishan Zheng, Jiena Wu. Volterra Integral Equations with Vanishing Delay. Applied and Computational Mathematics. Vol. 4, No. 3, 2015, pp. 152-161. doi: 10.11648/j.acm.20150403.18


Abstract: In this article, we use a Chebyshev spectral-collocation method to solve the Volterra integral equations with vanishing delay. Then a rigorous error analysis provided by the proposed method shows that the numerical error decay exponentially in the infinity norm and in the Chebyshev weighted Hilbert space norm. Numerical results are presented, which confirm the theoretical predicition of the exponential rate of convergence.

Keywords: Chebyshev Spectral-Collocation Method, Volterra Integral Equations, Vanishing Delay, Error Estimate, Convergence Analysis


1. Introduction

The Volterra integral equations (VIES) arise in many modeling problems in mathematical physics and chemical reaction, such as in the heat conduction, potential theory, fluid dynamic and radiative heat transfer problems. There are many methods to solve VIES, 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, according to [5], we use a Chebyshev spectral-collocation method, where the collocation points are Chebyshev Gauss, Chebyshev Gauss-Radau, Chebyshev Gauss-Lobatto points to solve Volterra integral equations with vanishing delay.

2. Definition

The Volterra integral equations with the vanishing delay are defined as

(1)

where , and

(2)

where and  is a constant,

3. Chebyshev Spectral-Collocation Method

For ease of analysis, we will transfer the integral interval  and  to fixed interval [1,1].

3.1. The Changes of Variables

Firstly, we use the changes of variables

Then (1) becomes

(3)

where

3.2. Set the Collocation Points

Now we assume that(see,e.g., [6]) are the set of Chebyshev Gauss, or Chebyshev Gauss-Radau, or Chebyshev Gauss-Lobatto points, then (3) holds at :

(4)

3.3. Linear Transformation

We make two simple linear transformations

 

Then (4) becomes

(5)

where

Applying appropriate  Gauss quadrature formula, we can obtain that

where  are the Legendre Gauss, or Legendre Gauss-Radau, or Legendre Gauss-Lobatto points, corresponding weight ,  (see, e.g.[6]). We use  to approximate the function value  and use  to approximate the function , where is the  Lagrange basic function.

First, deal with  

Similarly, = ,

then

(6)

The Chebyshev spectral-collocation method is to seek   such that satisfies the above equation.

3.4. Implementation of the Spectral Collocation Algorithm

(6) can be written in matrix form:

where

We now discuss an efficient computation of .Considering Chebyshev function

(7)

where  is called the discrete polynomial coefficients of  The inverse relation (see, e.g., [6]) is

(8)

and  is the weight corresponding to ,

and  In addition,  if are the  Chebyshev Gauss, or Chebyshev Gauss-Radau points,  if  are the  Chebyshev Gauss-Lobatto points.

4. Convergence Analysis

4.1. Some Spaces

For simplicity, we denote  by , .

For non-negative integer , we define  with the norm as .

For a nonnegative integer, we define the semi-norm

when , we denote  by . When , we denote  by .

The space  is the Banach space of the measurable functions  that is bounded outside a set of measure zero, equipped the norm .  is the space of all polynomials of degree not exceeding .

4.2. Lemmas

Lemma 1. [6,7] If ,  then

(9)

(10)

where is the interpolation operater associated with the Chebyshev Gauss, or Chebyshev Gauss-Radau, or Chebyshev Gauss-Lobatto points .

Lemma 2. [6,7] If and , then there exists a constant  independent of  such that

and

where  is the Chebyshev Gauss, or Chebyshev Gauss-Radau, or Chebyshev Gauss-Lobatto point, corresponding weight , and is -point Legendre Gauss, or Legendre Gauss-Radau, or Legendre Gauss-Lobatto point, corresponding weight , .

Lemma 3. If ,  are the Lagrange interpolation polynomials associated with the  Chebyshev Gauss, or Chebyshev Gauss-Radau, or Chebyshev Gauss-Lobatto points , then

.

Lemma 4. [1,8] (Gronwall inequality) Suppose  is a non-negative, locally integrable function satisfying  where is a constant,  is a integrable function, then there exists a constant  such that

Lemma 5. Assume that  is a non-negative integrable function and satisfies

 , () ,

where is also a non-negative integrable function, then

Proof. Since

Then

Lemma 6. [9] For all measurable function, the following generalized Hardy’s inequality  holds if and only if

whereis an operator of the form .

Lemma 7. [10] For every bounded function, there exists a constant  independent of  such that

4.3. Theorems

4.3.1. Convergence Analysis in Space

Theorem 1. Suppose  is the exact solution to (3) and  is the approximate solution obtained by using the spectral collocation schemes (6). Then for  sufficiently large, there is

(11)

where

Proof. Subtracting (6) from (5) gives

Then

(12)

where

By Lemma 2,

(13)

Multiplying  on both sides of the error equation (12) and summing up from  to  yield

besides,

So

(14)

Consequently,

(15)

where

By Lemma 5, we get

(16)

Using Lemma 1 for  yields

(17)

Using Lemma 7 and (13), we have

(18)

By Lemma 3,

(19)

Similarly,

(20)

According to Lemma 1 with  to  yields

(21)

Similarly,

(22)

So

(23)

Since  then, for  sufficiently large,

Hence

This completes the proof of this theorem.

4.3.2. Convergence Analysis in Space

Theorem 2. Suppose  is the exact solution to (3), and  is the approximate solution obtained by using the spectral collocation schemes (6). Then for  sufficiently large, we have

where

Proof. Applying Lemma 5, it follows from (15) that

By Lemma 6, we have

Using Lemma 1 for  yields

(24)

With Lemma 7, we get

(25)

From Theorem 1, let , then (11) becomes

,

This makes (25) become

(26)

Similarly,

(27)

As the same analysis in (21), we obtain that

From Theorem 1, we get

(28)

Similarly,

(29)

Hence, we have

This completes the proof of this Theorem.

5. Examples

5.1. Example 1

From (1), let.

The corresponding exact solution is .

We use the numerical scheme (6).Numerical errors versus several values of  are displayed in Table 1 and Figure . These results indicate that the desired spectral accuracy is obtained. Figure presents the approximate solution () and the exact solution, which are found in excellent agreement.

Table 1. The errors  versus the number of collocation points in and norms.

N 6 8 10 12 14

2.4058e-004 2.0357e-006 1.3388e-008 6.6231e-011 2.5646e-013

2.6350e-004 2.3946e-006 1.5653e-008 7.9650e-011 3.1722e-013
N 16 18 20 22 24

2.1316e-014 7.9936e-015 7.9936e-015 7.1054e-015 1.1546e-014

1.3511e-014 1.0471e-014 1.4066e-014 1.2207e-014 1.6227e-014

Figure 1. The errors  versus the number of collocation points in and norms.

Figure 2. Comparison between approximate solution  and the exact solution .

5.2. Example 2

From (1), let

The corresponding exact solution is

Figure plots the errors for  in both and  norms. The approximate solution () and the exact solution are displayed in Table 2. As expected, the errors decay exponentially which confirmed our theoretical predictions.

Table 2. The errors versus the number of collocation points in and norms.

N 2 4 6 8 10

2.9023e-002 2.3295e-004 9.6846e-007 2.6571e-009 5.8210e-012

3.5380e-002 2.6275e-004 1.0136e-006 2.8130e-009 5.8353e-012
N 12 14 16 18 20

8.9928e-015 3.3307e-015 2.5535e-015 2.2204e-015 2.1094e-015

8.8042e-015 2.1409e-015 1.7146e-015 2.4297e-015 2.4104e-015

Figure 3. Plots the errors for  in both and norms.

Figure 4. Comparison between approximate solution () and the exact solution.

6. Conclusion

We successfully solve the Volterra integral equation with vanishing delay by Chebyshev spectral-collocation method and provide a rigorous error analysis for this method. We get the conclusion that the error of the approximate solution decay exponentially in norm and norm. We also carry out the numerical experiment which confirm the theoretical predicition of the exponential rate of convergence.


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