Volterra Integral Equations with Vanishing Delay
Xiaoxuan Li, Weishan Zheng^{*}, Jiena Wu
Department of Mathematics and Statistics, Hanshan Normal University, Chaozhou, China
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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 ①. These results indicate that the desired spectral accuracy is obtained. Figure ② 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 ③ 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.
References