Advances in Mathematical Physics

Volume 2013 (2013), Article ID 821327, 14 pages

http://dx.doi.org/10.1155/2013/821327

Research Article

Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

Yin Yang¹ and Yunqing Huang²

¹College of Civil Engineering and Mechanics, Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China

²Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Xiangtan, Hunan 411105, China

Received 15 May 2013; Accepted 15 September 2013

Academic Editor: Varsha Daftardar-Gejji

Copyright © 2013 Yin Yang and Yunqing Huang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We propose and analyze a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The fractional derivative is described in the Caputo sense. We provide a rigorous error analysis for the collocation method, which shows that the error of approximate solution decays exponentially in norm and weighted -norm. The numerical examples are given to illustrate the theoretical results.

1. Introduction

Many phenomena in engineering, physics, chemistry, and other sciences can be described very successfully by models using mathematical tools from fractional calculus, that is, the theory of derivatives and integrals of fractional noninteger order. This allows one to describe physical phenomena more accurately. Moreover, fractional calculus is applied to the model frequency dependent damping behavior of many viscoelastic materials, economics, and dynamics of interfaces between nanoparticles and substrates. Recently, several numerical methods to solve fractional differential equations (FDEs) and fractional integrodifferential equations (FIDEs) have been proposed.

In this paper, we consider the general linear fractional pantograph delay-integrodifferential equations (FDIDEs) with proportional delays,

with , where ,   , and    are given functions and are assumed to be sufficiently smooth in the respective domains. In (1), denotes the fractional derivative of fractional order .

Differential and integral equations involving derivatives of noninteger order have shown to be adequate models for various phenomena arising in damping laws, diffusion processes, models of earthquake [1], fluid-dynamics traffic model [2], mathematical physics and engineering [3], fluid and continuum mechanics [4], chemistry, acoustics, and psychology [5].

Let denote the Gamma function. For any positive integer and , the Caputo derivative is defined as follows:

The Riemann-Liouville fractional integral of order is defined as

we note that

From (4), fractional integrodifferential equation (1) can be described as

Several analytical methods have been introduced to solve FDEs including various transformation techniques [6], operational calculus methods [7], the Adomian decomposition method [8], and the iterative and series-based methods [9]. A small number of algorithms for the numerical solution of FDEs have been suggested [10], and most of them are finite difference methods, which are generally limited to low dimensions and are of limited accuracy.

As we know, fractional derivatives are global (they are defined by an integral over the whole interval ), and therefore, global methods such as spectral methods are perhaps better suited for FDEs. Standard spectral methods possess an infinite order of accuracy for the equations with regular solutions, while failing for many complicated problems with singular solutions. So, it is relevant to be interested in how to enlarge the adaptability of spectral methods and construct certain simple approximation schemes without a loss of accuracy for more complicated problems.

Spectral methods have been proposed to solve fractional differential equations, such as the Legendre collocation method [11, 12], Legendre wavelets method [13, 14], and Jacobi-Gauss-Lobatto collocation method [15]. The authors in [16–18] constructed an efficient spectral method for the numerical approximation of fractional integrodifferential equations based on tau and pseudospectral methods. Moreover, Bhrawy et al. [19] introduced a quadrature shifted Legendre tau method based on the Gauss-Lobatto interpolation for solving multiorder FDEs with variable coefficients and in [20], shifted Legendre spectral methods have been developed for solving fractional-order multipoint boundary value problems. In [21], truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for the numerical integration of fractional differential equations. In [22], the authors derived a new explicit formula for the integral of shifted Chebyshev polynomials of any degree for any fractional-order. The shifted Chebyshev operational matrix [23] and shifted Jacobi operational matrix [24] of fractional derivatives have been developed, which are applied together with the spectral tau method for numerical solution of general linear and nonlinear multiterm fractional differential equations. However, very few theoretical results were provided to justify the high accuracy numerically obtained. Recently, Chen and Tang [25, 26] developed a novel spectral Jacobi-collocation method to solve second kind Volterra integral equations with a weakly singular kernel and provided a rigorous error analysis which theoretically justifies the spectral rate of convergence. Inspired by the work of [26], we extend the approach to fractional order delay-integrodifferential equations (1). However, it is difficult to apply the spectral approximations to the initial value problem and fractional order derivatives. To facilitate the use of the spectral methods, we restate the initial condition as an equivalent integral equation with singular kernel. Then, we get the discrete scheme by using Gauss quadrature formula. In this paper, we will provide a rigorous error analysis not only for approximate solutions but also for approximate fractional derivatives which theoretically justifies the spectral rate of convergence.

For ease of analysis, we will describe the spectral methods on the standard interval . Hence, we employ the transformation

then, the previous problem (5) becomes

where

This paper is organized as follows. In Section 2, we introduce the spectral approaches for pantograph FDIDEs. Some useful lemmas are provided in Section 3. These lemmas will play a key role in the derivation of the convergence analysis. We provide a rigorous error analysis for the spectral methods, which shows that both the errors of approximate solutions and the errors of approximate fractional derivatives of the solutions decay exponentially in norm and weighted -norm in Section 4, and Section 5 contains numerical results, which will be used to verify the theoretical results obtained in Section 4.

Throughout the paper, will denote a generic positive constant that is independent of but which will depend on , and on the bounds for the given functions , , and , .

2. Jacobi-Collocation Method

Let be a weight function in the usual sense, for . The set of Jacobi polynomials forms a complete -orthogonal system, where is a weighted space defined by

equipped with the norm

and the inner product

For a given , we denote by the Legendre points and by the corresponding Legendre weights (i.e., Jacobi weights ). Then, the Legendre-Gauss integration formula is

Similarly, we denote by the Jacobi-Gauss points and by the corresponding Jacobi weights. Then, the Jacobi-Gauss integration formula is

For a given positive integer , we denote the collocation points by , which is the set of Jacobi-Gauss points corresponding to the weight . Let denote the space of all polynomials of degree not exceeding . For any , we can define the Lagrange interpolating polynomial , satisfying

The Lagrange interpolating polynomial can be written in the form

where is the Lagrange interpolation basis function associated with .

Let . In order to obtain high order accuracy of the approximate solution, the main difficulty is to compute the integral terms in (7) and (8). In particular, for small values of , there is little information available for . To overcome this difficulty, we transfer the integration interval to a fixed interval by using the following variable changes

Set the collocation points as the set of Jacobi-Gauss points, associated with . Assume that (19) and (20) holds at :

Next, using a -point Gauss quadrature formula relative to the Jacobi weight , the integration term in (21) can be approximated by

The sets and coincide with the Jacobi-Gauss points corresponding Jacobi weights ; that is, and are Legendre-Gauss points.

Using a -point Gauss quadrature formula relative to the Jacobi weight , the integration term in (22) can be approximated by

where the set is the Jacobi-Gauss points corresponding to the weight .

We use , to approximate the function value , , , and expand and using Lagrange interpolation polynomials; that is,

where is the Lagrange interpolation basis function associated with which is the set of () Jacobi-Gauss points. The Jacobi collocation methods are to seek and such that the following collocation equations hold:

Writing and , we obtain the following of the matrix form from (26)-(27):

where

We can get the values of and by solving the system of linear system (28). Therefore, the expressions of and can be obtained.

3. Some Useful Lemmas

In this section, we will provide some elementary lemmas, which are important for the derivation of the main results in the subsequent section. Let .

Lemma 1 (see [27]). Assume that an -point Gauss quadrature formula relative to the Jacobi weight is used to integrate the product , where with for some and . Then, there exists a constant independent of N such that

where

Lemma 2 (see [26, 27]). Assume that and denote by its interpolation polynomial associated with the Jacobi-Gauss points ; namely,

Then, the following estimates hold:

where denotes the Chebyshev weight function.

Lemma 3 (see [28]). Assume that are the -th degree Lagrange basis polynomials associated with the Gauss points of the Jacobi polynomials. Then,

Lemma 4 (Gronwall inequality, see [29] Lemma ). Suppose that , , and and are a nonnegative, locally integrable functions defined on satisfying

Then, there exists a constant such that

Lemma 5 (see [30, 31]). For a nonnegative integer and , there exists a constant such that for any function , there exists a polynomial function such that

where is the standard norm in which is denoted by the space of functions whose th derivatives are Hölder continuous with exponent , endowed with the usual norm

is a linear operator from into .

Lemma 6 (see [32]). Let and let be defined by

Then, for any function , there exists a positive constant such that

under the assumption that , for any and . This implies that

Lemma 7 (see [33]). For every bounded function , there exists a constant , independent of such that

where ,  , are the Lagrange interpolation basis functions associated with the Jacobi collocation points .

Lemma 8 (see [34]). For all measurable function , the following generalized Hardy’s inequality

holds if and only if

for the case . Here, is an operator of the form

with a given kernel, , are nonnegative weight functions, and .

4. Convergence Analysis

This section is devoted to provide a convergence analysis for the numerical scheme. The goal is to show that the rate of convergence is exponential; that is, the spectral accuracy can be obtained for the proposed approximations. Firstly, we will carry our convergence analysis in space.

Theorem 9. Let be the exact solution of the fractional delay-integrodifferential equation (7)-(8), which is assumed to be sufficiently smooth. Assume that and are obtained by using the spectral collocation scheme (26)-(27) together with a polynomial interpolation (25). If associated with the weakly singular kernel satisfies and , then

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

Proof. We let

The numerical scheme (26)-(27) can be written as

which gives

where

Using the integration error estimates from Jacobi-Gauss polynomials quadrature in Lemma 1, we have

From (18), (53) can be rewritten as

Let and denote the error function

we have

Multiplying on both sides of (62) and (52), summing up from to , and using (7)-(8) yield

where

From (61)-(62), we have

where

Due to (64), we obtain

where

Using the Dirichlet’s formula which sates that

provided that the integral exists, we obtain

letting , , we have

Then, (67) gives

It follows from the Gronwall inequality that

From (65), we have

We now apply Lemma 7 to obtain that

Due to Lemma 2,

By virtue of Lemma 2 (33b) with ,

We now estimate the term . It follows from Lemmas 5 and 6 with that

where in the last step we have used Lemma 6 under the following assumption:

Provided that is sufficiently large, combining (75), (76), (77), and (78) gives

Using , we have the desired estimates (46) and (47).

Next, we will give the error estimates in space.

Theorem 10. If the hypotheses given in Theorem 9 hold, then

for any provided that is sufficiently large and is a constant independent of , where

Proof. By using the generalization of Gronwalls Lemma 4 and the Hardy inequality Lemma 8, it follows from (72) that

Now, using Lemma 7, we have

By the convergence result in Theorem 9 (), we have

So that

Due to Lemma 2 (33a),

By virtue of Lemma 2 (33a) with ,

Finally, it follows from Lemmas 5 and 7 that

where in the last step we used Lemma 6 for any . By the convergence result in Theorem 9, we obtain that

for sufficiently large and for any . The desired estimates (81) follows from the previous estimates and (83) with .

5. Algorithm Implementation and Numerical Experiments

In this subsection, we present the numerical results obtained by the proposed collocation spectral method. The estimates in Section 4 indicates that the convergence of numerical solutions is exponential if the exact solution is smooth. To confirm the theoretical prediction, a numerical experiment is carried out by considering the following example.

Example 1. Consider the following fractional integrodifferential equation:

The corresponding exact solution is given by .

Figure 1 presents the approximate and exact solutions on the left-hand side and presents the approximate and exact derivatives on the right-hand side, which are found in excellent agreement. In Figure 2, the numerical errors are plotted for in both and norms. As expected, an exponential rate of convergence is observed for the problem, which confirmed our theoretical predictions.

(a)

(b)

Figure 1: Example 1: comparison between approximate solution and exact solution (a) and approximate fraction derivative and exact derivative (b).

(a)

(b)

Figure 2: Example 1: the errors of numerical and exact solution (a) and the errors of numerical and exact solution (b) versus the number of collocation points in and norms.

Example 2. Consider the following fractional integrodifferential equation:

when , the exact solution of (92) is .

In the only case of , we know the exact solution. We have reported the obtained numerical results for and = 0.25, 0.5, 0.75, and 1 in Figure 3. We can see that, as approaches , the numerical solutions converge to the analytical solution ; that is, in the limit, the solution of fractional integrodifferential equations approach to that of the integer order integrodifferential equations. In Figure 4, we plot the resulting errors versus the number of the steps. This figure shows the exponential rate of convergence predicted by the proposed method.

(a)

(b)

Figure 3: Example 2: approximation solutions with different and exact solution of with (a). Comparison between approximate solution and exact solution of .

(a)

(b)

Figure 4: Example 2: the errors of numerical and exact solution (a) and the errors of numerical and exact solution (b) versus the number of collocation points in and norms.

6. Conclusions and Future Work

This paper proposes a spectral Jacobi-collocation approximation for fractional order integrodifferential equations of Volterra type with pantograph delay. The most important contribution of this work is that we are able to demonstrate rigorously that the errors of spectral approximations decay exponentially in both infinity and weighted norms, which is a desired feature for a spectral method.

We only investigated the case of pantograph delay in the present work, with the availability of this methodology, it will be possible to generalize this algorithm to solve the same problem in semi-infinite interval based on generalized Laguerre [35] and modified generalized Laguerre polynomials.

Acknowledgments

The authors are grateful to the referees for many useful suggestions. The work was supported by NSFC Project (11301446, 11031006), China Postdoctoral Science Foundation Grant (2013M531789), Project of Scientific Research Fund of Hunan Provincial Science and Technology Department (2013RS4057), and the Research Foundation of Hunan Provincial Education Department (13B116).

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