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Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations
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.
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 , fluid-dynamics traffic model , mathematical physics and engineering , fluid and continuum mechanics , chemistry, acoustics, and psychology .
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 , operational calculus methods , the Adomian decomposition method , and the iterative and series-based methods . A small number of algorithms for the numerical solution of FDEs have been suggested , 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 . 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.  introduced a quadrature shifted Legendre tau method based on the Gauss-Lobatto interpolation for solving multiorder FDEs with variable coefficients and in , shifted Legendre spectral methods have been developed for solving fractional-order multipoint boundary value problems. In , truncated Legendre series together with the Legendre operational matrix of fractional derivatives are used for the numerical integration of fractional differential equations. In , 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  and shifted Jacobi operational matrix  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 , 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
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 ). 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 ). Assume that are the -th degree Lagrange basis polynomials associated with the Gauss points of the Jacobi polynomials. Then,
Lemma 4 (Gronwall inequality, see  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 ). 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 ). 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 ). 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
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
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 thatDue 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, thenfor any provided that is sufficiently large and is a constant independent of , where