Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

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 𝐿^∞ norm and weighted 𝐿²-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, 𝐷^𝛾𝑦(𝑡)=𝑦(𝑡)+𝑦𝑞𝑡+𝑔(𝑡)+^𝑡₀𝐾₁()𝑦(𝑠)+𝑡,𝑠𝑑𝑠_0𝑞𝑡𝐾₂()𝑦(𝜏)̃[],𝑦(0)𝑡,𝜏𝑑𝜏,0<𝛾≤1,𝑡∈𝐼∶=0,𝑇=𝑦₀, (1) with 0<𝑞<1, where 𝑔∶𝐼̃→𝑅, 𝐾₁∶𝐷→𝑅  (𝐷∶={(𝑡,𝜏)∶0≤𝑠≤𝑡≤𝑇}), and 𝐾₂∶𝐷→𝑅  (𝐷∶={(𝑡,𝜏)∶0≤𝜏≤𝑞𝑡,𝑡∈𝐼}̃) are given functions and are assumed to be sufficiently smooth in the respective domains. In (1), 𝐷^𝛾 denotes the fractional derivative of fractional order 𝛾.

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 Γ(⋅) denote the Gamma function. For any positive integer 𝑛 and 𝑛−1<𝛾<𝑛, the Caputo derivative is defined as follows: 𝐷^𝛾𝑦(𝑡)=1Γ𝑛−𝛾^𝑡₀𝑦^(𝑛)(𝑠)(𝑡−𝑠)^{(𝛾−𝑛+1)}𝑑𝑠,𝑡∈[0,𝑇]. (2) The Riemann-Liouville fractional integral 𝐼^𝛾 of order 𝛾 is defined as 𝐼^𝛾𝑦(𝑡)=1Γ𝛾^𝑡₀(𝑡−𝑠)^𝛾−1𝑦(𝑠)𝑑𝑠, (3) we note that 𝐼^𝛾𝐷^𝛾𝑦(𝑡)=𝑦(𝑡)−^𝑛−1_𝑘=0𝑦^(𝑘)(0)𝑡^𝑘.𝑘! (4) From (4), fractional integrodifferential equation (1) can be described as 𝐷^𝛾𝑦(𝑡)=𝑦(𝑡)+𝑦𝑞𝑡+𝑔(𝑡)+^𝑡₀𝐾₁()𝑦(𝑠)+𝑡,𝑠𝑑𝑠_0𝑞𝑡𝐾₂()𝑦(𝜏)̃[],𝑦(𝑡)=1𝑡,𝜏𝑑𝜏,0<𝛾≤1,𝑡∈𝐼∶=0,𝑇Γ𝛾^𝑡₀(𝑡−𝜏)^𝛾−1𝐷^𝛾𝑦(𝜏)𝑑𝜏+𝑦(0). (5)

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 [0,𝑇]), 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 𝐼∶=[−1,1]. Hence, we employ the transformation 𝑇𝑡=2()𝑇()1+𝑥,𝑠=1+𝜃2𝑇,𝜏=1+𝜂2; (6) then, the previous problem (5) becomes 𝐷^𝛾𝑢(𝑥)(𝑥)(𝑥)+=𝑢+𝑢𝑞𝑥+𝑞−1+𝑔_𝑥−1𝐾₁(𝑥,𝜃)𝑢(𝜃)𝑑𝜃+^{𝑞𝑥+𝑞−1}₋₁𝐾₂𝑢𝜂𝑢(𝑥)=1𝑥,𝜂𝑑𝜂,𝑥∈𝐼,Γ𝛾𝑇2^𝛾𝑥₋₁(𝑥−𝑠)^𝛾−1𝐷^𝛾𝑢(𝑠)𝑑𝑠+𝑢(−1), (7)(8) where 𝑢(𝑥)𝑇=𝑦2(1+𝑥),𝐷^𝛾𝑢(𝑥)=𝐷^𝛾𝑦𝑇2(),𝑔(𝑥)=𝑇1+𝑥2𝑔𝑇2(),𝐾1+𝑥₁()=𝑇𝑥,𝜃4𝐾₁𝑇2(),𝑇1+𝑥2(,𝐾1+𝜃)₂=𝑇𝑥,𝜂4𝐾₂𝑇2(),𝑇1+𝑥21+𝜂. (9)(10)

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 𝐿^∞ norm and weighted 𝐿²-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, 𝐶 will denote a generic positive constant that is independent of 𝑁 but which will depend on 𝑞, 𝑇 and on the bounds for the given functions 𝑎, 𝑏, and 𝐾_𝑗, 𝑗=1,2.

2. Jacobi-Collocation Method

Let 𝜔^𝛼,𝛽(𝑥)=(1−𝑥)^𝛼(1+𝑥)^𝛽 be a weight function in the usual sense, for 𝛼,𝛽>−1. The set of Jacobi polynomials {𝐽_𝑛^𝛼,𝛽(𝑥)}^∞_𝑛=0 forms a complete 𝐿²_{𝜔𝛼,𝛽}(−1,1)-orthogonal system, where 𝐿²_{𝜔𝛼,𝛽}(−1,1) is a weighted space defined by 𝐿²_{𝜔𝛼,𝛽}(−1,1)=𝑣∶𝑣ismeasurableand‖𝑣‖_{𝜔𝛼,𝛽}<∞, (11) equipped with the norm ‖𝑣‖_{𝜔𝛼,𝛽}=¹₋₁|𝑣(𝑥)|²𝜔^𝛼,𝛽(𝑥)𝑑𝑥^1/2, (12) and the inner product (𝑢,𝑣)_{𝜔𝛼,𝛽}=¹₋₁𝑢(𝑥)𝑣(𝑥)𝜔^𝛼,𝛽(𝑥)𝑑𝑥,∀𝑢,𝑣∈𝐿²_{𝜔𝛼,𝛽}(−1,1). (13)

For a given 𝑁≥0, we denote by {𝜃_𝑘}^𝑁_𝑘=0 the Legendre points and by {𝜔_𝑘}^𝑁_𝑘=0 the corresponding Legendre weights (i.e., Jacobi weights {𝜔_𝑘^0,0}^𝑁_𝑘=0). Then, the Legendre-Gauss integration formula is ¹₋₁𝑓(𝑥)𝑑𝑥≈^𝑁_𝑘=0𝑓𝜃_𝑘𝜔_𝑘. (14) Similarly, we denote by {𝜃_𝑘}^𝑁_𝑘=0 the Jacobi-Gauss points and by {𝜔_𝑘^𝛼,𝛽}^𝑁_𝑘=0 the corresponding Jacobi weights. Then, the Jacobi-Gauss integration formula is ¹₋₁𝑓(𝑥)𝜔^𝛼,𝛽(𝑥)𝑑𝑥≈^𝑁_𝑘=0𝑓𝜃_𝑘𝜔_𝑘^𝛼,𝛽. (15)

For a given positive integer 𝑁, we denote the collocation points by {𝑥_𝑖^𝛼,𝛽}^𝑁_𝑖=0, which is the set of (𝑁+1) Jacobi-Gauss points corresponding to the weight 𝜔^𝛼,𝛽(𝑥). Let 𝒫_𝑁 denote the space of all polynomials of degree not exceeding 𝑁. For any 𝑣∈𝐶[−1,1], we can define the Lagrange interpolating polynomial 𝐼_𝑁^𝛼,𝛽𝑣∈𝒫_𝑁, satisfying 𝐼_𝑁^𝛼,𝛽𝑣𝑥_𝑖=𝑣𝑥_𝑖,0≤𝑖≤𝑁. (16) The Lagrange interpolating polynomial can be written in the form 𝐼_𝑁^𝛼,𝛽𝑣(𝑥)=^𝑁_𝑖−0𝑣𝑥_𝑖𝐹_𝑖(𝑥),0≤𝑖≤𝑁, (17) where 𝐹_𝑖(𝑥) is the Lagrange interpolation basis function associated with {𝑥_𝑖}^𝑁_𝑖=0.

Let −𝜇=𝛾−1. 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 𝑥, there is little information available for 𝑢. To overcome this difficulty, we transfer the integration interval to a fixed interval [−1,1] by using the following variable changes 𝜃=1+𝑥2𝜈+𝑥−12(),𝑞()≜𝜃𝑥,𝜈𝜂=1+𝑥2𝑞()𝜉+1+𝑥2(),𝐷−1≜𝜂𝑥,𝜉_𝛾𝑢(𝑥)(𝑥)(𝑥)+=𝑢+𝑢𝑞𝑥+𝑞−1+𝑔1+𝑥2¹₋₁𝐾₁((𝑢(𝜃(+𝑞()𝑥,𝜃𝑥,𝜈))𝑥,𝜈))𝑑𝜈1+𝑥2¹₋₁𝐾₂()𝑢𝜂()𝑢(𝑥)=1𝑥,𝜂𝑥,𝜉𝑥,𝜉𝑑𝜉,Γ𝛾1+𝑥2^𝛾1₋₁(1−𝜈)^−𝜇𝐷^𝛾𝑢×(𝜃(𝑥,𝜈))𝑑𝜈+𝑢(−1). (18)(19)(20)

Set the collocation points as the set of (𝑁+1) Jacobi-Gauss points, {𝑥_𝑖^{−𝜇,−𝜇}}^𝑁_𝑖=0 associated with 𝜔^{−𝜇,−𝜇}. Assume that (19) and (20) holds at 𝑥_𝑖^{−𝜇,−𝜇}: 𝐷^𝛾𝑢𝑥_𝑖^{−𝜇,−𝜇}=𝑢𝑥_𝑖^{−𝜇,−𝜇}+𝑢𝑞𝑥_𝑖^{−𝜇,−𝜇}𝑥+𝑞−1+𝑔_𝑖^{−𝜇,−𝜇}+1+𝑥_𝑖^{−𝜇,−𝜇}2×¹₋₁𝐾₁𝑥_𝑖^{−𝜇,−𝜇},𝜃𝑥_𝑖^{−𝜇,−𝜇}𝜃𝑥,𝜈×𝑢_𝑖^{−𝜇,−𝜇}+𝑞,𝜈𝑑𝜈1+𝑥_𝑖^{−𝜇,−𝜇}2×¹₋₁𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑥_𝑖^{−𝜇,−𝜇}𝜂𝑥,𝜉×𝑢_𝑖^{−𝜇,−𝜇}𝑢𝑥,𝜉𝑑𝜉,_{𝑖−𝜇,−𝜇}=1Γ𝛾1+𝑥_𝑖^{−𝜇,−𝜇}𝑇4^𝛾×¹₋₁(1−𝜈)^−𝜇𝐷^𝛾𝑢𝜃𝑥_𝑖^{−𝜇,−𝜇}().,𝜈𝑑𝜈+𝑢−1 (21)(22)

Next, using a (𝑁+1)-point Gauss quadrature formula relative to the Jacobi weight {𝜔_𝑘^0,0}^𝑁_𝑖=0, the integration term in (21) can be approximated by ¹₋₁𝐾₁𝑥^{−𝜇,−𝜇},𝜃𝑥_𝑖^{−𝜇,−𝜇},𝜈𝑢𝜃𝑥_𝑖^{−𝜇,−𝜇}=,𝜈𝑑𝜈_𝑁𝑘=0𝐾₁𝑥_𝑖,𝜃𝑥_𝑖,𝜈_𝑘𝑢𝜃𝑥_𝑖,𝜈_𝑘𝜔_𝑘^0,0,₁₋₁𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑥_𝑖^{−𝜇,−𝜇},𝜉𝑢𝜂𝑥_𝑖^{−𝜇,−𝜇}=,𝜉𝑑𝜉_𝑁𝑘=0𝐾₂𝑥_𝑖,𝜂𝑥_𝑖,𝜉_𝑘𝑢𝜂𝑥_𝑖,𝜉_𝑘𝜔_𝑘^0,0. (23) The sets {𝜈_𝑘}^𝑁_𝑘=0 and {𝜉_𝑘}^𝑁_𝑘=0 coincide with the Jacobi-Gauss points corresponding Jacobi weights {𝜔_𝑘^0,0}^𝑁_𝑘=0; that is, {𝜈_𝑘}^𝑁_𝑘=0 and {𝜉_𝑘}^𝑁_𝑘=0 are Legendre-Gauss points.

Using a (𝑁+1)-point Gauss quadrature formula relative to the Jacobi weight 𝜔^−𝜇,0, the integration term in (22) can be approximated by ¹₋₁(1−𝜈)^𝜇𝐷^𝛾𝑢𝜃𝑥_𝑖^{−𝜇,−𝜇}=,𝜈𝑑𝜃_𝑁𝑘=0𝐷^𝛾𝑢𝜃𝑥_𝑖^{−𝜇,−𝜇},𝜈_𝑘𝜔_𝑘^−𝜇,0, (24) where the set {𝜈_𝑘}^𝑁_𝑘=0 is the Jacobi-Gauss points corresponding to the weight 𝜔^−𝜇,0.

We use 𝑢_𝑖, 𝑢^𝛾_𝑖 to approximate the function value 𝑢(𝑥_𝑖^{−𝜇,−𝜇}), 𝐷^𝛾𝑢(𝑥_𝑖^{−𝜇,−𝜇}), 0≤𝑖≤𝑁, and expand 𝑢 and 𝑢^𝛾 using Lagrange interpolation polynomials; that is, 𝑈(𝑥)=^𝑁_𝑗=0𝑢_𝑗𝐹_𝑗(𝑥),𝑈^𝛾(𝑥)=^𝑁_𝑗=0𝑢^𝛾_𝑗𝐹_𝑗(𝑥), (25) where 𝐹_𝑗(𝑥) is the Lagrange interpolation basis function associated with {𝑥_𝑖^{−𝜇,−𝜇}}^𝑁_𝑖=0 which is the set of (𝑁+1) Jacobi-Gauss points. The Jacobi collocation methods are to seek {𝑢^𝛾_𝑖}^𝑁_𝑖=0 and {𝑢_𝑖}^𝑁_𝑖=0 such that the following collocation equations hold: 𝑢^𝛾_𝑖=𝑢_𝑖+^𝑁_𝑗=0𝑢_𝑗𝐹_𝑗𝑞𝑥_𝑖^{−𝜇,−𝜇}𝑥+𝑞−1+𝑔_𝑖^{−𝜇,−𝜇}+1+𝑥_𝑖^{−𝜇,−𝜇}2^𝑁_𝑗=0𝑢_𝑗^𝑁_𝑘=0𝐾₁𝑥_𝑖^{−𝜇,−𝜇},𝜃𝑥_𝑖,𝜈_𝑘×𝐹_𝑗𝜃𝑥_𝑖^{−𝜇,−𝜇},𝜈_𝑘𝜔_𝑘^0,0+𝑞1+𝑥_𝑖^{−𝜇,−𝜇}2^𝑁_𝑗=0𝑢_𝑗^𝑁_𝑘=0𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑥_𝑖,𝜈_𝑘×𝐹_𝑗𝜂𝑥_𝑖^{−𝜇,−𝜇},𝜈_𝑘𝜔_𝑘^0,0,𝑢_𝑖=1Γ𝛾1+𝑥_𝑖^{−𝜇,−𝜇}𝑇4^𝛾𝑁_𝑗=0𝑢^𝛾_𝑗×^𝑁_𝑘=0𝐹_𝑗𝑠𝑥_𝑖^{−𝜇,−𝜇},𝜈_𝑘𝜔_𝑘^−𝜇,0+𝑢(−1). (26)(27)

Writing 𝑈_𝑁=(𝑢₀,𝑢₁,…,𝑢_𝑁)^𝑇 and 𝑈^𝛾_𝑁=(𝑢^𝛾₀,𝑢^𝛾₁,…,𝑢^𝛾_𝑁)^𝑇, we obtain the following of the matrix form from (26)-(27): 𝑈^𝛾_𝑁=(𝐸+𝐿+𝐶+𝐷)𝑈_𝑁𝑈+𝐺,_𝑁=𝑈₋₁+𝐵𝑈^𝛾_𝑁, (28) where 𝑔𝑥𝐸istheidentitymatrix,𝐺=₀^{−𝜇,−𝜇},𝑔𝑥₁^{−𝜇,−𝜇},…,𝑔𝑥_𝑁^{−𝜇,−𝜇𝑇},𝐿_𝑖𝑗=𝐹_𝑗𝑞𝑥_𝑖^{−𝜇,−𝜇},𝐶+𝑞−1_𝑖𝑗=1+𝑥_𝑖^{−𝜇,−𝜇}2^𝑁_𝑘=0𝐾₁𝑥_𝑖^{−𝜇,−𝜇},𝜃𝑥_𝑖^{−𝜇,−𝜇},𝜈_𝑘𝐹_𝑗×𝜃𝑥_𝑖,𝜈_𝑘𝜔_𝑘^0,0,𝐷_𝑖𝑗=𝑞1+𝑥_𝑖^{−𝜇,−𝜇}2^𝑁_𝑘=0𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑥_𝑖^{−𝜇,−𝜇},𝜈_𝑘𝐹_𝑗×𝜂𝑥_𝑖^{−𝜇,−𝜇},𝜈_𝑘𝜔_𝑘^0,0,𝐵_𝑖𝑗=1Γ𝛾1+𝑥_𝑖^{−𝜇,−𝜇}𝑇4^𝛾×^𝑁_𝑘=0𝐹_𝑗𝑠𝑥_𝑖^{−𝜇,−𝜇},𝜈_𝑘𝜔_𝑘^−𝜇,0,𝑈₋₁=(𝑢(−1),𝑢(−1),…,𝑢(−1))^𝑇. (29) We can get the values of {𝑢_𝑖}^𝑁_𝑖=0 and {𝑢^𝛾_𝑖}^𝑁_𝑖=0 by solving the system of linear system (28). Therefore, the expressions of 𝑈(𝑥) and 𝑈^𝛾(𝑥) 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 𝐼∶=(−1,1).

Lemma 1 (see [27]). Assume that an (𝑁+1)-point Gauss quadrature formula relative to the Jacobi weight is used to integrate the product 𝑢𝜑, where 𝑢∈𝐻^𝑚(𝐼) with 𝐼 for some 𝑚≥1 and 𝜑∈𝒫_𝑁. Then, there exists a constant 𝐶 independent of N such that ¹₋₁𝑢(𝑥)𝜑(𝑥)𝑑𝑥−𝑢,𝜑_𝑁≤𝐶𝑁^−𝑚|𝑢|_𝐻𝜔^𝑚,𝑁_{𝛼,𝛽(𝐼)}𝜑_{𝐿2𝜔𝛼,𝛽(𝐼)}, (30) where |𝑢|_𝐻𝜔^𝑚,𝑁_{𝛼,𝛽(𝐼)}=^𝑚_{()𝑗=min𝑚,𝑁+1}𝑢^(𝑗)2_{𝐿2𝜔𝛼,𝛽(𝐼)}^1/2,𝑢,𝜑_𝑁=^𝑁_𝑗=0𝑢𝑥_𝑗𝜑𝑥_𝑗𝜔_𝑗. (31)

Lemma 2 (see [26, 27]). Assume that 𝑢∈𝐻_𝜔^𝑚,𝑁_{−𝜇,−𝜇}(𝐼) and denote by 𝐼_𝑁^{−𝜇,−𝜇}𝑢 its interpolation polynomial associated with the (𝑁+1) Jacobi-Gauss points {𝑥_𝑗}^𝑁_𝑗=0; namely, 𝐼_𝑁^{−𝜇,−𝜇}𝑢=^𝑁_𝑖=0𝑢𝑥_𝑖𝐹𝑥_𝑖. (32) Then, the following estimates hold:𝑢−𝐼_𝑁^{−𝜇,−𝜇}𝑢_{𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−𝑚|𝑢|_𝐻𝜔^𝑚,𝑁_{𝛼,𝛽(𝐼)},𝑢−𝐼_𝑁^{−𝜇,−𝜇}𝑢_{𝐿∞(𝐼)}≤𝐶𝑁^{1−𝜇−𝑚}|𝑢|_𝐻𝜔𝑐^𝑚,𝑁_(𝐼)1,0≤𝜇<2,𝐶𝑁^1/2−𝑚log𝑁|𝑢|_𝐻𝜔𝑐^𝑚,𝑁_(𝐼),12≤𝜇<1, (33a)(33b)where 𝜔^𝑐=𝜔^{−1/2,−1/2} denotes the Chebyshev weight function.

Lemma 3 (see [28]). Assume that {𝐹_𝑗(𝑥)}^𝑁_𝑗=0 are the 𝑁-th degree Lagrange basis polynomials associated with the Gauss points of the Jacobi polynomials. Then, 𝐼_𝑁^𝛼,𝛽_{𝐿∞(𝐼)}≤max_[]^𝑁_{𝑥∈−1,1𝑗=0}𝐹_𝑗(𝑥=𝒪1)log𝑁,−1<𝛼,𝛽≤−2,𝒪𝑁^𝛾+1/2,𝛾=max𝛼,𝛽,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (34)

Lemma 4 (Gronwall inequality, see [29] Lemma 7.1.1). Suppose that 𝐿≥0, 0<𝜇<1, and 𝑢 and 𝑣 are a nonnegative, locally integrable functions defined on [−1,1] satisfying 𝑢(𝑥)≤𝑣(𝑥)+𝐿^𝑥₋₁(𝑥−𝜏)^−𝜇𝑢(𝜏)𝑑𝜏. (35) Then, there exists a constant 𝐶=𝐶(𝜇) such that 𝑢(𝑥)≤𝑣(𝑥)+𝐶𝐿^𝑥₋₁(𝑥−𝜏)^−𝜇𝑣(𝜏)𝑑𝜏,𝑓𝑜𝑟−1≤𝑥<1. (36)

Lemma 5 (see [30, 31]). For a nonnegative integer 𝑟 and 𝜅∈(0,1), there exists a constant 𝐶_𝑟,𝜅>0 such that for any function 𝑣∈𝐶^𝑟,𝜅([−1,1]), there exists a polynomial function 𝒯_𝑁𝑣∈𝒫_𝑁 such that 𝑣−𝒯_𝑁𝑣_{𝐿∞(𝐼)}≤𝐶_𝑟,𝜅𝑁^{−(𝑟+𝜅)}‖𝑣‖_𝑟,𝜅, (37) where ‖⋅‖_𝑟,𝜅 is the standard norm in 𝐶^𝑟,𝜅([−1,1]) which is denoted by the space of functions whose 𝑟th derivatives are Hölder continuous with exponent 𝜅, endowed with the usual norm ‖𝑣‖_𝑟,𝜅=max_{0≤𝜅≤𝑟}max_{[]𝑥∈−1,1}𝜕^𝜅_𝑥𝑣(𝑥)+max_{0≤𝜅≤𝑟}sup_{[]𝑥,𝑦∈−1,1,𝑥≠𝑦}𝜕^𝜅_𝑥𝑣(𝑥)−𝜕^𝜅_𝑥𝑣(𝑥)𝑥−𝑦^𝜅. (38)𝒯_𝑁 is a linear operator from 𝐶^𝑟,𝜅([−1,1]) into 𝒫_𝑁.

Lemma 6 (see [32]). Let 𝜅∈(0,1) and let ℳ be defined by (ℳ𝑣)(𝑥)=^𝑥₋₁(𝑥−𝜏)^−𝜇𝐾(𝑥,𝜏)𝑣(𝜏)𝑑𝜏. (39) Then, for any function 𝑣∈𝐶([−1,1]), there exists a positive constant 𝐶 such that ℳ𝑣𝑥^′−ℳ𝑣𝑥^′′𝑥^′−𝑥^′′≤𝐶max_{[]𝑥∈−1,1}|𝑣(𝑥)|, (40) under the assumption that 0<𝜅<1−𝜇, for any 𝑥^′,𝑥^′′∈[−1,1] and 𝑥^′≠𝑥^′′. This implies that ‖ℳ𝑣‖_0,𝜅≤𝐶max_{[]𝑥∈−1,1}|𝑣(𝑥)|,0<𝜅<1−𝜇. (41)

Lemma 7 (see [33]). For every bounded function 𝑣, there exists a constant 𝐶, independent of 𝑣 such that sup_𝑁^𝑁_𝑗=0𝑣𝑥_𝑗𝐹_𝑗(𝑥)_{𝐿2𝜔𝛼,𝛽(𝐼)}≤𝐶max_{[]𝑥∈−1,1}|𝑣(𝑥)|, (42) where 𝐹_𝑗(𝑥),  𝑗=0,1,…,𝑁, are the Lagrange interpolation basis functions associated with the Jacobi collocation points {𝑥_𝑗}^𝑁_𝑗=0.

Lemma 8 (see [34]). For all measurable function 𝑓≥0, the following generalized Hardy’s inequality 𝑇𝑓(^𝑏_𝑎𝑥)^𝑞𝑢(𝑥)𝑑𝑥^1/𝑞≤^𝑏_𝑎𝑓(𝑥)^𝑝𝑣(𝑥)𝑑𝑥^1/𝑝 (43) holds if and only if sup_{𝑎<𝑥<𝑏}^𝑏_𝑥𝑢(𝑡)𝑑𝑡^1/𝑞𝑥_𝑎𝑣^1−𝑝′(𝑡)𝑑𝑡^1/𝑝′<∞,𝑝^′=𝑝𝑝−1 (44) for the case 1<𝑝≤𝑞<∞. Here, 𝑇 is an operator of the form (𝑇𝐹)(𝑥)=^𝑥_𝑎𝑘(𝑥,𝑡)𝑓(𝑡)𝑑𝑡, (45) with 𝑘(𝑥,𝑡) a given kernel, 𝑢, 𝑣 are nonnegative weight functions, and −∞≤𝑎<𝑏≤∞.

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 𝐿^∞ space.

Theorem 9. Let 𝑢(𝑥) be the exact solution of the fractional delay-integrodifferential equation (7)-(8), which is assumed to be sufficiently smooth. Assume that 𝑈(𝑥) and 𝑈^𝛾(𝑥) are obtained by using the spectral collocation scheme (26)-(27) together with a polynomial interpolation (25). If 𝛾 associated with the weakly singular kernel satisfies 0<𝛾<1 and 𝑢∈𝐻_𝜔^𝑚+1_{−𝜇,−𝜇}(𝐼), then 𝑈^𝛾−𝐷^𝛾𝑢_{𝐿∞(𝐼)}≤𝐶𝑁^{𝛾−1/2−𝑚}𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰,12<𝛾<1,𝐶𝑁^−𝑚log𝑁𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰1,0<𝛾≤2,‖𝑈−𝑢‖_𝐿^∞_(𝐼)≤𝐶𝑁^{𝛾−1/2−𝑚}𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰,12<𝛾<1,𝐶𝑁^−𝑚log𝑁𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰1,0<𝛾≤2, (46)(47) provided that 𝑁 is sufficiently large, where 𝐶 is a constant independent of 𝑁 but which will depend on the bounds of the functions 𝐾(𝑥,𝑠) and the index 𝜇, 𝐾^∗=max_{[]𝑥∈−1,1}𝐾₁(𝑥,𝜃(𝑥,⋅))_𝐻𝜔^𝑚,𝑁_0,0(𝐼)+max_{[]𝑥∈−1,1}𝐾₂𝑥,𝜂(𝑥,⋅)_𝐻𝜔^𝑚,𝑁_0,0(𝐼),𝒰=𝐷^𝛾𝑢_𝐻𝜔𝑐^𝑚,𝑁_(𝐼)+|𝑢|_𝐻𝜔𝑐^𝑚,𝑁_(𝐼). (48)(49)

Proof. We let 𝐾𝑠𝑥_𝑖,𝜈,𝜙𝑠𝑥_𝑖,𝜈_𝑁=^𝑁_𝑘=0𝐾𝑠𝑥_𝑖,𝜈_𝑘𝜙𝑠𝑥_𝑖,𝜈_𝑘𝜔_𝑘^0,0. (50) The numerical scheme (26)-(27) can be written as 𝑢^𝛾_𝑖=𝑢_𝑖+^𝑁_𝑗=0𝑢_𝑗𝐹_𝑗𝑞𝑥_𝑖^{−𝜇,−𝜇}+𝑞−1+𝑔𝑥_𝑖^{−𝜇,−𝜇}+1+𝑥_𝑖^{−𝜇,−𝜇}2𝐾₁𝑥_𝑖^{−𝜇,−𝜇},𝜃𝑥_𝑖^{−𝜇,−𝜇},𝑈𝜃𝑥,𝜈_{𝑖−𝜇,−𝜇},𝜈_𝑁+𝑞1+𝑥_𝑖^{−𝜇,−𝜇}2𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑥_𝑖^{−𝜇,−𝜇},𝑈𝜂𝑥,𝜈_{𝑖−𝜇,−𝜇},𝜈_𝑁,𝑢_𝑖=1Γ𝛾𝑇2^𝛾𝑥𝑖₋₁𝑥_𝑖−𝜃^−𝜇𝑈^𝛾(𝜃)𝑑𝜃+𝑢₋₁, (51)(52) which gives 𝑢^𝛾_𝑖=𝑢_𝑖+^𝑁_𝑗=0𝑢_𝑗𝐹_𝑗𝑞𝑥_𝑖^{−𝜇,−𝜇}+𝑞−1+𝑔𝑥_𝑖^{−𝜇,−𝜇}+1+𝑥_𝑖^{−𝜇,−𝜇}2¹₋₁𝐾₁𝑥_𝑖^{−𝜇,−𝜇},𝜃𝑥_𝑖^{−𝜇,−𝜇}𝜃𝑥,𝜈×𝑈_𝑖^{−𝜇,−𝜇},𝜈𝑑𝜈+𝐼₁𝑥_𝑖^{−𝜇,−𝜇}+𝑞1+𝑥_𝑖^{−𝜇,−𝜇}2¹₋₁𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑥_𝑖^{−𝜇,−𝜇}𝜂𝑥,𝜉×𝑈_𝑖^{−𝜇,−𝜇},𝜉𝑑𝜉+𝐼₂𝑥_𝑖^{−𝜇,−𝜇}, (53) where 𝐼₁(𝑥)=1+𝑥2¹₋₁𝐾₁((𝑥(𝜃(−𝑥,𝜃)),𝑈𝑥,𝜈))𝑑𝜈1+𝑥2𝐾₁(𝑥,𝜃(𝑥,𝜈)),𝑈(𝜃(𝑥,𝜈))_𝑁,𝐼₂(𝑥)=𝑞(1+𝑥)2¹₋₁𝐾₂()𝑈𝜂()−𝑞()𝑥,𝜂𝑥,𝜉𝑥,𝜉𝑑𝜉1+𝑥2𝐾₂𝑥,𝜂(𝑥,𝜈),𝑈𝜂(𝑥,𝜈)_𝑁. (54) Using the integration error estimates from Jacobi-Gauss polynomials quadrature in Lemma 1, we have 𝐼₁(𝑥)≤𝐶𝑁^−𝑚𝐾₁(𝑥,𝜃(𝑥,⋅))_{𝐻𝜔𝑚;𝑁0,0(𝐼)}‖𝑈(𝜃(𝑥,⋅))‖_𝐿²_(𝐼)≤𝐶𝑁^−𝑚𝐾^∗₁‖𝑈‖_𝐿²_(𝐼),𝐼₂(𝑥)≤𝐶𝑁^−𝑚𝐾₂𝑥,𝜂(𝑥,⋅)_{𝐻𝜔𝑚;𝑁0,0(𝐼)}𝑈𝜂(𝑥,⋅)_𝐿2(𝐼)≤𝐶𝑁^−𝑚𝐾^∗₂‖𝑈‖_𝐿²_(𝐼),𝐾^∗₁=max_{[]𝑥∈−1,1}𝐾₁(𝑥,𝜃(𝑥,⋅))_𝐻𝜔^𝑚,𝑁_0,0(𝐼),𝐾^∗₂=max_{[]𝑥∈−1,1}𝐾₁(𝑥,𝜃(𝑥,⋅))_𝐻𝜔^𝑚,𝑁_0,0(𝐼). (55)(56)(57) From (18), (53) can be rewritten as 𝑢^𝛾_𝑖=𝑎𝑥_𝑖^{−𝜇,−𝜇}𝑢_𝑖+𝑏𝑥_𝑖^{−𝜇,−𝜇}×^𝑁_𝑗=0𝑢_𝑗𝐹_𝑗𝑞𝑥_𝑖^{−𝜇,−𝜇}+𝑞−1+𝑔𝑥_𝑖^{−𝜇,−𝜇}+^{𝑥𝑖−𝜇,−𝜇}₋₁𝐾₁𝑥_𝑖^{−𝜇,−𝜇},𝜃𝑈(𝜃)𝑑𝜃+𝐼₁𝑥_𝑖^{−𝜇,−𝜇}+^{𝑞𝑥𝑖−𝜇,−𝜇+𝑞−1}₋₁𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑈𝜂𝑑𝜂+𝐼₂𝑥_𝑖^{−𝜇,−𝜇}. (58) Let 𝑒 and 𝑒^𝛾 denote the error function 𝑒(𝑥)=𝑈(𝑥)−𝑢(𝑥),𝑒^𝛾(𝑥)=𝑈^𝛾(𝑥)−𝑢^𝛾(𝑥), (59) we have 𝑢^𝛾_𝑖=𝑎𝑥_𝑖^{−𝜇,−𝜇}𝑢_𝑖+𝑏𝑥_𝑖^{−𝜇,−𝜇}×^𝑁_𝑗=0𝑢_𝑗𝐹_𝑗𝑞𝑥_𝑖^{−𝜇,−𝜇}+𝑞−1+𝑔𝑥_𝑖^{−𝜇,−𝜇}+^{𝑥𝑖−𝜇,−𝜇}₋₁𝐾₁𝑥_𝑖^{−𝜇,−𝜇}𝑢(𝜃)+,𝜃𝑑𝜃_{𝑥𝑖−𝜇,−𝜇−1}𝐾₁𝑥_𝑖^{−𝜇,−𝜇}𝑒(𝜃)+,𝜃𝑑𝜃_{𝑞𝑥𝑖−𝜇,−𝜇+𝑞−1−1}𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑢𝜂𝑑𝜂+^{𝑞𝑥𝑖−𝜇,−𝜇+𝑞−1}₋₁𝐾₂𝑥_𝑖^{−𝜇,−𝜇},𝜂𝑒𝜂𝑑𝜂+𝐼₁𝑥_𝑖^{−𝜇,−𝜇}+𝐼₂𝑥_𝑖^{−𝜇,−𝜇}. (60)
Multiplying 𝐹_𝑖(𝑥) on both sides of (62) and (52), summing up from 0 to 𝑁, and using (7)-(8) yield 𝑈^𝛾(𝑥)=𝑈(𝑥)+𝐼_𝑁^{−𝜇,−𝜇𝑁}_𝑗=0𝑢_𝑗𝐹_𝑗𝑞𝑥+𝑞−1+𝐼_𝑁^{−𝜇,−𝜇}𝑔(𝑥)+𝐼_𝑁^{−𝜇,−𝜇}𝐷^𝛾𝑢(𝑥)(𝑥)𝑞(𝑥)(𝑥−𝑢−𝑢+𝑞−1−𝑔)+𝐼_𝑁^{−𝜇,−𝜇𝑥}₋₁𝐾₁(𝑥,𝜃)𝑒(𝜃)𝑑𝜃+𝐼_𝑁^{−𝜇,−𝜇𝑞𝑥+𝑞−1}₋₁𝐾₂𝑥,𝜂𝑒𝜂𝑑𝜂+𝐽₁(𝑥)+𝐽₂(𝑥),𝑈(𝑥)=𝐼_𝑁^{−𝜇,−𝜇}𝑢(𝑥)−𝑢₋₁+𝐼_𝑁^{−𝜇,−𝜇}1Γ𝛾^𝑥₋₁(𝑥−𝜃)^−𝜇𝑒^𝛾(𝜃)𝑑𝜃+𝐼_𝑁^{−𝜇,−𝜇}𝑢₋₁, (61)(62) where 𝐽₁(𝑥)=𝐼_𝑁^{−𝜇,−𝜇}𝐼₁(𝑥),𝐽₂(𝑥)=𝐼_𝑁^{−𝜇,−𝜇}𝐼₂(𝑥). (63) From (61)-(62), we have 𝑒^𝛾(𝑥)=𝑒(𝑥)+^𝑥₋₁𝐾₁()𝑒(𝜃)+𝑥,𝜃𝑑𝜃_{𝑞𝑥+𝑞−1−1}𝐾₂𝑥,𝜂𝑒𝜂𝑑𝜂+𝐽₁(𝑥)+𝐽₂(𝑥)+𝐽₃(𝑥)+𝐽₄(𝑥)+𝐽₅(𝑥)+𝐽₆(𝑥)+𝐽₇(𝑥),𝑒(𝑥)=1Γ𝛾^𝑥₋₁(𝑥−𝜃)^−𝜇𝑒^𝛾(𝜃)𝑑𝜃+𝐽₄(𝑥)+𝐽₈(𝑥), (64)(65) where 𝐽₃(𝑥)=𝐼_𝑁^{−𝜇,−𝜇}𝐷^𝛾𝑢(𝑥)−𝐷^𝛾𝑢(𝑥),𝐽₄(𝑥)=𝐼_𝑁^{−𝜇,−𝜇}𝑢(𝑥)(𝑥),𝐽−𝑢₅(𝑥)=𝐼_𝑁^{−𝜇,−𝜇}𝑢,𝐽𝑞𝑥+𝑞−1−𝑢𝑞𝑥+𝑞−1₆(𝑥)=𝐼_𝑁^{−𝜇,−𝜇𝑥}₋₁𝐾₁()𝑒(𝜃)−𝑥,𝜃𝑑𝜃_𝑥−1𝐾()𝑒(𝑠)𝐽𝑥,𝑠𝑑𝑠,₇(𝑥)=𝐼_𝑁^{−𝜇,−𝜇𝑞𝑥+𝑞−1}₋₁𝐾₂𝑥,𝜂𝑒𝜂𝑑𝜂−^{𝑞𝑥+𝑞−1}₋₁𝐾₂𝑒𝜂𝐽𝑥,𝜂𝑑𝜂,₈(𝑥)=𝐼_𝑁^{−𝜇,−𝜇}1Γ𝛾^𝑥₋₁(𝑥−𝜃)^−𝜇𝑒^𝛾(𝜃)−1𝑑𝜃Γ𝛾^𝑥₋₁(𝑥−𝜃)^−𝜇𝑒^𝛾(𝜃)𝑑𝜃. (66)
Due to (64), we obtain 𝑒^𝛾(𝑥)=1Γ𝛾^𝑥₋₁(𝑥−𝜃)^−𝜇𝑒^𝛾(𝜃)+1𝑑𝜃Γ𝛾^𝑥₋₁𝐾₁(𝑥,𝜃)^𝜃₋₁(𝜃−𝜏)^−𝜇𝑒^𝛾(𝜏)+1𝑑𝜏𝑑𝜃Γ𝛾^{𝑞𝑥+𝑞−1}₋₁𝐾₂𝑥,𝜂^𝜂₋₁𝜂−𝑧^−𝜇×𝑒^𝛾(𝑧)𝑑𝑧𝑑𝜂+𝐻(𝑥)+𝐽₁(𝑥)+𝐽₂(𝑥)+𝐽₃(𝑥)+𝐽₄(𝑥)+𝐽₅(𝑥)+𝐽₆(𝑥)+𝐽₇(𝑥), (67) where 𝐻(𝑥)=^𝑥₋₁𝐾₁(𝑥,𝜃)^𝜃₋₁𝐽₄(𝜃)+𝐽₈(𝜃)+1𝑑𝜃Γ𝛾^{𝑞𝑥+𝑞−1}₋₁𝐾₂𝑥,𝜂^𝜂₋₁𝐽₄𝜂+𝐽₈𝜂𝑑𝜂+𝐽₄(𝑥)+𝐽₈(𝑥). (68) Using the Dirichlet’s formula which sates that ^𝑥₋₁^𝜃₋₁Φ(𝜃,𝜏)𝑑𝜏𝑑𝜃=^𝑥₋₁^𝑥_𝜏Φ(𝜃,𝜏)𝑑𝜃𝑑𝜏, (69) provided that the integral exists, we obtain 1Γ𝛾^𝑥₋₁𝐾₁(𝑥,𝜃)^𝜃₋₁(𝜃−𝜏)^−𝜇𝑒^𝛾(𝜏)=1𝑑𝜏𝑑𝜃Γ𝛾^𝑥₋₁^𝑥_𝜃𝐾₁(𝑥,𝜃)(𝜃−𝜏)^−𝜇𝑑𝜃𝑒^𝛾(𝜏)≤1𝑑𝜏Γ𝛾2^1−𝜇max_{[]𝑥,𝜏∈−1,1}𝐾₁(𝑥,𝜏)^𝑥₋₁𝑒^𝛾(𝜏)𝑑𝜏≜𝑀₁^𝑥₋₁𝑒^𝛾(𝜏)𝑑𝜏, (70) letting 𝜂=𝑞𝜏+𝑞−1, 𝑧=𝑞𝜃+𝑞−1, we have 1Γ𝛾^{𝑞𝑥+𝑞−1}₋₁𝐾₂𝑥,𝜂^𝜂₋₁𝜂−𝑧^−𝜇𝑒^𝛾(𝑧)=1𝑑𝑧𝑑𝜂Γ𝛾𝑞^𝑥₋₁𝐾₂×𝑥,𝑞𝜏+𝑞−1_{𝑞𝜏+𝑞−1−1}(𝑥−𝑧)^−𝜇𝑒^𝛾(𝑧)=1𝑑𝑧𝑑𝜏Γ𝛾𝑞^1+𝛾𝑥₋₁𝐾₂×𝑥,𝑞𝜏+𝑞−1_𝜏−1(1−𝜃)^−𝜇𝑒^𝛾=1𝑞𝜃+𝑞−1𝑑𝜃𝑑𝜏Γ𝛾𝑞^1+𝛾𝑥₋₁^𝑥_𝜃𝐾₂×()𝑥,𝑞𝜏+𝑞−11−𝜃^−𝜇𝑑𝜏𝑒^𝛾×≤1𝑞𝜃+𝑞−1𝑑𝜃Γ𝛾2^1−𝜇𝑞^1+𝛾max_{[]𝑥,𝜏∈−1,1}𝐾₂(×𝑥,𝜏)_𝑥−1𝑒^𝛾𝑞𝜃+𝑞−1𝑑𝜃≜𝑀₂^𝑥₋₁𝑒^𝛾𝑞𝜃+𝑞−1𝑑𝜃. (71) Then, (67) gives 𝑒^𝛾(𝑥)≤1Γ𝛾^𝑥₋₁(𝑥−𝜃)^−𝜇𝑒^𝛾(𝜃)𝑑𝜃+𝑀₁^𝑥₋₁𝑒^𝛾(𝜏)𝑑𝜏+𝑀₂^𝑥₋₁𝑒^𝛾(𝑥)𝑞𝜃+𝑞−1𝑑𝜃+𝐻+𝐽₁(𝑥)+𝐽₂(𝑥)+𝐽₃(𝑥)+𝐽₄(𝑥)+𝐽₅(𝑥)+𝐽₆(𝑥)+𝐽₇(𝑥). (72) It follows from the Gronwall inequality that 𝑒^𝛾(𝑥)_{𝐿∞(𝐼)}≤⁸_𝑖=1𝐽_𝑖(𝑥)_{𝐿∞(𝐼)}. (73) From (65), we have ‖𝑒(𝑥)‖_𝐿^∞_(𝐼)≤⁸_𝑖=1𝐽_𝑖(𝑥)_{𝐿∞(𝐼)}. (74)
We now apply Lemma 7 to obtain that𝐽_{1𝐿∞(𝐼)}≤𝐶𝑁^(1/2)−𝜇max_{0≤𝑖≤𝑁}𝐼_𝑖,11,0<𝜇<2,𝐶log𝑁max_{0≤𝑖≤𝑁}𝐼_𝑖,1,12≤≤𝜇<1,𝐶𝑁^{(1/2)−𝜇−𝑚}𝐾^∗₁‖𝑢‖_𝐿²_(𝐼)+𝑒^𝛾(𝑥)_{𝐿∞(𝐼)}1,0<𝜇<2,𝐶𝑁^−𝑚log𝑁𝐾^∗₁‖𝑢‖_𝐿²_(𝐼)+𝑒^𝛾(𝑥)_{𝐿∞(𝐼)},12𝐽≤𝜇<1,_{2𝐿∞(𝐼)}≤𝐶𝑁^(1/2)−𝜇max_{0≤𝑖≤𝑁}𝐼_𝑖,21,0<𝜇<2,𝐶log𝑁max_{0≤𝑖≤𝑁}𝐼_𝑖,1,12≤≤𝜇<1,𝐶𝑁^{(1/2)−𝜇−𝑚}𝐾^∗₂‖𝑢‖_𝐿²_(𝐼)+𝑒^𝛾(𝑥)_{𝐿∞(𝐼)}1,0<𝜇<2,𝐶𝑁^−𝑚log𝑁𝐾^∗₂‖𝑢‖_𝐿²_(𝐼)+𝑒^𝛾(𝑥)_{𝐿∞(𝐼)},12≤𝜇<1. (75)Due to Lemma 2, 𝐽_{3𝐿∞(𝐼)}≤𝐶𝑁^{1−𝜇−𝑚}𝐷^𝛾𝑢_𝐻𝜔𝑐^𝑚,𝑁_(𝐼)1,0<𝜇<2,𝐶𝑁^(1/2)−𝑚log𝑁𝐷^𝛾𝑢_𝐻𝜔𝑐^𝑚,𝑁_(𝐼),12𝐽≤𝜇<1,_{4𝐿∞(𝐼)}≤𝐶𝑁^{1−𝜇−𝑚}|𝑢|_𝐻𝜔𝑐^𝑚,𝑁_(𝐼)1,0<𝜇<2,𝐶𝑁^(1/2)−𝑚log𝑁|𝑢|_𝐻𝜔𝑐^𝑚,𝑁_(𝐼),12𝐽≤𝜇<1,_{5𝐿∞(𝐼)}≤𝐶𝑁^{1−𝜇−𝑚}|𝑢|_𝐻𝜔𝑐^𝑚,𝑁_(𝐼)1,0<𝜇<2,𝐶𝑁^(1/2)−𝑚log𝑁|𝑢|_𝐻𝜔𝑐^𝑚,𝑁_(𝐼),12≤𝜇<1. (76)
By virtue of Lemma 2 (33b) with 𝑚=1, 𝐽_{6𝐿∞(𝐼)}≤𝐶𝑁^−𝜇‖𝑒‖_𝐿^∞_(𝐼)1,0<𝜇<2,𝐶𝑁^−1/2‖𝑒‖_𝐿^∞_(𝐼),12𝐽≤𝜇<1,_{7𝐿∞(𝐼)}≤𝐶𝑁^−𝜇‖𝑒‖_𝐿^∞_(𝐼)1,0<𝜇<2,𝐶𝑁^−1/2‖𝑒‖_𝐿^∞_(𝐼),12≤𝜇<1. (77)
We now estimate the term 𝐽₈(𝑥). It follows from Lemmas 5 and 6 with 𝐾(𝑥,𝜏)=1/Γ(𝛾) that 𝐽_{5𝐿∞(𝐼)}=𝐼_𝑁^{−𝜇,−𝜇}−𝐼ℳ𝑒^𝛾_{𝐿∞(𝐼)}=𝐼_𝑁^{−𝜇,−𝜇}−𝐼ℳ𝑒^𝛾−𝒯_𝑁ℳ𝑒^𝛾_{𝐿∞(𝐼)}≤1+𝐼_𝑁^{−𝜇,−𝜇}_{𝐿∞(𝐼)}𝐶𝑁^−𝑘ℳ𝑒^𝛾_0,𝜅≤𝐶𝑁^{(1/2)−𝜇−𝜅}𝑒^𝛾_{𝐿∞(𝐼)}1,0<𝜇<2,𝐶𝑁^−𝜅log𝑁𝑒^𝛾_{𝐿∞(𝐼)},12≤𝜇<1, (78) where in the last step we have used Lemma 6 under the following assumption: 121−𝜇<𝜅<1−𝜇,when0<𝜇<2,10<𝜅<1−𝜇,when2≤𝜇<1. (79)
Provided that 𝑁 is sufficiently large, combining (75), (76), (77), and (78) gives 𝑈^𝛾(𝑥)−𝑢^𝛾(𝑥)_{𝐿∞(𝐼)}≤𝐶𝑁^{1/2−𝜇−𝑚}𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰1,0<𝜇<2,𝐶𝑁^−𝑚log𝑁𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰,12‖𝑈(𝑥)(𝑥≤𝜇<1,−𝑢)‖_𝐿^∞_(𝐼)≤𝐶𝑁^{1/2−𝜇−𝑚}𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰1,0<𝜇<2,𝐶𝑁^−𝑚log𝑁𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰,12≤𝜇<1. (80) Using 𝛾=1−𝜇, we have the desired estimates (46) and (47).

Next, we will give the error estimates in 𝐿²_{𝜔−𝜇,−𝜇} space.

Theorem 10. If the hypotheses given in Theorem 9 hold, then𝑈^𝛾(𝑥)−𝑢^𝛾(𝑥)_{𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−𝑚𝑉₁+𝑁^{𝛾−1/2−𝜅}𝑉₂+𝑁^𝛾−𝜅𝒰,12<𝛾<1,𝐶𝑁^−𝑚𝑉₁+𝑁^−𝜅log𝑁𝑉₂+𝑁^1/2−𝜅1log𝑁𝒰,0<𝛾≤2,‖𝑈(𝑥)−𝑢(𝑥)‖_𝐿²_{𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−𝑚𝑉₁+𝑁^{𝛾−1/2−𝜅}𝑉₂+𝑁^𝛾−𝜅𝒰,12<𝛾<1,𝐶𝑁^−𝑚𝑉₁+𝑁^−𝜅log𝑁𝑉₂+𝑁^1/2−𝜅1log𝑁𝒰,0<𝛾≤2, (81)for any 𝜅∈(0,𝛾) provided that 𝑁 is sufficiently large and 𝐶 is a constant independent of 𝑁, where 𝑉₁=𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝐷^𝛾𝑢_𝐻𝜔𝑐^𝑚;𝑁_(𝐼)+|𝑢|_𝐻𝜔𝑐^𝑚;𝑁_(𝐼),𝑉₂=𝐾^∗‖𝑢‖_𝐿²_(𝐼). (82)

Proof. By using the generalization of Gronwalls Lemma 4 and the Hardy inequality Lemma 8, it follows from (72) that 𝑒^𝛾_{𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶⁸_𝑖=1𝐽_{𝑖𝐿2𝜔−𝜇,−𝜇(𝐼)},‖𝑒‖_𝐿²_{𝜔−𝜇,−𝜇(𝐼)}≤𝐶⁸_𝑖=1𝐽_{𝑖𝐿2𝜔−𝜇,−𝜇(𝐼)}. (83) Now, using Lemma 7, we have 𝐽_{1𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶max_{[]𝑥∈−1,1}|𝐼(𝑥)|≤𝐶𝑁^−𝑚𝐾^∗₁‖𝑢‖_𝐿²_(𝐼)+‖𝑒‖_𝐿^∞_(𝐼),𝐽_{2𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶max_{[]𝑥∈−1,1}|𝐼(𝑥)|≤𝐶𝑁^−𝑚𝐾^∗₂‖𝑢‖_𝐿²_(𝐼)+‖𝑒‖_𝐿^∞_(𝐼). (84) By the convergence result in Theorem 9 (𝑚=1), we have ‖𝑒‖_𝐿^∞_(𝐼)≤𝐶𝐷^𝛾𝑢_𝐻𝜔𝑐^𝑚;𝑁_(𝐼)+|𝑢|_𝐻𝜔𝑐^𝑚;𝑁_(𝐼)+‖𝑢‖_𝐿²_(𝐼). (85) So that 𝐽_{1𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−𝑚𝐾^∗₁𝐷^𝛾𝑢_𝐻𝜔𝑐^𝑚;𝑁_(𝐼)+|𝑢|_𝐻𝜔𝑐^𝑚;𝑁_(𝐼)+‖𝑢‖_𝐿²_(𝐼),𝐽_{2𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−𝑚𝐾^∗₂𝐷^𝛾𝑢_𝐻𝜔𝑐^𝑚;𝑁_(𝐼)+|𝑢|_𝐻𝜔𝑐^𝑚;𝑁_(𝐼)+‖𝑢‖_𝐿²_(𝐼). (86) Due to Lemma 2 (33a), 𝐽_{3𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−𝑚𝐷^𝛾𝑢_𝐻𝜔𝑐^𝑚;𝑁_(𝐼),𝐽_{4𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−𝑚|𝑢|_𝐻𝜔𝑐^𝑚;𝑁_(𝐼),𝐽_{5𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−𝑚|𝑢|_𝐻𝜔𝑐^𝑚;𝑁_(𝐼). (87) By virtue of Lemma 2 (33a) with 𝑚=1, 𝐽_{6𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^−1𝑥₋₁𝐾₁(𝑥,𝜃)𝑒(𝜃)𝑑𝜃_{𝐻𝜔1;𝑁−𝜇,−𝜇(𝐼)}≤𝐶𝑁⁻¹‖𝑒‖_𝐿²_{𝜔−𝜇,−𝜇(𝐼)},𝐽_{7𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^{−1𝑞𝑥+𝑞−1}₋₁𝐾₂𝑥,𝜂𝑒𝜂𝑑𝜂_{𝐻𝜔1;𝑁−𝜇,−𝜇(𝐼)}≤𝐶𝑁⁻¹‖𝑒‖_𝐿²_{𝜔−𝜇,−𝜇(𝐼)}. (88) Finally, it follows from Lemmas 5 and 7 that 𝐽_{8𝐿2𝜔−𝜇,−𝜇(𝐼)}=𝐼_𝑁^{−𝜇,−𝜇}−𝐼ℳ𝑒^𝛾_{𝐿2𝜔−𝜇,−𝜇(𝐼)}=𝐼_𝑁^{−𝜇,−𝜇}−𝐼(ℳ𝑒^𝛾−𝒯_𝑁𝑒^𝛾_{𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐼_𝑁^{−𝜇,−𝜇}ℳ𝑒^𝛾−𝒯_𝑁𝑒^𝛾_{𝐿2𝜔−𝜇,−𝜇(𝐼)}+ℳ𝑒^𝛾−𝒯_𝑁𝑒^𝛾_{𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶ℳ𝑒^𝛾−𝒯_𝑁𝑒^𝛾_{𝐿∞(𝐼)}≤𝐶𝑁^−𝜅ℳ𝑒^𝛾_0,𝜅≤𝐶𝑁^−𝜅𝑒^𝛾_{𝐿∞(𝐼)}, (89) where in the last step we used Lemma 6 for any 𝜅∈(0,1−𝜇). By the convergence result in Theorem 9, we obtain that 𝐽_{8𝐿2𝜔−𝜇,−𝜇(𝐼)}≤𝐶𝑁^{1/2−𝜇−𝑚−𝜅}𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰1,0<𝜇<2,𝐶𝑁^{−𝑚−𝜅}log𝑁𝐾^∗‖𝑢‖_𝐿²_(𝐼)+𝑁^1/2𝒰,12≤𝜇<1, (90) for 𝑁 sufficiently large and for any 𝜅∈(0,1−𝜇). The desired estimates (81) follows from the previous estimates and (83) with 𝛾=1−𝜇.

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: 𝐷^0.75𝑦(𝑡)=6𝑡^2.25Γ()−𝑡3.25²𝑒^𝑡5𝑦(𝑡)−𝑞𝑡²5𝑦+𝑞𝑡_𝑡0𝑒^𝑡𝑠𝑦(𝑠)𝑑𝑠+₀^𝑞𝑡(𝜏)[],𝑦(0)𝜏𝑦𝑑𝜏,𝑡∈0,1=0. (91) The corresponding exact solution is given by 𝑦(𝑡)=𝑡³.

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 2≤𝑁≤20 in both 𝐿^∞ and 𝐿²_{𝜔−𝜇,−𝜇} 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 𝑦(𝑡) (a) and approximate fraction derivative and exact derivative 𝐷0.75𝑦(𝑡) (b).

(a)

(b)

Figure 2 

Example 1: the errors of numerical and exact solution 𝑦(𝑡) (a) and the errors of numerical and exact solution 𝐷0.75𝑦(𝑡) (b) versus the number of collocation points in 𝐿∞ and 𝐿2𝜔 norms.

Example 2. Consider the following fractional integrodifferential equation: 𝐷^𝛼𝑦(𝑡)=2𝑡−𝑦(𝑡)+𝑦𝑞𝑡+𝑡(1+2𝑡)^𝑡₀𝑒^{𝑠(𝑡−𝑠)}𝑦(𝑠)+𝑑𝑠_0𝑞𝑡𝑞𝑡𝑒^{𝜏(𝑞𝑡−𝜏)}𝑦(𝜏)[],𝑦(0)𝑑𝜏,𝑡∈0,1=1, (92) when 𝛼=1, the exact solution of (92) is 𝑦(𝑡)=𝑒^𝑡2.

In the only case of 𝛼=1, we know the exact solution. We have reported the obtained numerical results for 𝑁=20 and 𝛼 = 0.25, 0.5, 0.75, and 1 in Figure 3. We can see that, as 𝛼 approaches 1, the numerical solutions converge to the analytical solution 𝑦(𝑡)=𝑒^𝑡2; 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 𝑁 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 𝛼 and exact solution of 𝑦(𝑡) with 𝛼=1 (a). Comparison between approximate solution and exact solution of 𝑦′(𝑡).

(a)

(b)

Figure 4 

Example 2: the errors of numerical and exact solution 𝑦(𝑡) (a) and the errors of numerical and exact solution 𝑦′(𝑡) (b) versus the number of collocation points in 𝐿∞ and 𝐿2𝜔 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).