# Spectral-Collocation Methods for Fractional Pantograph Delay-Integrodifferential Equations

Authors: Yin Yang; Yunqing Huang
Journal: Advances in Mathematical Physics
 (2013)

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## 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 inL∞ norm and weighted L2-norm. The numerical examples are given to illustrate the theoretical results.


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## Body

## 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,(1)Dγy(t)=y(t)+y(qt)+g^(t)+∫0t‍K^1(t,s)y(s)dsDγy(t)=+∫0qt‍K^2(t,τ)y(τ)dτ,Dγy(t)0<γ≤1,t∈I~≔[0,T],y(0)=y0,
with 0<q<1, where g~:I~→R, K^1:D→R  (D:={(t,τ):0≤s≤t≤T}), and K^2: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), Dγ 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].LetΓ(·) denote the Gamma function. For any positive integer n and n-1<γ<n, the Caputo derivative is defined as follows:
(2)Dγy(t)=1Γ(n-γ)∫0t‍y(n)(s)(t-s)(γ-n+1)ds,t∈[0,T].
The Riemann-Liouville fractional integral Iγ of order γ is defined as
(3)Iγy(t)=1Γ(γ)∫0t‍(t-s)γ-1y(s)ds,
we note that
(4)Iγ(Dγy(t))=y(t)-∑k=0n-1‍y(k)(0)tkk!.
From (4), fractional integrodifferential equation (1) can be described as
(5)Dγy(t)=y(t)+y(qt)+g^(t)+∫0t‍K^1(t,s)y(s)dsDγy(t)=+∫0qt‍K^2(t,τ)y(τ)dτ,Dγy(t)0<γ≤1,t∈I~≔[0,T],77y(t)=1Γ(γ)∫0t‍(t-τ)γ-1Dγy(τ)dτ+y(0).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 intervalI:=[-1,1]. Hence, we employ the transformation
(6)t=T2(1+x),s=T(1+θ)2,τ=T(1+η)2;
then, the previous problem (5) becomes
(7)Dγu(x)=u(x)+u(qx+q-1)+g(x)+∫-1x‍K1(x,θ)u(θ)dθ+∫-1qx+q-1‍K2(x,η)u(η)dη,x∈I,(8)u(x)=1Γ(γ)(T2)γ∫-1x‍(x-s)γ-1Dγu(s)ds+u(-1),
where
(9)u(x)=y(T2(1+x)),Dγu(x)=Dγy(T2(1+x)),g(x)=T2g^(T2(1+x)),(10)K1(x,θ)=T4K^1(T2(1+x),T2(1+θ)),K2(x,η)=T4K^2(T2(1+x),T2(1+η)).This paper is organized as follows. In Section2, 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 Kj, j=1,2.


## 2. Jacobi-Collocation Method

Letωα,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 {Jnα,β(x)}n=0∞ forms a complete Lωα,β2(-1,1)-orthogonal system, where Lωα,β2(-1,1) is a weighted space defined by
(11)Lωα,β2(-1,1)={v:vismeasurableand∥v∥ωα,β<∞},
equipped with the norm
(12)∥v∥ωα,β=(∫-11‍|v(x)|2ωα,β(x)dx)1/2,
and the inner product
(13)(u,v)ωα,β=∫-11‍u(x)v(x)ωα,β(x)dx,∀u,v∈Lωα,β2(-1,1).For a givenN≥0, we denote by {θ~k}k=0N the Legendre points and by {ωk}k=0N the corresponding Legendre weights (i.e., Jacobi weights {ωk0,0}k=0N). Then, the Legendre-Gauss integration formula is
(14)∫-11‍f(x)dx≈∑k=0N‍f(θk)ωk.
Similarly, we denote by {θk}k=0N the Jacobi-Gauss points and by {ωkα,β}k=0N the corresponding Jacobi weights. Then, the Jacobi-Gauss integration formula is
(15)∫-11‍f(x)ωα,β(x)dx≈∑k=0N‍f(θ~k)ωkα,β.For a given positive integerN, we denote the collocation points by {xiα,β}i=0N, which is the set of (N+1) Jacobi-Gauss points corresponding to the weight ωα,β(x). Let 𝒫N denote the space of all polynomials of degree not exceeding N. For any v∈C[-1,1], we can define the Lagrange interpolating polynomial INα,βv∈𝒫N, satisfying
(16)INα,βv(xi)=v(xi),0≤i≤N.
The Lagrange interpolating polynomial can be written in the form
(17)INα,βv(x)=∑i-0N‍v(xi)Fi(x),0≤i≤N,
where Fi(x) is the Lagrange interpolation basis function associated with {xi}i=0N.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
(18)θ=1+x2ν+x-12≜θ(x,ν),η=q(1+x)2ξ+q(1+x)2-1≜η(x,ξ),(19)Dγu(x)=u(x)+u(qx+q-1)+g(x)+1+x2∫-11‍K1(x,θ(x,ν))u(θ(x,ν))dν+q(1+x)2∫-11‍K2(x,η(x,ξ))u(η(x,ξ))dξ,(20)u(x)=1Γ(γ)(1+x2)γ∫-11‍(1-ν)-μDγu×(θ(x,ν))dν+u(-1).Set the collocation points as the set of(N+1) Jacobi-Gauss points, {xi-μ,-μ}i=0N associated with ω-μ,-μ. Assume that (19) and (20) holds at xi-μ,-μ:
(21)Dγu(xi-μ,-μ)=u(xi-μ,-μ)+u(qxi-μ,-μ+q-1)∫-11‍+g(xi-μ,-μ)+1+xi-μ,-μ2∫-11‍×∫-11‍K1(xi-μ,-μ,θ(xi-μ,-μ,ν))dd∫-11‍×u(θ(xi-μ,-μ,ν))dν∫-11‍+q(1+xi-μ,-μ)2∫-11‍×∫-11‍K2(xi-μ,-μ,η(xi-μ,-μ,ξ))dd∫-11‍×u(η(xi-μ,-μ,ξ))dξ,(22)u(xi-μ,-μ)=1Γ(γ)((1+xi-μ,-μ)T4)γ×∫-11‍(1-ν)-μDγu(θ(xi-μ,-μ,ν))dν+u(-1).Next, using a(N+1)-point Gauss quadrature formula relative to the Jacobi weight {ωk0,0}i=0N, the integration term in (21) can be approximated by
(23)∫-11‍K1(x-μ,-μ,θ(xi-μ,-μ,ν))u(θ(xi-μ,-μ,ν))dν=∑k=0N‍K1(xi,θ(xi,νk))u(θ(xi,νk))ωk0,0,∫-11‍K2(xi-μ,-μ,η(xi-μ,-μ,ξ))u(η(xi-μ,-μ,ξ))dξ=∑k=0N‍K2(xi,η(xi,ξk))u(η(xi,ξk))ωk0,0.
The sets {νk}k=0N and {ξk}k=0N coincide with the Jacobi-Gauss points corresponding Jacobi weights {ωk0,0}k=0N; that is, {νk}k=0N and {ξk}k=0N 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
(24)∫-11‍(1-ν)μDγu(θ(xi-μ,-μ,ν))dθ=∑k=0N‍Dγu(θ(xi-μ,-μ,ν~k))ωk-μ,0,
where the set {ν~k}k=0N is the Jacobi-Gauss points corresponding to the weight ω-μ,0.We useui, uiγ to approximate the function value u(xi-μ,-μ), Dγu(xi-μ,-μ), 0≤i≤N, and expand u and uγ using Lagrange interpolation polynomials; that is,
(25)U(x)=∑j=0N‍ujFj(x),Uγ(x)=∑j=0N‍ujγFj(x),
where Fj(x) is the Lagrange interpolation basis function associated with {xi-μ,-μ}i=0N which is the set of (N+1) Jacobi-Gauss points. The Jacobi collocation methods are to seek {uiγ}i=0N and {ui}i=0N such that the following collocation equations hold:
(26)uiγ=ui+(∑j=0N‍ujFj(qxi-μ,-μ+q-1))+g(xi-μ,-μ)+1+xi-μ,-μ2∑j=0N‍uj(∑k=0N‍K1(xi-μ,-μ,θ(xi,νk))hhhhhhhhhhhhhhhh×Fj(θ(xi-μ,-μ,νk))ωk0,0∑k=0N‍)+q(1+xi-μ,-μ)2∑j=0N‍uj(∑k=0N‍K2(xi-μ,-μ,η(xi,νk))hhhhhhffhhhhhhhhhhh×Fj(η(xi-μ,-μ,νk))ωk0,0∑k=0N‍),(27)ui=1Γ(γ)((1+xi-μ,-μ)T4)γ∑j=0N‍ujγ×(∑k=0N‍Fj(s(xi-μ,-μ,ν~k))ωk-μ,0)+u(-1).WritingUN=(u0,u1,…,uN)T and UNγ=(u0γ,u1γ,…,uNγ)T, we obtain the following of the matrix form from (26)-(27):
(28)UNγ=(E+L+C+D)UN+G,UN=U-1+BUNγ,
where
(29)Eistheidentitymatrix,G=(g~(x0-μ,-μ),g~(x1-μ,-μ),…,g~(xN-μ,-μ))T,Lij=Fj(qxi-μ,-μ+q-1),Cij=1+xi-μ,-μ2∑k=0N‍K~1(xi-μ,-μ,θ(xi-μ,-μ,νk))Fjjjjjj×(θ(xi,νk))ωk0,0,Dij=q(1+xi-μ,-μ)2∑k=0N‍K~2(xi-μ,-μ,η(xi-μ,-μ,νk))Fjjjjjjj×(η(xi-μ,-μ,νk))ωk0,0,Bij=1Γ(γ)((1+xi-μ,-μ)T4)γjjjjj×∑k=0N‍Fj(s(xi-μ,-μ,ν~k))ωk-μ,0,U-1=(u(-1),u(-1),…,u(-1))T.
We can get the values of {ui}i=0N and {uiγ}i=0N by solving the system of linear system (28). Therefore, the expressions of U(x) and Uγ(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. LetI:=(-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∈Hm(I) with I for some m≥1 and φ∈𝒫N. Then, there exists a constant C independent of N such that
(30)|∫-11‍u(x)φ(x)dx-(u,φ)N|≤CN-m|u|Hωα,βm,N(I)∥φ∥Lωα,β2(I),
where
(31)|u|Hωα,βm,N(I)=(∑j=min(m,N+1)m‍∥u(j)∥Lωα,β2(I)2)1/2,(u,φ)N=∑j=0N‍u(xj)φ(xj)ωj.Lemma 2 (see [26, 27]).
Assume thatu∈Hω-μ,-μm,N(I) and denote by IN-μ,-μu its interpolation polynomial associated with the (N+1) Jacobi-Gauss points {xj}j=0N; namely,
(32)IN-μ,-μu=∑i=0N‍u(xi)F(xi).
Then, the following estimates hold:(33a)∥u-IN-μ,-μu∥Lω-μ,-μ2(I)≤CN-m|u|Hωα,βm,N(I),(33b)∥u-IN-μ,-μu∥L∞(I)≤{CN1-μ-m|u|Hωcm,N(I),0≤μ<12,CN1/2-mlogN|u|Hωcm,N(I),12≤μ<1,where ωc=ω-1/2,-1/2 denotes the Chebyshev weight function.Lemma 3 (see [28]).
Assume that{Fj(x)}j=0N are the N-th degree Lagrange basis polynomials associated with the Gauss points of the Jacobi polynomials. Then,
(34)∥INα,β∥L∞(I)≤maxx∈[-1,1]∑j=0N‍|Fj(x)|={𝒪(logN),-1<α,β≤-12,𝒪(Nγ+1/2),γ=max(α,β),otherwise.Lemma 4 (Gronwall inequality, see [29] Lemma 7.1.1).
Suppose thatL≥0, 0<μ<1, and u and v are a nonnegative, locally integrable functions defined on [-1,1] satisfying
(35)u(x)≤v(x)+L∫-1x‍(x-τ)-μu(τ)dτ.
Then, there exists a constant C=C(μ) such that
(36)u(x)≤v(x)+CL∫-1x‍(x-τ)-μv(τ)dτ,for-1≤x<1.Lemma 5 (see [30, 31]).
For a nonnegative integerr and κ∈(0,1), there exists a constant Cr,κ>0 such that for any function v∈Cr,κ([-1,1]), there exists a polynomial function 𝒯Nv∈𝒫N such that
(37)∥v-𝒯Nv∥L∞(I)≤Cr,κN-(r+κ)∥v∥r,κ,
where ∥·∥r,κ is the standard norm in Cr,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
(38)∥v∥r,κ=max0≤κ≤rmaxx∈[-1,1]|∂xκv(x)|+max0≤κ≤rsupx,y∈[-1,1],x≠y|∂xκv(x)-∂xκv(x)||x-y|κ.𝒯N is a linear operator from Cr,κ([-1,1]) into 𝒫N.Lemma 6 (see [32]).
Letκ∈(0,1) and let ℳ be defined by
(39)(ℳv)(x)=∫-1x‍(x-τ)-μK(x,τ)v(τ)dτ.
Then, for any function v∈C([-1,1]), there exists a positive constant C such that
(40)|ℳv(x′)-ℳv(x′′)||x′-x′′|≤Cmaxx∈[-1,1]|v(x)|,
under the assumption that 0<κ<1-μ, for any x′,x′′∈[-1,1] and x′≠x′′. This implies that
(41)∥ℳv∥0,κ≤Cmaxx∈[-1,1]|v(x)|,0<κ<1-μ.Lemma 7 (see [33]).
For every bounded functionv, there exists a constant C, independent of v such that
(42)supN∥∑j=0N‍v(xj)Fj(x)∥Lωα,β2(I)≤Cmaxx∈[-1,1]|v(x)|,
where Fj(x),  j=0,1,…,N, are the Lagrange interpolation basis functions associated with the Jacobi collocation points {xj}j=0N.Lemma 8 (see [34]).
For all measurable functionf≥0, the following generalized Hardy’s inequality
(43)(∫ab‍|(Tf)(x)|qu(x)dx)1/q≤(∫ab‍|f(x)|pv(x)dx)1/p
holds if and only if
(44)supa<x<b(∫xb‍u(t)dt)1/q(∫ax‍v1-p′(t)dt)1/p′<∞,p′=pp-1
for the case 1<p≤q<∞. Here, T is an operator of the form
(45)(TF)(x)=∫ax‍k(x,t)f(t)dt,
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 inL∞ space.Theorem 9.
Letu(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 Uγ(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∈Hω-μ,-μm+1(I), then
(46)∥Uγ-Dγu∥L∞(I)≤{CNγ-1/2-m(K*∥u∥L2(I)+N1/2𝒰),12<γ<1,CN-mlogN(K*∥u∥L2(I)+N1/2𝒰),0<γ≤12,(47)∥U-u∥L∞(I)≤{CNγ-1/2-m(K*∥u∥L2(I)+N1/2𝒰),12<γ<1,CN-mlogN(K*∥u∥L2(I)+N1/2𝒰),0<γ≤12,
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,
(48)K*=maxx∈[-1,1]|K1(x,θ(x,·))|H~ω0,0m,N(I)+maxx∈[-1,1]|K2(x,η(x,·))|H~ω0,0m,N(I),(49)𝒰=|Dγu|Hωcm,N(I)+|u|Hωcm,N(I).Proof.
We let(50)(K(s(xi,ν)),ϕ(s(xi,ν)))N=∑k=0N‍K(s(xi,νk))ϕ(s(xi,νk))ωk0,0.
The numerical scheme (26)-(27) can be written as
(51)uiγ=ui+(∑j=0N‍ujFj(qxi-μ,-μ+q-1))+g(xi-μ,-μ)+1+xi-μ,-μ2(K1(xi-μ,-μ,θ(xi-μ,-μ,ν)),U(θ(xi-μ,-μ,ν)))N+q(1+xi-μ,-μ)2(K2(xi-μ,-μ,η(xi-μ,-μ,ν)),U(η(xi-μ,-μ,ν)))N,(52)ui=1Γ(γ)(T2)γ∫-1xi‍(xi-θ)-μUγ(θ)dθ+u-1,
which gives
(53)uiγ=ui+(∑j=0N‍ujFj(qxi-μ,-μ+q-1))+g(xi-μ,-μ)+1+xi-μ,-μ2∫-11‍K1(xi-μ,-μ,θ(xi-μ,-μ,ν))×U(θ(xi-μ,-μ,ν))dν+I1(xi-μ,-μ)+q(1+xi-μ,-μ)2∫-11‍K2(xi-μ,-μ,η(xi-μ,-μ,ξ))×U(η(xi-μ,-μ,ξ))dξ+I2(xi-μ,-μ),
where
(54)I1(x)=1+x2∫-11‍K1(x,θ(x)),U(θ(x,ν))dν-1+x2(K1(x,θ(x,ν)),U(θ(x,ν)))N,I2(x)=q(1+x)2∫-11‍K2(x,η(x,ξ))U(η(x,ξ))dξ-q(1+x)2(K2(x,η(x,ν)),U(η(x,ν)))N.
Using the integration error estimates from Jacobi-Gauss polynomials quadrature in Lemma 1, we have
(55)|I1(x)|≤CN-m|K~1(x,θ(x,·))|Hω0,0m;N(I)∥U(θ(x,·))∥L2(I)≤CN-mK1*∥U∥L2(I),(56)|I2(x)|≤CN-m|K~2(x,η(x,·))|Hω0,0m;N(I)∥U(η(x,·))∥L2(I)≤CN-mK2*∥U∥L2(I),(57)K1*=maxx∈[-1,1]|K1(x,θ(x,·))|H~ω0,0m,N(I),K2*=maxx∈[-1,1]|K1(x,θ(x,·))|H~ω0,0m,N(I).
From (18), (53) can be rewritten as
(58)uiγ=a(xi-μ,-μ)ui+b(xi-μ,-μ)×(∑j=0N‍ujFj(qxi-μ,-μ+q-1))+g(xi-μ,-μ)+∫-1xi-μ,-μ‍K1(xi-μ,-μ,θ)U(θ)dθ+I1(xi-μ,-μ)+∫-1qxi-μ,-μ+q-1‍K2(xi-μ,-μ,η)U(η)dη+I2(xi-μ,-μ).
Let e and eγ denote the error function
(59)e(x)=U(x)-u(x),eγ(x)=Uγ(x)-uγ(x),
we have
(60)uiγ=a(xi-μ,-μ)ui+b(xi-μ,-μ)×(∑j=0N‍ujFj(qxi-μ,-μ+q-1))+g(xi-μ,-μ)+∫-1xi-μ,-μ‍K1(xi-μ,-μ,θ)u(θ)dθ+∫-1xi-μ,-μ‍K1(xi-μ,-μ,θ)e(θ)dθ+∫-1qxi-μ,-μ+q-1‍K2(xi-μ,-μ,η)u(η)dη+∫-1qxi-μ,-μ+q-1‍K2(xi-μ,-μ,η)e(η)dη+I1(xi-μ,-μ)+I2(xi-μ,-μ).
MultiplyingFi(x) on both sides of (62) and (52), summing up from 0 to N, and using (7)-(8) yield
(61)Uγ(x)=U(x)+IN-μ,-μ(∑j=0N‍ujFj(qx+q-1))+IN-μ,-μg(x)+IN-μ,-μ[Dγu(x)-u(x)ggggggggg-u(q(x)+q-1)-g(x)]+IN-μ,-μ∫-1x‍K1(x,θ)e(θ)dθ+IN-μ,-μ∫-1qx+q-1‍K2(x,η)e(η)dη+J1(x)+J2(x),(62)U(x)=IN-μ,-μ(u(x)-u-1)+IN-μ,-μ(1Γ(γ)∫-1x(x-θ)-μeγ(θ)dθ)+IN-μ,-μ(u-1),
where
(63)J1(x)=IN-μ,-μI1(x),J2(x)=IN-μ,-μI2(x).
From (61)-(62), we have
(64)eγ(x)=e(x)+∫-1x‍K1(x,θ)e(θ)dθ+∫-1qx+q-1‍K2(x,η)e(η)dη+J1(x)+J2(x)+J3(x)+J4(x)+J5(x)+J6(x)+J7(x),(65)e(x)=1Γ(γ)∫-1x‍(x-θ)-μeγ(θ)dθ+J4(x)+J8(x),
where
(66)J3(x)=IN-μ,-μDγu(x)-Dγu(x),J4(x)=IN-μ,-μu(x)-u(x),J5(x)=IN-μ,-μu(qx+q-1)-u(qx+q-1),J6(x)=IN-μ,-μ∫-1x‍K1(x,θ)e(θ)dθggggg-∫-1x‍K(x,s)e(s)ds,J7(x)=IN-μ,-μ∫-1qx+q-1‍K2(x,η)e(η)dηggggg-∫-1qx+q-1‍K2(x,η)e(η)dη,J8(x)=IN-μ,-μ(1Γ(γ)∫-1x‍(x-θ)-μeγ(θ)dθ)ggggg-1Γ(γ)∫-1x‍(x-θ)-μeγ(θ)dθ.
Due to (64), we obtain
(67)eγ(x)=1Γ(γ)∫-1x‍(x-θ)-μeγ(θ)dθ+1Γ(γ)∫-1x‍K1(x,θ)∫-1θ‍(θ-τ)-μeγ(τ)dτdθ+1Γ(γ)∫-1qx+q-1‍K2(x,η)∫-1η‍(η-z)-μ×eγ(z)dzdη+H(x)+J1(x)+J2(x)+J3(x)+J4(x)+J5(x)+J6(x)+J7(x),
where
(68)H(x)=∫-1x‍K1(x,θ)∫-1θ‍(J4(θ)+J8(θ))dθ+1Γ(γ)∫-1qx+q-1‍K2(x,η)∫-1η‍(J4(η)+J8(η))dη+J4(x)+J8(x).
Using the Dirichlet’s formula which sates that
(69)∫-1x‍∫-1θ‍Φ(θ,τ)dτdθ=∫-1x‍∫τx‍Φ(θ,τ)dθdτ,
provided that the integral exists, we obtain
(70)1Γ(γ)∫-1x‍K1(x,θ)∫-1θ‍(θ-τ)-μeγ(τ)dτdθ=1Γ(γ)∫-1x‍(∫θx‍K1(x,θ)(θ-τ)-μdθ)eγ(τ)dτ≤1Γ(γ)21-μmaxx,τ∈[-1,1]|K~1(x,τ)|∫-1x‍|eγ(τ)|dτ≜M1∫-1x‍|eγ(τ)|dτ,
letting η=qτ+q-1, z=qθ+q-1, we have
(71)1Γ(γ)∫-1qx+q-1K2(x,η)∫-1η‍(η-z)-μeγ(z)dzdη‍=1Γ(γ)q∫-1xK2(x,qτ+q-1)‍×(∫-1qτ+q-1‍(x-z)-μeγ(z)dz)dτ=1Γ(γ)q1+γ∫-1xK2(x,qτ+q-1)‍×(∫-1τ‍(1-θ)-μeγ(qθ+q-1)dθ)dτ=1Γ(γ)q1+γ∫-1x(∫θx‍K2(x,qτ+q-1)‍gggggggggggggggggg∫θx‍×(1-θ)-μdτ)eγ×(qθ+q-1)dθ≤1Γ(γ)21-μq1+γmaxx,τ∈[-1,1]|K2(x,τ)|×∫-1x|eγ(qθ+q-1)|dθ‍≜M2∫-1x‍|eγ(qθ+q-1)|dθ.
Then, (67) gives
(72)eγ(x)≤1Γ(γ)∫-1x(x-θ)-μeγ(θ)dθ‍+M1∫-1x‍|eγ(τ)|dτ+M2∫-1x‍|eγ(qθ+q-1)|dθ+H(x)+J1(x)+J2(x)+J3(x)+J4(x)+J5(x)+J6(x)+J7(x).
It follows from the Gronwall inequality that
(73)∥eγ(x)∥L∞(I)≤∑i=18‍∥Ji(x)∥L∞(I).
From (65), we have
(74)∥e(x)∥L∞(I)≤∑i=18‍∥Ji(x)∥L∞(I).
We now apply Lemma7 to obtain that(75)∥J1∥L∞(I)≤{CN(1/2)-μmax0≤i≤N|Ii,1|,0<μ<12,ClogNmax0≤i≤N|Ii,1|,12≤μ<1,≤{CN(1/2)-μ-mK1*(∥u∥L2(I)+∥eγ(x)∥L∞(I)),0<μ<12,CN-mlogNK1*(∥u∥L2(I)+∥eγ(x)∥L∞(I)),12≤μ<1,∥J2∥L∞(I)≤{CN(1/2)-μmax0≤i≤N|Ii,2|,0<μ<12,ClogNmax0≤i≤N|Ii,1|,12≤μ<1,≤{CN(1/2)-μ-mK2*(∥u∥L2(I)+∥eγ(x)∥L∞(I)),0<μ<12,CN-mlogNK2*(∥u∥L2(I)+∥eγ(x)∥L∞(I)),12≤μ<1.Due to Lemma 2,
(76)∥J3∥L∞(I)≤{CN1-μ-m|Dγu|Hωcm,N(I),0<μ<12,CN(1/2)-mlogN|Dγu|Hωcm,N(I),12≤μ<1,∥J4∥L∞(I)≤{CN1-μ-m|u|Hωcm,N(I),0<μ<12,CN(1/2)-mlogN|u|Hωcm,N(I),12≤μ<1,∥J5∥L∞(I)≤{CN1-μ-m|u|Hωcm,N(I),0<μ<12,CN(1/2)-mlogN|u|Hωcm,N(I),12≤μ<1.
By virtue of Lemma2 (33b) with m=1,
(77)∥J6∥L∞(I)≤{CN-μ∥e∥L∞(I),0<μ<12,CN-1/2∥e∥L∞(I),12≤μ<1,∥J7∥L∞(I)≤{CN-μ∥e∥L∞(I),0<μ<12,CN-1/2∥e∥L∞(I),12≤μ<1.
We now estimate the termJ8(x). It follows from Lemmas 5 and 6 with K(x,τ)=1/Γ(γ) that
(78)∥J5∥L∞(I)=∥(IN-μ,-μ-I)ℳeγ∥L∞(I)=∥(IN-μ,-μ-I)(ℳeγ-𝒯Nℳeγ)∥L∞(I)≤(1+∥IN-μ,-μ∥L∞(I))CN-k∥ℳeγ∥0,κ≤{CN(1/2)-μ-κ∥eγ∥L∞(I),0<μ<12,CN-κlogN∥eγ∥L∞(I),12≤μ<1,
where in the last step we have used Lemma 6 under the following assumption:
(79)12-μ<κ<1-μ,when0<μ<12,0<κ<1-μ,when12≤μ<1.
Provided thatN is sufficiently large, combining (75), (76), (77), and (78) gives
(80)∥Uγ(x)-uγ(x)∥L∞(I)≤{CN1/2-μ-m(K*∥u∥L2(I)+N1/2𝒰),0<μ<12,CN-mlogN(K*∥u∥L2(I)+N1/2𝒰),12≤μ<1,∥U(x)-u(x)∥L∞(I)≤{CN1/2-μ-m(K*∥u∥L2(I)+N1/2𝒰),0<μ<12,CN-mlogN(K*∥u∥L2(I)+N1/2𝒰),12≤μ<1.
Using γ=1-μ, we have the desired estimates (46) and (47).Next, we will give the error estimates inLω-μ,-μ2 space.Theorem 10.
If the hypotheses given in Theorem9 hold, then(81)∥Uγ(x)-uγ(x)∥Lω-μ,-μ2(I)≤{CN-m(V1+Nγ-1/2-κV2+Nγ-κ𝒰),12<γ<1,CN-m(V1+N-κlogNV2+N1/2-κlogN𝒰),0<γ≤12,∥U(x)-u(x)∥Lω-μ,-μ2(I)≤{CN-m(V1+Nγ-1/2-κV2+Nγ-κ𝒰),12<γ<1,CN-m(V1+N-κlogNV2+N1/2-κlogN𝒰),0<γ≤12,for any κ∈(0,γ) provided that N is sufficiently large and C is a constant independent of N, where
(82)V1=K*(∥u∥L2(I)+|Dγu|Hωcm;N(I)+|u|Hωcm;N(I)),V2=K*∥u∥L2(I).Proof.
By using the generalization of Gronwalls Lemma4 and the Hardy inequality Lemma 8, it follows from (72) that
(83)∥eγ∥Lω-μ,-μ2(I)≤C∑i=18‍∥Ji∥Lω-μ,-μ2(I),∥e∥Lω-μ,-μ2(I)≤C∑i=18‍∥Ji∥Lω-μ,-μ2(I).
Now, using Lemma 7, we have
(84)∥J1∥Lω-μ,-μ2(I)≤Cmaxx∈[-1,1]|I(x)|≤CN-mK1*(∥u∥L2(I)+∥e∥L∞(I)),∥J2∥Lω-μ,-μ2(I)≤Cmaxx∈[-1,1]|I(x)|≤CN-mK2*(∥u∥L2(I)+∥e∥L∞(I)).
By the convergence result in Theorem 9 (m=1), we have
(85)∥e∥L∞(I)≤C(|Dγu|Hωcm;N(I)+|u|Hωcm;N(I)+∥u∥L2(I)).
So that
(86)∥J1∥Lω-μ,-μ2(I)≤CN-mK1*(|Dγu|Hωcm;N(I)+|u|Hωcm;N(I)+∥u∥L2(I)),∥J2∥Lω-μ,-μ2(I)≤CN-mK2*(|Dγu|Hωcm;N(I)+|u|Hωcm;N(I)+∥u∥L2(I)).
Due to Lemma 2 (33a),
(87)∥J3∥Lω-μ,-μ2(I)≤CN-m|Dγu|Hωcm;N(I),∥J4∥Lω-μ,-μ2(I)≤CN-m|u|Hωcm;N(I),∥J5∥Lω-μ,-μ2(I)≤CN-m|u|Hωcm;N(I).
By virtue of Lemma 2 (33a) with m=1,
(88)∥J6∥Lω-μ,-μ2(I)≤CN-1|∫-1x‍K1(x,θ)e(θ)dθ|Hω-μ,-μ1;N(I)∥J6∥Lω-μ,-μ2(I)≤CN-1∥e∥Lω-μ,-μ2(I),∥J7∥Lω-μ,-μ2(I)≤CN-1|∫-1qx+q-1‍K2(x,η)e(η)dη|Hω-μ,-μ1;N(I)∥J6∥Lω-μ,-μ2(I)≤CN-1∥e∥Lω-μ,-μ2(I).
Finally, it follows from Lemmas 5 and 7 that
(89)∥J8∥Lω-μ,-μ2(I)=∥(IN-μ,-μ-I)ℳeγ∥Lω-μ,-μ2(I)=∥(IN-μ,-μ-I)(ℳeγ-𝒯Neγ)∥Lω-μ,-μ2(I)≤∥IN-μ,-μ(ℳeγ-𝒯Neγ)∥Lω-μ,-μ2(I)+∥ℳeγ-𝒯Neγ∥Lω-μ,-μ2(I)≤C∥ℳeγ-𝒯Neγ∥L∞(I)≤CN-κ∥ℳeγ∥0,κ≤CN-κ∥eγ∥L∞(I),
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
(90)∥J8∥Lω-μ,-μ2(I)≤{CN1/2-μ-m-κ(K*∥u∥L2(I)+N1/2𝒰),0<μ<12,CN-m-κlogN(K*∥u∥L2(I)+N1/2𝒰),12≤μ<1,
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 Section4 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:(91)D0.75y(t)=6t2.25Γ(3.25)-t2et5y(t)-(qt)25y(qt)D0.75y(t)=+∫0t‍etsy(s)ds+∫0qt‍τy(τ)dτ,t∈[0,1],y(0)=0.
The corresponding exact solution is given by y(t)=t3.Figure1 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 Lω-μ,-μ2 norms. As expected, an exponential rate of convergence is observed for the problem, which confirmed our theoretical predictions.Example1: comparison between approximate solution and exact solution y(t) (a) and approximate fraction derivative and exact derivative D0.75y(t) (b).
(a)
(b)Example1: 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 Lω2 norms.
(a)
(b)Example 2.
Consider the following fractional integrodifferential equation:(92)Dαy(t)=2t-y(t)+y(qt)+t(1+2t)∫0t‍es(t-s)y(s)ds+∫0qt‍qteτ(qt-τ)y(τ)dτ,t∈[0,1],y(0)=1,
when α=1, the exact solution of (92) is y(t)=et2.In the only case ofα=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)=et2; 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.Example2: 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)Example2: 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 Lω2 norms.
(a)
(b)


## 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.



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