Numerical Solution of a Class of Functional-Differential Equations Using Jacobi Pseudospectral Method

Abstract

The shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method is applied to neutral functional-differential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of Gauss-Lobatto quadrature. The shifted Legendre-Gauss-Lobatto Pseudo-spectral and Chebyshev-Gauss-Lobatto Pseudo-spectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear high-order NFDE with proportional delay. Some examples are displayed for implicit and explicit forms of NFDEs to demonstrate the computation accuracy of the proposed method. We also compare the performance of the method with variational iteration method, one-leg 𝜃-method, continuous Runge-Kutta method, and reproducing kernel Hilbert space method.

1. Introduction

In the last four decades, spectral method has become increasingly popular and been successfully applied in solving all types of differential equations owing to its high order of accuracy (see, for instance, [1–5]). Recently there has been a growing interest in applying spectral methods for the numerical solution of fractional differential equations and delay differential equations (DDEs). Moreover, the principal difficulty in studying NFDEs with proportional delays lies in their special transcendental nature. Thus we propose an efficient technique to solve such differential equations numerically based on shifted Jacobi polynomials.

In the last two decades, some numerical approaches for treating several types of DDEs were presented in [6–12]. In this direction, Zhao et al. [13] studied the stability of Runge-Kutta approach and applied it for solving the NFDEs with proportional delays. Tohidi et al. [12] derived the operational matrix of Bernoulli polynomial and used it with collocation method to solve the pantograph type equation. The one-leg 𝜃 approach was implemented to solve nonlinear NFDEs in [14]. Meanwhile, Trif [15] proposed a direct solution technique by using the Chebyshev Tau operational matrix method for solving the pantograph type equation. More recently, Bhrawy et al. [16] proposed the Legendre pseudospectral algorithm with studying the error analysis and stability of the proposed algorithm for a class of DDEs. Sun and Zhang [17] proposed a compact difference method for solving nonlinear partial DDE. Cordero and Escalante [18] extended the application of segmented tau approach for solving a class of NFDEs, meanwhile, the history-valued NFDEs were approximated in [19]. Isik et al. [20] introduced a new Bernstein collocation method for the numerical solution of DDEs of pantograph-type with retarded case. The existence of solutions of neutral functional-differential equations with proportional delays had been discussed in [21–24].

The aim of this paper is to develop a direct solution technique to approximate the linear high-order NFDEs with proportional delays using the shifted Jacobi polynomials on the interval [0,𝐿]; we dedicate the shifted Jacobi-Gauss-Lobatto pseudospectral (SJGLP) method to find the approximate solution 𝑢_𝑁(𝑥). Approximate semianalytical solution with high accuracy can be obtained by selecting a limited number of Gauss-Lobatto collocation points for the linear and nonlinear high-order NFDEs with proportional delays. For suitable collocation points we use the (𝑁−𝑚+1) nodes of the shifted Jacobi-Gauss-Lobatto interpolation on (0,𝐿) in which the nodes are distinct and lie between 0 and 𝐿, and the two endpoints of the domain (0 and 𝐿) are used as the first and last collocation points, respectively. The algorithms given in [16] can be obtained as special cases from the proposed algorithms. Finally, the accuracy of the proposed method is showed by test problems. From the results, these algorithms are extremely efficient and accurate for solving NFDEs.

The paper is arranged in the following way. In the next section, some basic properties of Jacobi polynomials which are required in the present paper are given, and in Section 3, the way of constructing the pseudospectral technique for NFDEs with proportional delays is described using the shifted Jacobi polynomials. In Section 4, we investigate the shifted Jacobi-Gauss-Lobatto pseudo-spectral (SJGLP) method for solving nonlinear high-order NFDEs with proportional delays. Some numerical results exhibiting the accuracy of the proposed algorithm are given in Section 5. Finally, a conclusion is given in Section 6.

2. Preliminaries

In this section, we briefly recall some properties of the Jacobi polynomials (𝐽_𝑘^(𝜃,𝜗)(𝑥), 𝑘=0,1,…, 𝜃>−1, 𝜗>−1), which satisfy the following relations: 𝐽_𝑘^(𝜃,𝜗)(−𝑥)=(−1)^𝑘𝐽_𝑘^(𝜃,𝜗)(𝑥),𝐽_𝑘^(𝜃,𝜗)()=()−1−1^𝑘Γ(𝑘+𝜗+1)(),𝐽𝑘!Γ𝜗+1_{𝑘(𝜃,𝜗)}(1)=Γ(𝑘+𝜃+1)().𝑘!Γ𝜃+1 (1) The 𝑞th derivative of Jacobi polynomials of degree 𝑘 can be given by 𝐷^(𝑞)𝐽_𝑘^(𝜃,𝜗)(𝑥)=Γ𝑗+𝜃+𝜗+𝑞+12^𝑞Γ𝐽𝑗+𝜃+𝜗+1^{(𝜃+𝑞,𝜗+𝑞)𝑘−𝑞}(𝑥). (2) These polynomials are the only polynomials arising as eigenfunctions of the following singular Sturm-Liouville equation: 1−𝑥²𝜙^′′(𝑥)+[𝜗−𝜃+(𝜃+𝜗+2)𝑥]𝜙^′(𝑥)+𝑛(𝑛+𝜃+𝜗+1)𝜙(𝑥)=0. (3) Let 𝑤^(𝜃,𝜗)(𝑥)=(1−𝑥)^𝜃(1+𝑥)^𝜗; then we define the weighted space 𝐿²_{𝑤(𝜃,𝜗)} as usual. The inner product and the norm of 𝐿²_{𝑤(𝜃,𝜗)} with respect to the weight function are defined as follows: (𝑢,𝑣)_{𝑤(𝜃,𝜗)}=¹₋₁𝑢(𝑥)𝑣(𝑥)𝑤^(𝜃,𝜗)(𝑥)‖𝑢‖𝑑𝑥,_{𝑤(𝜃,𝜗)}=(𝑢,𝑢)_𝑤^1/2_(𝜃,𝜗). (4)

The set of Jacobi polynomials forms a complete 𝐿²_{𝑤(𝜃,𝜗)}-orthogonal system, and 𝐽_𝑘^(𝜃,𝜗)_{𝑤(𝜃,𝜗)}∶=ℎ_𝑘=2^𝜃+𝜗+1Γ(𝑘+𝜃+1)Γ(𝑘+𝜗+1)()Γ()Γ().2𝑘+𝜃+𝜗+1𝑘+1𝑘+𝜃+𝜗+1 (5)

Let us denote 𝑃^(𝛼,𝛽)_𝐿,𝑘(𝑥)=𝐽_𝑘^(𝛼,𝛽)((2𝑥/𝐿)−1), 𝐿>0. By the shifted Jacobi polynomial of degree 𝑘 and by using (1) and (2); then we deduce that 𝑃^(𝛼,𝛽)_𝐿,𝑘(0)=(−1)^𝑘Γ𝑘+𝛽+1Γ𝐷𝛽+1𝑘!_𝑞,𝑃^(𝛼,𝛽)_𝐿,𝑘(0)=(−1)^𝑘−𝑞Γ𝑘+𝛽+1𝑘+𝛼+𝛽+1_𝑞𝐿^𝑞ΓΓ,𝐷𝑘−𝑞+1𝑞+𝛽+1_𝑚𝑃^(𝛼,𝛽)_𝐿,𝑘(𝑥)=Γ𝑚+𝑘+𝛼+𝛽+1𝐿^𝑚Γ𝑃𝑘+𝛼+𝛽+1^{(𝛼+𝑚,𝛽+𝑚)𝐿,𝑘−𝑚}(𝑥). (6)(7)(8)

The following inner product and norm (𝑢,𝑣)_𝑤𝐿^(𝛼,𝛽)=^𝐿₀𝑢(𝑥)𝑣(𝑥)𝑤_𝐿^(𝛼,𝛽)(𝑥)‖𝑣‖𝑑𝑥,_{𝑤𝐿(𝛼,𝛽)}=(𝑣,𝑣)_𝑤^1/2_{𝐿(𝛼,𝛽)}, (9) are defined on the weighted space 𝐿²_{𝑤𝐿(𝛼,𝛽)}(0,𝐿) subject to the weight function 𝑤_𝐿^(𝛼,𝛽)(𝑥)=(𝐿−𝑥)^𝛼𝑥^𝛽. Moreover, 𝑃^(𝛼,𝛽)_𝐿,𝑘(𝑥) forms a complete 𝐿²_{𝑤𝐿(𝛼,𝛽)}(0,𝐿)-orthogonal system.

According to (5), we get 𝑃^(𝛼,𝛽)_𝐿,𝑘²_{𝑤𝐿(𝛼,𝛽)}=𝐿2^𝛼+𝛽+1ℎ_𝑘^(𝛼,𝛽)∶=ℎ^(𝛼,𝛽)_𝐿,𝑘. (10)

3. Linear High-Order NFDE with Proportional Delay

In this section, we shall investigate solutions to NFDEs with proportional delays of the form 𝑢(𝑥)+𝑎(𝑥)𝑢𝛾_𝑚𝑥^(𝑚)=𝛽𝑢(𝑥)+^𝑚−1_𝑛=0𝑏_𝑛(𝑥)𝑢^(𝑛)𝛾_𝑛𝑥+𝑓(𝑥),𝑥≥0, (11) with the initial conditions ^𝑚−1_𝑛=0𝑐_𝑖𝑛𝑢^(𝑛)(0)=𝜆_𝑖,𝑖=0,1,…,𝑚−1, (12) where 𝑎 and 𝑏_𝑛 (𝑛=0,1,…,𝑚−1) are given functions, meanwhile, 𝛽, 𝛾_𝑛, 𝑐_𝑖𝑛, and 𝜆_𝑖 are constants with 0<𝛾_𝑛<1 (𝑛=0,1,…,𝑚). This class of equations plays an important role in modeling phenomena of the real world.

In the pseudo-spectral methods [25–28], one needs to exactly satisfy the differential equation at specified collocation points in the domain of solution. Generally, the distribution of the collocation nodes can be freely chosen, but an accurate approximations are obtained by selecting the collocation nodes as the zeros of the orthogonal polynomials. For shifted Jacobi polynomials, two commonly used quadrature and collocation nodes, namely, (i) shifted Jacobi-Gauss nodes (in the interior of the domain) and (ii) shifted Jacobi-Gauss-Lobatto nodes (in the interior and at the two endpoints of the domain).

Now, we will present the shifted Jacobi-Gauss-Lobatto type quadratures. Let 𝑥^(𝛼,𝛽)_𝑁,𝑗, 0⩽𝑗⩽𝑁, be the nodes of the Jacobi-Gauss-Lobatto interpolation on (−1,1), and let 𝜛^(𝛼,𝛽)_𝑁,𝑗, 0⩽𝑗⩽𝑁, be the corresponding weights. Throughout this paper, we assume that 𝑥_{𝐿,𝑁,𝑗}, 0⩽𝑗⩽𝑁, stands for the nodes of the shifted Jacobi-Gauss-Lobatto interpolation on the interval (0,𝐿). Thus 𝑥^(𝛼,𝛽)_{𝐿,𝑁,𝑗}, 0⩽𝑗⩽𝑁, and their corresponding weights are 𝜛^(𝛼,𝛽)_{𝐿,𝑁,𝑗}=(𝐿/2)^𝛼+𝛽+1𝜛^(𝛼,𝛽)_𝑁,𝑗, 0⩽𝑗⩽𝑁. Let 𝑆_𝑁(0,𝐿) be the set of all polynomials of degree =𝑁. One gets for any 𝜙∈𝑆_2𝑁−1(0,𝐿), ^𝐿₀(𝐿−𝑥)^𝛼𝑥^𝛽𝜙(𝑥)=𝐿𝑑𝑥2^𝛼+𝛽+11₋₁(1−𝑥)^𝛼(1+𝑥)^𝛽𝜙𝐿2()=𝐿𝑥+1𝑑𝑥2^{𝑁𝛼+𝛽+1}_𝑗=0𝜛^(𝛼,𝛽)_𝑁,𝑗𝜙𝐿2𝑥^(𝛼,𝛽)_𝑁,𝑗=+1_𝑁𝑗=0𝜛^(𝛼,𝛽)_{𝐿,𝑁,𝑗}𝜙𝑥^(𝛼,𝛽)_{𝐿,𝑁,𝑗}. (13)

Let us first introduce some basic notations that will be used in the sequel. We set 𝑆_𝑁(0,𝐿)=span𝑃^(𝛼,𝛽)_𝐿,0(𝑥),𝑃^(𝛼,𝛽)_𝐿,1(𝑥),…,𝑃^(𝛼,𝛽)_𝐿,𝑁(𝑥). (14) The discrete inner product and norm are defined by (𝑢,𝑣)_𝑤𝐿^(𝛼,𝛽)_,𝑁=^𝑁_𝑗=0𝑢𝑥^(𝛼,𝛽)_{𝐿,𝑁,𝑗}𝑣𝑥^(𝛼,𝛽)_{𝐿,𝑁,𝑗}𝜛^(𝛼,𝛽)_{𝐿,𝑁,𝑗},‖𝑢‖_𝑤𝐿^(𝛼,𝛽)_,𝑁=(𝑢,𝑢)_𝑤𝐿^(𝛼,𝛽)_,𝑁, (15) where 𝑥^(𝛼,𝛽)_{𝐿,𝑁,𝑗} and 𝜛^(𝛼,𝛽)_{𝐿,𝑁,𝑗} are the nodes and the corresponding weights of the shifted Jacobi-Gauss-Lobatto quadrature formula on the interval (0,𝐿), respectively. Obviously, (𝑢,𝑣)_𝑤𝐿^(𝛼,𝛽)_,𝑁=(𝑢,𝑣)_𝑤𝐿^(𝛼,𝛽),∀𝑢,𝑣∈𝑆_2𝑁−1. (16)

The shifted Jacobi-Gauss-Lobatto pseudo-spectral method for solving (28) and (29) is to seek 𝑢_𝑁(𝑥)∈𝑆_𝑁(0,𝐿), such that 𝑢𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}+𝑎𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}𝑢𝛾_𝑚𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}^(𝑚)=𝛽𝑢𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}+^𝑚−1_𝑛=0𝑏_𝑛𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}𝑢^(𝑛)𝛾_𝑛𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}+𝑓𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑘=0,1,…,𝑁−𝑚,𝑘=0,1,…,𝑁−𝑚,_{𝑚−1𝑛=0}𝑐_𝑖𝑛𝑢^(𝑛)(0)=𝜆_𝑖,𝑖=0,1,…,𝑚−1, (17) where the 𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}, 𝑘=1,2,…,𝑁−𝑚−1, are distinct and lie between 0 and 𝐿, 𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,0}=0, and 𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑁−𝑚}=𝐿. For simplicity in presentation and without loss of generality, assume that 𝑎(𝑥)≡1. We now derive the collocation algorithm for solving (28) and (29). To do this, consider the solution is approximated by a truncated Jacobi expansion 𝑢_𝑁(𝑥)=^𝑁_ℎ=0𝑎_ℎ𝑃^(𝛼,𝛽)_𝐿,ℎ(𝑥),𝐚=𝑎₀,𝑎₁,…,𝑎_𝑁^𝑇. (18)

We first approximate 𝑢(𝑥) and 𝑢^𝑛(𝑥) as (32). By substituting these approximation in (28), we get ^𝑁_ℎ=0𝑎_ℎ𝐷^(𝑚)𝑃^(𝛼,𝛽)_𝐿,ℎ(𝑥)+^𝑁_ℎ=0𝑎_ℎ𝐷^(𝑚)𝑃^(𝛼,𝛽)_𝐿,ℎ𝛾_𝑚𝑥=𝛽^𝑁_ℎ=0𝑎_ℎ𝑃^(𝛼,𝛽)_𝐿,ℎ(𝑥)+^𝑚−1𝑁_𝑛=0ℎ=0𝑏_𝑛(𝑥)𝑎_ℎ𝐷^(𝑛)𝑃^(𝛼,𝛽)_𝐿,ℎ𝛾_𝑛𝑥+𝑓(𝑥). (19)

Then, by virtue of (8), we deduce that ^𝑁_ℎ=0𝑎_ℎΓℎ+𝛼+𝛽+𝑚+1𝐿^𝑛Γ𝑃ℎ+𝛼+𝛽+1^{(𝛼+𝑚,𝛽+𝑚)𝐿,ℎ−𝑚}(𝑥)+𝛾^𝑚_𝑚Γℎ+𝛼+𝛽+𝑚+1𝐿^𝑚Γ𝑃ℎ+𝛼+𝛽+1^{(𝛼+𝑚,𝛽+𝑚)𝐿,ℎ−𝑚}𝛾_𝑚𝑥=𝛽^𝑁_ℎ=0𝑎_ℎ𝑃^(𝛼,𝛽)_𝐿,ℎ(𝑥)+^𝑚−1𝑁_𝑛=0ℎ=0𝛾_𝑛^(𝑛)𝑏_𝑛(𝑥)𝑎_ℎΓℎ+𝛼+𝛽+𝑛+1𝐿^𝑛Γℎ+𝛼+𝛽+1×𝑃^{(𝛼+𝑛,𝛽+𝑛)}_{𝐿,ℎ−𝑛}𝛾_𝑛𝑥+𝑓(𝑥). (20)

Also, by substituting (32) in (29), we obtain ^𝑚−1𝑁_𝑛=0ℎ=0𝑐_𝑖𝑛𝑎_ℎ𝐷^(𝑛)𝑃^(𝛼,𝛽)_𝐿,ℎ(0)=𝜆_𝑖. (21)

To find the solution 𝑢_𝑁(𝑥), we first collocate (20) at the (𝑁−𝑚+1) shifted Jacobi roots that yields ^𝑁_𝑗=0𝑎_𝑗Γ𝑗+𝛼+𝛽+𝑚+1𝐿^𝑛Γ𝑃𝑗+𝛼+𝛽+1^{(𝛼+𝑚,𝛽+𝑚)𝐿,𝑗−𝑚}𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}+𝛾^𝑚_𝑚Γ𝑗+𝛼+𝛽+𝑚+1𝐿^𝑚Γ𝑗+𝛼+𝛽+1×𝑃^{(𝛼+𝑚,𝛽+𝑚)}_{𝐿,𝑗−𝑚}𝛾_𝑚𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}=𝛽^𝑁_𝑗=0𝑎_𝑗𝑃^(𝛼,𝛽)_𝐿,𝑗𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}+^𝑚−1𝑁_𝑛=0𝑗=0𝛾_𝑛^(𝑛)𝑏_𝑛𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}𝑎_𝑗×Γ𝑗+𝛼+𝛽+𝑛+1𝐿^𝑛Γ𝑃𝑗+𝛼+𝛽+1^{(𝛼+𝑛,𝛽+𝑛)𝐿,𝑗−𝑛}𝛾_𝑛𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}+𝑓𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑘=0,1,…,𝑁−𝑚. (22)

Next, (21), after using (7), can be written as ^𝑚−1𝑁_𝑛=0𝑗=0(−1)^𝑗−𝑛𝑐_𝑖𝑛𝑎_𝑗Γ𝑗+𝛽+1𝑗+𝛼+𝛽+1_𝑛𝐿^𝑛Γ𝑗−𝑛+1Γ𝑛+𝛽+1=𝜆_𝑖,𝑖=0,1,…,𝑚−1. (23)

Thus (22) with relation (23) can be written as a linear algebraic system. To do this, let us consider 𝐚=𝑎₀,𝑎₁,…,𝑎_𝑁^𝑇,𝑓_𝑘=𝑓𝑥_{𝐿,𝑁−𝑚,𝑘}𝑓,𝑘=0,1,…,𝑁−𝑚,𝐟=₀,𝑓₁,…,𝑓_𝑁−𝑚,𝜆₀,…,𝜆_𝑚−1^𝑇. (24)

The matrix system associated with (22) and (23) becomes 𝐴+𝛾^𝑚_𝑚𝐵+𝛽𝐶+^𝑚−1_𝑛=0𝛾^𝑛_𝑛𝐷_𝑛+𝐸𝐚=𝐟, (25) where the matrices 𝐴, 𝐵, 𝐶, 𝐷_𝑖, 𝑖=1,2,…,𝑚−1, and 𝐸 are given explicitly in the following. If we denote that 𝐴=(𝑎_𝑘𝑗)_{0<𝑘,𝑗<𝑁}, 𝐵=(𝑏_𝑘𝑗)_{0<𝑘,𝑗<𝑁}, 𝐶=(𝑐_𝑘𝑗)_{0<𝑘,𝑗<𝑁}, 𝐷_𝑛=(𝑑^𝑛_𝑘𝑗)_{0<𝑘,𝑗<𝑁}, 𝑛=1,2,…,𝑚−1, and 𝐸=(𝑒_𝑘𝑗)_{0<𝑘,𝑗<𝑁}, then the elements 𝑎_𝑘𝑗, 𝑏_𝑘𝑗, 𝑐_𝑘𝑗, and 𝑑^𝑛_𝑘𝑗 are given by 𝑎_𝑘𝑗=Γ𝑗+𝛼+𝛽+𝑚+1𝐿^𝑛Γ𝑗+𝛼+𝛽+1×𝑃^{(𝛼+𝑚,𝛽+𝑚)}_{𝐿,𝑗−𝑚}𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}𝑏,𝑘=0,1,…,𝑁−𝑚,𝑗=0,1,…,𝑁,0,𝑘=𝑁−𝑚+1,…,𝑁,𝑗=0,1,…,𝑁,_𝑘𝑗=Γ𝑗+𝛼+𝛽+𝑚+1𝐿^𝑚Γ𝑗+𝛼+𝛽+1×𝑃^{(𝛼+𝑚,𝛽+𝑚)}_{𝐿,𝑗−𝑚}𝛾_𝑚𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}𝑐,𝑘=0,1,…,𝑁−𝑚,𝑗=0,1,…,𝑁,0,𝑘=𝑁−𝑚+1,…,𝑁,𝑗=0,1,…,𝑁,_𝑘𝑗=−𝑃^(𝛼,𝛽)_𝐿,𝑗×𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}𝑑,𝑘=0,1,…,𝑁−𝑚,𝑗=0,1,…,𝑁,0,𝑘=𝑁−𝑚+1,…,𝑁,𝑗=0,1,…,𝑁,_𝑛𝑘𝑗=−𝑏_𝑛𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}×Γ𝑗+𝛼+𝛽+𝑛+1𝐿^𝑛Γ𝑗+𝛼+𝛽+1×𝑃^{(𝛼+𝑛,𝛽+𝑛)}_{𝐿,𝑗−𝑛}𝛾_𝑛𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑘=0,1,…,𝑁−𝑚,𝑗=0,1,…,𝑁,0,𝑘=𝑁−𝑚+1,…,𝑁,𝑗=0,1,…,𝑁. (26) Moreover, the elements of the matrix corresponding to the mixed initial conditions are given by 𝑒_𝑘𝑗=0,𝑘=0,1,…,𝑁−𝑚,𝑗=0,1,…,𝑁,_{𝑚−1𝑛=0}(−1)^𝑗−𝑛×𝑐_{𝑘−𝑁−𝑚+1,𝑛}×Γ𝑗+𝛽+1𝑗+𝛼+𝛽+1_𝑛𝐿^𝑛Γ𝑗−𝑛+1Γ𝑛+𝛽+1,𝑘=𝑁−𝑚+1,…,𝑁,𝑗=0,1,…,𝑁. (27)

In the case of 𝑎(𝑥)≠0, 𝑏_𝑛(𝑥)≠0, 𝑛=0,1,…,𝑚−1, and 𝛽≠0, the linear system (25) can be solved by forming explicitly the LU factorization; that is, 𝐴+𝛾^𝑚_𝑚𝐵+𝛽𝐶+∑^𝑚−1_𝑛=0𝛾^𝑛_𝑛𝐷_𝑛+𝐸=LU. The expense of calculating LU factorization is 𝑂(𝑁³) operations, and the expense of solving the linear system (25), provided that the factorization is known, is 𝑂(𝑁²).

4. Nonlinear High-Order NFDE with Proportional Delay

In this section, we investigate the shifted Jacobi-Gauss-Lobatto pseudospectral method to numerically approximate the nonlinear high-order NFDE with proportional delay; namely, 𝑑^𝑚𝑢(𝑥)𝑑𝑥^𝑚=𝐺𝑥,𝑢(𝑥),𝑢𝛾₀𝑥,𝑑𝑢𝛾₁𝑥𝑑𝑑𝑥,…,^𝑚𝑢𝛾_𝑚𝑥𝑑𝑥^𝑚,0≤𝑥≤𝐿, (28) subject to ^𝑚−1_𝑛=0𝑐_𝑖𝑛𝑢^(𝑛)(0)=𝜆_𝑖,𝑖=0,1,…,𝑚−1, (29) where 𝛾_𝑛, 𝑐_𝑖𝑛, and 𝜆_𝑖 are constants with 0<𝛾_𝑛<1 (𝑛=0,1,…,𝑚), meanwhile, 𝐺 is nonlinear in general. This equation is a generalized form of the proportional delay differential equations given in [14, 29, 30] and plays an important role in modeling phenomena of the real world.

The shifted Jacobi-Gauss-Lobatto pseudospectral approximation for (28) is to find 𝑢_𝑁(𝑥)∈𝑆_𝑁(0,𝐿) such that 𝑑^𝑚𝑑𝑥^𝑚𝑢𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}=𝐺𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑢𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑢𝛾₀𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑑𝑢𝛾𝑑𝑥¹𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}𝑑,…,_𝑚𝑑𝑥^𝑚𝑢𝛾_𝑚𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑘=0,1,…,𝑁−𝑚. (30)

Now, we approximate the numerical solution as a truncated series expansion of shifted Jacobi polynomial in the form 𝑢_𝑁(𝑥)=^𝑁_𝑗=0𝑎_𝑗𝑃^(𝛼,𝛽)_𝐿,𝑗(𝑥). (31) Accordingly, (30) can be written as ^𝑁_𝑗=0𝑎_𝑗𝑑^𝑚𝑑𝑥^𝑚𝑃^(𝛼,𝛽)_𝐿,𝑗𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}=𝐺𝑥,^𝑁_𝑗=0𝑎_𝑗𝑃^(𝛼,𝛽)_𝐿,𝑗𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},_𝑁𝑗=0𝑎_𝑗𝑃^(𝛼,𝛽)_𝐿,𝑗𝛾₀𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},_𝑁𝑗=0𝑎_𝑗𝑑𝑃𝑑𝑥^{(𝛼,𝛽)𝐿,𝑗}𝛾₁𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},…,_𝑁𝑗=0𝑎_𝑗𝑑^𝑚𝑑𝑥^𝑚𝑃^(𝛼,𝛽)_𝐿,𝑗𝛾_𝑚𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑘=0,1,…,𝑁−𝑚. (32)

Next, making use of relation (8), thus the high-order derivatives of the proposed solution with proportional delays, (𝑑𝑢(𝛾₁𝑥)/𝑑𝑥),…,(𝑑^𝑚𝑢(𝛾_𝑚𝑥)/𝑑𝑥^𝑚), can be expressed explicitly in terms of the shifted Jacobi polynomials and the expansion coefficients 𝑎_𝑗; consequently ^𝑁_𝑗=0𝑎_𝑗Γ𝑗+𝛼+𝛽+𝑚+1𝐿^𝑛Γ𝑃𝑗+𝛼+𝛽+1^{(𝛼+𝑚,𝛽+𝑚)𝐿,𝑗−𝑚}𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘}=𝐺𝑥,^𝑁_𝑗=0𝑎_𝑗𝑃^(𝛼,𝛽)_𝐿,𝑗𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},^𝑁_𝑗=0𝑎_𝑗𝑃^(𝛼,𝛽)_𝐿,𝑗𝛾₀𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},_𝑁𝑗=0𝑎_𝑗𝛾₁Γ𝑗+𝛼+𝛽+𝑚+1𝐿^𝑛Γ𝑗+𝛼+𝛽+1×𝑃^{(𝛼+𝑚,𝛽+𝑚)}_{𝐿,𝑗−𝑚}𝛾₁𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},…,_𝑁𝑗=0𝑎_𝑗𝛾^𝑚_𝑚Γ𝑗+𝛼+𝛽+𝑚+1𝐿^𝑛Γ𝑗+𝛼+𝛽+1×𝑃^{(𝛼+𝑚,𝛽+𝑚)}_{𝐿,𝑗−𝑚}𝛾_𝑚𝑥^(𝛼,𝛽)_{𝐿,𝑁−𝑚,𝑘},𝑘=0,1,…,𝑁−𝑚. (33)

Finally, to find the unknown expansion coefficients 𝑎_𝑗, 𝑗=0,1,…,𝑁, we implement any iteration technique to solve a system of 𝑁+1 nonlinear algebraic equations resulting from the combination of 𝑁−𝑚+1 nonlinear algebraic equations resulting from (33) and 𝑚 linear algebraic equations resulting from (29).

5. Numerical Results

In this section, we will carry out three test examples to study the validity and effectiveness of the proposed method and also show that high accurate solutions are achieved using a few number of the Jacobi Gauss-Lobatto points. Moreover, comparisons with other methods reveal that the present method is accurate and convenient. All the numerical computations have been performed by the symbolic computation software Mathematica 8.0.

Example 1. Consider the first-order NFDE with proportional delay considered in [31] 𝑢^′(𝑥)=−𝑢(𝑥)+0.1𝑢(0.8𝑥)+0.5𝑢^′()+()𝑒0.8𝑥0.32𝑥−0.5^−0.8𝑥+𝑒^−𝑥𝑢(0),𝑥≥0,=0, (34) which has the exact solution 𝑥𝑒^−𝑥.

Table 1 lists the absolute error using Jacobi pseudospectral method for three choices of 𝛼,𝛽 at 𝑁=20 in the interval [0,1]. We compare the errors obtained by the proposed method with variational iteration (VI) method [32], the one-leg 𝜃 method [14, 33] with 𝜃=0.8, and RKHSM method [31]. The graph of analytical solution and approximate solution in long interval [0,10] for 𝑁=28 and 𝛼=−1/2, 𝛽=1/2 is displayed in Figure 1 to make it easier to compare with analytical solution. Moreover, in this case the graph of the error is given in Figure 2. Consequently, we conclude that the approximate solution by Jacobi pseudospectral method agree very well with the exact solution.

Table 1 

Absolute errors using SJGLP method with various choices of 𝑁, 𝛼, and 𝛽 for Example 1.

Figure 1 

Comparison of the approximate solution with the exact solution for 𝛼=−1/2, 𝛽=1/2 at 𝑁=28 for Example 1.

Figure 2 

The error between the approximate solution and the exact solution in the interval [0,10] for 𝛼=−1/2, 𝛽=1/2 at 𝑁=28 for Example 1.

Example 2. Let us consider the second-order NFDE with proportional delay 𝑢^′′(𝑥)=𝑢^′𝑥2−12𝑥𝑢^′′𝑥2−𝜋2×2cos𝜋𝑥2+𝜋𝑥sin𝜋𝑥2()[],𝑢(0)+2𝜋sin𝜋𝑥,𝑥∈0,6=0,𝑢^′(0)=𝜋, (35) which enjoys exact solution 𝑢(𝑥)=sin(𝜋𝑥).

In Table 2, we introduce the absolute error using the proposed method at 𝑁=24 with various choices of 𝛼 and 𝛽. The resulting graph of (35) for the presented method in the case of 𝛼=𝛽=1 at 𝑁=24 and the analytic solution are shown in Figure 3.

Table 2 

Absolute errors using SJGLP method with various choices of 𝛼 and 𝛽 for Example 2.

Figure 3 

Graph of exact solution and approximate solution for 𝛼=𝛽=1 and 𝑁=24 for Example 2.

Example 3. Consider the third-order NFDE with proportional delays 𝑢^′′′(𝑥)=(𝑢(𝑥))²+𝑢^′𝑥2+𝑢^′′𝑥3+12𝑢^′′′𝑥4(𝑥)𝑢(0)+𝑓,𝑥≥0,=1,𝑢^′(0)=13,𝑢^′′(0)=−5759, (36) where 𝑓(𝑥)=1𝑒54^𝑥/12−13752sin(2𝑥)−54𝑒^7𝑥/12cos²()()8𝑥+1727cos2𝑥+6𝑒^𝑥/36×48sin8𝑥3+575cos8𝑥3−18𝑒^𝑥/12(cos(4𝑥)−24sin(4𝑥))+2𝑒^𝑥/4(13752sin(8𝑥)−1727cos(8𝑥)). (37) The exact solution of the problem is 𝑢(𝑥)=𝑒^𝑥/3cos(8𝑥).

Table 3 lists the maximum absolute errors using SJGLP method in the interval [0,1] at 𝛼=−𝛽=0.5 and different choices of 𝛼 and 𝛽.

Table 3 

Maximum absolute errors using SJGLP method with various choices of 𝛼, 𝛽, and 𝑁 for Example 3.

In case of Chebyshev polynomials of the second kind (𝛼=𝛽=1/2), the graph of exact solution and approximate solution for 𝑁=24 is plotted in Figure 4 in the interval [0,3].

Figure 4 

Graph of exact solution and approximate solution for 𝛼=𝛽=1/2 and 𝑁=24 for Example 3.

6. Conclusion

In this paper, we have demonstrated the feasibility of SJGLP for solving linear NFDEs with proportional delays. We also have discussed the resulting linear system. Moreover, we have implemented the SJGLP method to numerically approximate the nonlinear high-order NFDE with proportional delay.

All the given examples reveal that the results of SJGLP method are in excellent agreement with the analytical solutions. It is concluded from the aforementioned tables and figures that SJGLP method is an accurate and efficient method to solve NFDEs when compared with those generated by some other methods.

In the future work, we address the Jacobi pseudospectral approximation for the solution of linear and nonlinear delay partial differential equations in two and three dimensions (see, e.g., [17, 34]).