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A Legendre-Gauss collocation method for neutral functional-differential equations with proportional delays
© Bhrawy et al.; licensee Springer 2013
Received: 5 January 2013
Accepted: 28 February 2013
Published: 20 March 2013
In this paper, we present a unified framework for analyzing the spectral collocation method for neutral functional-differential equations with proportional delays using shifted Legendre polynomials. The proposed collocation technique is based on shifted Legendre-Gauss quadrature nodes as collocation knots. Error analysis and stability of the proposed algorithm are theoretically investigated under several mild conditions. The accuracy of the proposed method has been compared with a variational iteration method, a one-leg θ-method, a particular Runge-Kutta method, and a reproducing kernel Hilbert space method. Numerical results show that the proposed methods are of high accuracy and are efficient for solving such an equation. Also, the results demonstrate that the proposed method is a powerful algorithm for solving other delay differential equations.
One of the fundamental classes of delay differential equations (DDEs) is that of neutral functional-differential equations (NFDEs) with proportional delays. Such equations arise in many areas of science and engineering and play an important role in the modeling of real-life phenomena in other fields of science (cf. [1–5]). For these reasons, NFDEs have received much attention in the last decades. The principal difficulty in studying DDEs lies in their special transcendental nature. Obviously, most of NFDEs cannot be solved by the well-known exact methods. Therefore, it is highly desirable to design accurate numerical approaches to approximate the solutions of NFDEs.
The theory of DDEs with multiple delays has been analyzed by many authors, and we briefly review some of them. In  Jackiewicz and Lo proposed and developed the Adams predictor-corrector method for the numerical solution of NFDEs. In , the authors investigated the Adomian decomposition method for solving a special class of DDEs and established the convergence of this approach.
Some analytical and numerical solutions of a family of DDEs were presented in . The authors of  investigated the Runge-Kutta (RK) method for NFDEs with different proportional delays and proved the stability of RK for this equation. Wang and Li  proposed the one-leg-methods for solving nonlinear neutral functional differential equations. We also refer to the articles [11–13] and the references therein for applying the waveform relaxation method to solve neutral type functional-differential systems.
Over the years, it was found that the spectral methods were a valid method to obtain approximations for differential equations. The solution of a DDE globally depends on its history due to the delay variable, so a global spectral method could be a good candidate for numerical DDEs. In this direction, Ishiwata and Muroya  proposed the rational approximation algorithm based on Legendre polynomials and Ishiwata et al.  developed the collocation approximation for solving DDEs. Meanwhile, in  the authors discussed the tau approach for solving a class of NFDEs, and the solution was expanded by a shifted Legendre polynomial, and the unknown expansion coefficients were obtained by a segmented Lanczos-tau formulation. The authors of  proposed a Bernoulli operational matrix method for solving a generalized pantograph equation. Wang and Wang  proposed and analyzed the Legendre collocation algorithm for nonlinear DDEs with variable delay. The theory and the numerical results given in  and  explained that the proposed collocation approximations converge exponentially when the data in the given DDEs are smooth. Recently, Ali et al.  implemented a spectral Legendre approach for solving pantograph-type differential and integral equations and studied the error analysis of the method. More recently, the work of Trif  discussed the application of the tau method based on the operational matrix of Chebyshev polynomials for solving DDEs of pantograph type.
Our fundamental goal of this paper is to develop a suitable way to approximate the neutral functional-differential equations with proportional delays on the interval by using the shifted Legendre polynomials. In other words, we propose the spectral shifted Legendre-Gauss collocation (SLC) method to find the solution . For suitable collocation points, we use the nodes of the shifted Legendre-Gauss interpolation on the interval . These equations together with m initial conditions generate algebraic equations which can be solved. It should be noted that the basic requirement for using any spectral base (e.g., Legendre polynomials) is the smoothness of the solution of the considered problem. This may be guaranteed by the smoothness of known functions in the neutral equation. By these assumptions (i.e., being smooth), exponential convergence behavior of spectral approximations is exhibited in any test problem.
The paper is organized as follows. Section 2 is devoted to preliminaries needed hereafter. In Section 3, we design the shifted Legendre-Gauss collocation technique for NFDEs with proportional delays. The error analysis and stability of the solution are provided in Section 4 under several mild conditions. In Section 5, we present some numerical results demonstrating the efficiency of the suggested numerical algorithm. Concluding remarks are given in Section 6.
3 Shifted Legendre-Gauss collocation method
Here, and () are given analytical functions, and , β, are constants with ().
where is defined in (14).
Thus Eq. (21) with relation (22) generate of a set of algebraic equations which can be solved for the unknown coefficients , , by using any standard solver technique.
4 Error analysis and stability of the solution
This section is divided into two subsections. The first subsection is related to presenting a bound for the error of the proposed method; meanwhile, in the second subsection, the stability of the numerical solution is investigated briefly.
4.1 Error bound of the method
In this part, the error bound of the method will be provided under several mild conditions such as solution boundedness of the main neutral differential equation. However, some definitions and lemmas should be provided for clarifying the main theorem of this subsection.
and stands for any finite dimensional norm in .
where C is a constant independent of N, m is the order of smoothness of ξ, and . Here, with the smallest norm is called the Nth order best polynomial approximation of in the norm of .
where and .
In the following theorem, we show that the approximate solution which was expressed in terms of Legendre polynomials converges to the exact solution under several mild conditions.
Theorem 4.2 Consider Eq. (23) again. Assume and are the exact and approximate solutions of (23). Also, let the approximations of , , and be , , and , respectively. Moreover, suppose that , , , where . Then under the condition .
If , then (or ). This conclusion is made because of the smoothness of , , and . These assumptions imply that , , and (see Lemma 4.1). This completes the proof. □
4.2 Stability of the solution
are the Lebesgue constant and the Lebesgue function, respectively. Also, stands for the jth Lagrange polynomial which is constructed by using shifted Gaussian points.
The Lebesgue function and the Lebesgue constant both depend only on the choice of interpolation nodes. The above theorem indicates that a good set of grid points is one which minimizes the Lebesgue constant for interpolation. We know that on equally spaced interpolation points, the Lebesgue constant blows up exponentially with the increasing of N (as made apparent by the classical Runge phenomenon). On the contrary, the best possible Lebesgue constant among all distributions of N prohibits only a logarithmic growth with N. Fortunately, the main families of Gaussian points (Gauss, Gauss-Radau, Gauss-Lobatto) for the Legendre or Chebyshev weights have Lebesgue constants that grow logarithmically or sub-linearly with N.
Now we can deduce that for large N, Gaussian interpolation points with respect to equally spaced points have better stability, and for equally spaced points, an interpolation polynomial can become unstable for large N.
5 Numerical results
To illustrate the effectiveness of the proposed method in the present paper, several test examples are carried out in this section. Comparisons of the results obtained by the present method with those obtained by other methods reveal that the present method is very effective and convenient. We consider the following examples.
which has the exact solution .
Absolute errors using SLC method at for Example 1
SLC method N = 17
VI method 
RKT method 
n = 5
n = 6
the analytic solution of the aforementioned problem is .
Absolute errors using SLC method at for Example 2
θ-method with θ = 0.8
VI method m = 8
SLC method N = 14
the exact solution of which is .
Absolute errors using SLC method at for Example 3
θ-method with θ = 0.8
VI method m = 8
SLC method N = 10
and the function is chosen such that the analytical solution .
Absolute errors using SLC method for Example 4
6 Conclusions and future works
In this paper, we have proposed a new collocation method based on the shifted Legendre polynomials to numerically solve the neutral functional-differential equations with studying the error analysis of the proposed method. The comparison of the obtained results with those based on other methods shows that the present method is a powerful mathematical tool for finding the numerical solutions of such equations. High accuracy in long computational intervals and the stability of the approximated solutions encourage us to apply a similar method for solving other applied mathematics problems (see, for instance, ) in the future.
This study was supported by the Deanship of Scientific Research of King Abdulaziz University. The authors would like to thank the reviewers for their constructive comments and suggestions to improve the quality of the article.
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