The shifted JacobiGaussLobatto pseudospectral (SJGLP) method is applied to neutral functionaldifferential equations (NFDEs) with proportional delays. The proposed approximation is based on shifted Jacobi collocation approximation with the nodes of GaussLobatto quadrature. The shifted LegendreGaussLobatto Pseudospectral and ChebyshevGaussLobatto Pseudospectral methods can be obtained as special cases of the underlying method. Moreover, the SJGLP method is extended to numerically approximate the nonlinear highorder 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, oneleg
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, [
In the last two decades, some numerical approaches for treating several types of DDEs were presented in [
The aim of this paper is to develop a direct solution technique to approximate the linear highorder NFDEs with proportional delays using the shifted Jacobi polynomials on the interval
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
In this section, we briefly recall some properties of the Jacobi polynomials (
The set of Jacobi polynomials forms a complete
Let us denote
The following inner product and norm
According to (
In this section, we shall investigate solutions to NFDEs with proportional delays of the form
In the pseudospectral methods [
Now, we will present the shifted JacobiGaussLobatto type quadratures. Let
Let us first introduce some basic notations that will be used in the sequel. We set
The shifted JacobiGaussLobatto pseudospectral method for solving (
We first approximate
Then, by virtue of (
Also, by substituting (
To find the solution
Next, (
Thus (
The matrix system associated with (
In the case of
In this section, we investigate the shifted JacobiGaussLobatto pseudospectral method to numerically approximate the nonlinear highorder NFDE with proportional delay; namely,
The shifted JacobiGaussLobatto pseudospectral approximation for (
Now, we approximate the numerical solution as a truncated series expansion of shifted Jacobi polynomial in the form
Next, making use of relation (
Finally, to find the unknown expansion coefficients
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 GaussLobatto 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.
Consider the firstorder NFDE with proportional delay considered in [
Table
Absolute errors using SJGLP method with various choices of



SJGLP method 
In [ 
In [ 
In [  




0.1 







0  0 


 
0.2 







0  0 


 
0.3 







0  0 


 
0.4 







0  0 


 
0.5 







0  0 


 
0.6 







0  0 


 
0.7 







0  0 


 
0.8 







0  0 


 
0.9 







0  0 


 
1.0 







0  0 

Comparison of the approximate solution with the exact solution for
The error between the approximate solution and the exact solution in the interval
Let us consider the secondorder NFDE with proportional delay
In Table
Absolute errors using SJGLP method with various choices of





0.0 



0.1 



0.2 



0.3 



0.4 



0.5 



1.0 



2.0 



3.0 



4.0 



5.0 



6.0 



Graph of exact solution and approximate solution for
Consider the thirdorder NFDE with proportional delays
Table
Maximum absolute errors using SJGLP method with various choices of





12 



16 



20 



24 



28 



In case of Chebyshev polynomials of the second kind (
Graph of exact solution and approximate solution for
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 highorder 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., [