Abstract
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.
Highlights
One of the fundamental classes of delay differential equations (DDEs) is that of neutral functional-differential equations (NFDEs) with proportional delays
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 [, T] by using the shifted Legendre polynomials
3 Shifted Legendre-Gauss collocation method we develop a spectral Legendre-Gauss collocation approach to analyze the following NFDEs with proportional delays: m
Summary
One of the fundamental classes of delay differential equations (DDEs) is that of neutral functional-differential equations (NFDEs) with proportional delays. 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 [ , T] by using the shifted Legendre polynomials. 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. We show that the approximate solution which was expressed in terms of Legendre polynomials converges to the exact solution under several mild conditions. The logarithmic graph of absolute coefficients of shifted Legendre polynomials of a neutral functional-differential equation with proportional delay is shown in Figure , which shows that the method has exponential convergence rate.
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