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A Legendre-Gauss collocation method for neutral functional-differential equations with proportional delays. (English) Zbl 1380.65116

Summary: 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 \(\theta\)-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.

MSC:

65L03 Numerical methods for functional-differential equations
34K40 Neutral functional-differential equations
65L60 Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations
33C90 Applications of hypergeometric functions
34K07 Theoretical approximation of solutions to functional-differential equations

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