Legendre–Gauss–Lobatto spectral collocation method for nonlinear delay differential equations

Yi, Lijun; Zi-qiang, Liang; Wang, Zhong-Qing

doi:10.1002/mma.2769

Cited by 10 publications

(1 citation statement)

References 45 publications

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“…In this section, we provide the error analysis of the spectral collocation method for SVIDE. We will particularly determine the spectral accuracy for the numerical solution y N (t) [26].…”

Section: Error Analysismentioning

confidence: 99%

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

Khan

Ali

2019

Adv Differ Equ

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Volterra integro-differential equations arise in the modeling of natural systems where the past influence the present and future, for example pollution, population growth, mechanical systems and financial market. Furthermore, as many real-world phenomena are subject to perturbations or random noise, it is natural to move from deterministic models to stochastic models. Generally exact solutions of such models are not available and numerical methods are used to obtain the approximate solutions. Therefore the efficiency and long-term behavior of approximate solutions for these systems is an important area of investigation. This paper presents a new numerical approach for the approximate solution of stochastic Volterra integro-differential (SVID) equations based on the Legendre-spectral collocation method. In order to fully use the properties of orthogonal polynomials, we use some function and a variable transformation to change the given SVID equation into a new equation, which is defined on the standard interval [-1, 1]. For the evaluation of the integral term efficiently a Legendre-Gauss quadrature formula will be used. A rigorous error analysis of the proposed scheme will be provided under the assumption that the solution of the given SVID is sufficiently smooth. For the illustration of our theoretical results a number of numerical experiments will be performed.

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“…In this section, we provide the error analysis of the spectral collocation method for SVIDE. We will particularly determine the spectral accuracy for the numerical solution y N (t) [26].…”

Section: Error Analysismentioning

confidence: 99%

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

Khan

Ali

2019

Adv Differ Equ

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On the discontinuous solutions of explicit neutral delay differential equations

Davari

Maleki

2023

Math Methods in App Sciences

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This paper considers explicit neutral delay differential equations (NDDE) with piecewise continuous initial functions. We explain how the discontinuities in the solutions arise and present a perturbing scheme, in combination with an adaptive Legendre–Gauss–Radau collocation method, to deal with this type of problems computationally. The pointwise and mean convergence of the continuous solution of the perturbed NDDE to the discontinuous solution of the original NDDE are proved. Our new method for discontinuous NDDEs and the rigorous theoretical analysis provided are particularly important since explicit NDDEs have received little attention in the literature. Numerical results are given to show that the proposed method can be implemented in an efficient and accurate manner.

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An h–p version of the Chebyshev spectral collocation method for nonlinear delay differential equations

Meng

Wang

2018

Numerical Methods Partial

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We develop and analyze a spectral collocation method based on the Chebyshev-Gauss-Lobatto points for nonlinear delay differential equations with vanishing delays. We derive an a priori error estimate in the H 1 -norm that is completely explicit with respect to the local time steps and the local polynomial degrees. Several numerical examples are provided to illustrate the theoretical results. KEYWORDS h-p version, Chebyshev-Gauss-Lobatto points, nonlinear delay differential equations, spectral collocation method 1 where the delay function satisfying the following conditions:We always assume that f is chosen such that the problem (1.1) possesses a unique solution u ∈ C 1 (I). In particular, the problem (1.1) includes a special important case when (t) = qt (0 < q < 1) is a linear proportional delay function satisfying (C1) and (C2).DDEs arise in many areas of mathematical modelings. During the past few decades, many numerical methods have been proposed and studied for the DDEs; see, for example, the collocation Numer Methods Partial Differential Eq. 2019;35:664-680. wileyonlinelibrary.com/journal/num

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Legendre–Gauss–Lobatto spectral collocation method for nonlinear delay differential equations

Cited by 10 publications

References 45 publications

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

A spectral collocation method for stochastic Volterra integro-differential equations and its error analysis

On the discontinuous solutions of explicit neutral delay differential equations

An h–p version of the Chebyshev spectral collocation method for nonlinear delay differential equations

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