A Legendre–Gauss–Radau spectral collocation method for first order nonlinear delay differential equations

Yi, Lijun; Wang, Zhong-Qing

doi:10.1007/s10092-015-0169-5

Cited by 12 publications

(4 citation statements)

References 26 publications

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“…By the orthogonality condition (22) and the Peano theorem, it is obvious that quadrature errors satisfy…”

Section: Theorem 3 Let the Given Function Inmentioning

confidence: 99%

“…Collocation methods as numerical methods have a wide range of applications in the treatment of integral-algebraic equations [13][14][15][16], Volterra integral equations [17][18][19] and delay differential equations [20][21][22]. Specifically, the convergence of the collocation methods has received a lot of attention, such as the convergence of collocation methods for weakly singular Volterra integral equations [23], the superconvergence of collocation methods for first-kind Volterra integral equations [24], the convergence of collocation methods for Volterra integral equations [25], the convergence of multistep collocation methods for integral-algebraic equations [16], etc.…”

Section: Introductionmentioning

confidence: 99%

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Convergence of Collocation Methods for One Class of Impulsive Delay Differential Equations

Wang

Zhang

Sun

2023

Axioms

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This paper is concerned with collocation methods for one class of impulsive delay differential equations (IDDEs). Some results for the convergence, global superconvergence and local superconvergence of collocation methods are given. We choose a suitable piecewise continuous collocation space to obtain high-order numerical methods. Some illustrative examples are given to verify the theoretical results.

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“…By the orthogonality condition (22) and the Peano theorem, it is obvious that quadrature errors satisfy…”

Section: Theorem 3 Let the Given Function Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Convergence of Collocation Methods for One Class of Impulsive Delay Differential Equations

Wang

Zhang

Sun

2023

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“…Various kinds of spectral methods for DDEs were introduced and analyzed; see, for example, for a Legendre‐tau method for linear DDEs with constant delays, for Legendre spectral collocation discretizations of linear DDEs with vanishing proportional delays. For nonlinear DDEs with nonlinear delays, several Legendre spectral collocation methods were proposed and analyzed, see and the references therein. However, to the best of our knowledge, only limited efforts have been devoted to the Chebyshev spectral methods for DDEs (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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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“…[27]) If conditions Eq.(2)-Eq. (5) hold, R r T,1 (Φ) is a finite function for integers 1 ≤ r ≤ N , and T N -1 1 +…”

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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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A Legendre–Gauss–Radau spectral collocation method for first order nonlinear delay differential equations

Cited by 12 publications

References 26 publications

Convergence of Collocation Methods for One Class of Impulsive Delay Differential Equations

Convergence of Collocation Methods for One Class of Impulsive Delay Differential Equations

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

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

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