Numer. Math. Theory Methods Appl.

2016

DOI: 10.4208/nmtma.2016.m1429

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A Multiple Interval Chebyshev-Gauss-Lobatto Collocation Method for Ordinary Differential Equations

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Abstract: We introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

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The Cgl Spectral Collocation Methods For Ddes3

The Multistep Spectral Methods1

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Cited by 9 publications

(4 citation statements)

References 29 publications

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“…and the constant M k is of the order of the Chebyshev polynomial of approximation u k defined on Λ k . The proof can be found in [12,27].…”

Section: The Multistep Spectral Methodsmentioning

confidence: 99%

Comparative performance of time spectral methods for solving hyperchaotic finance and cryptocurrency systems

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2021

Chaos, Solitons & Fractals

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“…and the constant M k is of the order of the Chebyshev polynomial of approximation u k defined on Λ k . The proof can be found in [12,27].…”

Section: The Multistep Spectral Methodsmentioning

confidence: 99%

Comparative performance of time spectral methods for solving hyperchaotic finance and cryptocurrency systems

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²

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³

2021

Chaos, Solitons & Fractals

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“…Moreover, for any

φ \in {scriptP}_{p_{n} + 1} (I_{n})

, there holds (cf. Theorem 2.2 in )

{(‖‖, φ)}_{p_{n}, w_{n}} \leq \{\begin{matrix} left \\ \sqrt{2} {(‖‖, φ)}_{L_{ω_{n}}^{2} (I_{n})}, p_{n} > 1, \\ 2 {(‖‖, φ)}_{L_{ω_{n}}^{2} (I_{n})}, p_{n} = 1 . \end{matrix}

…”

Section: The Cgl Spectral Collocation Methods For Ddesmentioning

confidence: 99%

“…Then, we have the following interpolation errors.Lemma For any v ∈ H r (I n ) with integer 1 ≤ r ≤ p n + 1, there hold

\begin{matrix} {‖v - {normalℐ}_{p_{n}} v‖}_{L^{2} ((), I_{n})} & \leq {Ch}_{n}^{r} p_{n}^{- r} {‖\partial_{t}^{r} v‖}_{L^{2} ((), I_{n})}, \end{matrix}

\begin{matrix} {|v - {normalℐ}_{p_{n}} v|}_{H^{1} ((), I_{n})} & \leq {Ch}_{n}^{r - 1} p_{n}^{1 - r} {‖\partial_{t}^{r} v‖}_{L^{2} ((), I_{n})}, \end{matrix}

and

\begin{array}{c} {‖v - {normalℐ}_{p_{n}} v‖}_{L_{ω_{n}}^{2} ((), I_{n})} \leq {Ch}_{n}^{r - \frac{1}{2}} p_{n}^{- r} {‖\partial_{t}^{r} v‖}_{L^{2} ((), I_{n})} . \end{array}

Proof The proofs of and are given in Theorem 2.1 of . Let

{(|, u ((), x) = v ((), t))}_{t = \frac{h_{n} x + t_{n - 1} + t_{n}}{2}}

and

π_{p_{n}} u

be the standard CGL interpolation of u on the interval [−1, 1].…”

Section: The Cgl Spectral Collocation Methods For Ddesmentioning

confidence: 99%

“…In view of (2.18)–(2.20) and (2.4), a direct computation shows that (cf. )

\begin{array}{l} \sum_{j = 0}^{p_{n} + 1} U_{j}^{n} T_{n, j} ((), t) & = U^{n} ((), t_{n - 1}) + \sum_{j = 0}^{p_{n}} f_{n, j} \int_{t_{n - 1}}^{t} T_{n, j} ((), s) italicds \\ = U^{n} ((), t_{n - 1}) + \frac{h_{n}}{2} f_{n, 0} ((), T_{n, 1} ((), t) + 1) + \frac{h_{n}}{8} f_{n, 1} ((), T_{n, 2} ((), t) - 1) \\ + \frac{h_{n}}{4} \sum_{j = 2}^{p_{n}} f_{n, j} \int_{t_{n - 1}}^{t} ((), \frac{T_{n, j + 1}^{'} ((), s)}{j + 1} - \frac{T_{n, j - 1}^{'} ((), s)}{j - 1}) \\ = U^{n} ((), t_{n - 1}) + \frac{h_{n}}{2} f_{n, 0} - \frac{h_{n}}{8} f_{n, 1} + \frac{h_{n}}{2} \sum_{j = 2}^{p_{n}} f_{n, j} \frac{{((), - 1)}^{j - 1}}{j^{2} - 1} \\ + \frac{h_{n}}{4} \sum_{j = 1}^{p_{n} - 1} c_{j - 1...} \end{array}

…”

Section: The Cgl Spectral Collocation Methods For Ddesmentioning

confidence: 99%

See 1 more Smart Citation

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

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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

Introduction to Interval Analysis and Solving the Problems with Interval Uncertainties

2023

Interval Analysis

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No abstract

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