One- And Two-Stage Implicit Interval Methods of Runge-Kutta Type

Marciniak, Andrzej; Szyszka, Barbara

doi:10.12921/cmst.1999.05.01.53-65

Cited by 19 publications

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“…. , n − 1) one can obtain by applying an interval one-step method, for example an interval method of Runge-Kutta type (see, e.g., [7,8,20,27,31,43]) or interval methods based on Taylor series (see, e.g., [2,3,5,15,21,39]). Let us note that in (10), we cannot write γ * n + γ * * n n instead of γ * n n + γ * * n n , because in general γ * n + γ * * n may be different from γ * n + γ * * n .…”

Section: And Where (T Y ) Denotes An Interval Extension Of F (N) (Tmentioning

confidence: 99%

“…The first one was described by R. E. Moore in 1965 [33][34][35]. There are also known interval methods based on high-order Taylor series (see, e.g., [2,3,5,15,21,39,41]), explicit and implicit Runge-Kutta methods [7,8,20,24,27,31,43], explicit and implicit multistep methods [17-19, 25-27, 30, 43]. The last ones concern interval methods based on conventional methods of Adams-Bashforth, Adams-Moulton, Nyström, and Milne-Simpson types.…”

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confidence: 99%

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Interval versions of Milne’s multistep methods

Marciniak

Jankowska

2017

Numer Algor

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The paper presents explicit interval multistep methods of Milne type, which may be considered as alternative methods to other known explicit interval multistep methods (of Adams-Bashforth and Nyström). It is proved that enclosures of solutions (in the form of intervals) obtained by these methods contain the exact solutions of the initial value problem. Numerical examples show that the widths of intervals obtained by proposed methods are smaller than those obtained by explicit interval multistep methods known so far.

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Section: And Where (T Y ) Denotes An Interval Extension Of F (N) (Tmentioning

confidence: 99%

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confidence: 99%

Interval versions of Milne’s multistep methods

Marciniak

Jankowska

2017

Numer Algor

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“…We can obtain such Y i by applying interval one-step method, for example an interval method of Runge-Kutta type (see [19] or [20]). Then the explicit interval method of Adams-Bashforth type constructed by Šokin is given by the formula.…”

Section: ) ψ(T Y) -An Interval Extension Of ψ(T Y)mentioning

confidence: 99%

On Explicit Interval Methods of Adams-Bashforth Type

Jankowska¹,

Marciniak²

2002

CMST

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Abstract. In our previous paper [1] we have considered implicit interval multistep methods of AdamsMoulton type for solving the initial value problem. On the basis of these methods and the explicit ones introduced by Sokin [2] we wanted to construct predictor-corrector (explicit-implicit) interval methods. However, it turned out that the formulas given by Šokin are incorrect even in the simplest case. Therefore, in this paper we direct our attention to the explicit interval methods of Adams-Bashforth type and modify the formulas of Šokin. For the modified explicit interval methods it is proved, like for the implicit interval methods considered in [1], that the exact solution of the problem belongs to interval-solutions obtained by these methods. Moreover, it is shown an estimation of the widths of such interval-solutions.

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“…An interval method for ordinary differential equations using interval arithmetic was described first by R. E. Moore in 1965 [32,33]. There are also interval methods based on explicit Runge-Kutta methods [21,28,41] and implicit ones [10,11,25,28,31]. In [41], Yu.…”

Section: Introductionmentioning

confidence: 99%

On interval predictor-corrector methods

2016

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One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of RungeKutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.

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One- And Two-Stage Implicit Interval Methods of Runge-Kutta Type

Cited by 19 publications

References 0 publications

Interval versions of Milne’s multistep methods

Interval versions of Milne’s multistep methods

On Explicit Interval Methods of Adams-Bashforth Type

On interval predictor-corrector methods

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