Implicit Interval Multistep Methods for Solving the Initial Value Problem

Jankowska, Małgorzata A.; Marciniak, Andrzej

doi:10.12921/cmst.2002.08.01.17-30

Cited by 16 publications

(17 citation statements)

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“…In [18,20,26,28], we have proved that the exact solution of the initial value problem (1) belongs to the interval solutions Y k obtained by the implicit interval multistep methods considered in this section. In the same papers, we have estimated the widths of these solutions.…”

Section: As Previously Let Us Denote By T and Y The Sets In Which Thmentioning

confidence: 90%

“…Unfortunately, it can be shown that his formula fails in the simplest case, but it can be easily corrected [19,28]. Other explicit interval multistep methods have been considered in [26][27][28] and implicit ones in [18,20,[26][27][28]. In recent years, many studies have been conducted on a variety of interval methods based on high-order Taylor series (see, e.g., [2-4, 7, 17, 22, 36, 38, 40]).…”

Section: Introductionmentioning

confidence: 99%

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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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Section: As Previously Let Us Denote By T and Y The Sets In Which Thmentioning

confidence: 90%

Section: Introductionmentioning

confidence: 99%

On interval predictor-corrector methods

2016

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“…Let the assumptions about F(T,Y) and Ψ(T,Y) be the same as in [1] and [2], Moreover let us assume that for i = 1, 2,..., k -1 are and the intervals Y i such as known. 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]).…”

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

confidence: 99%

“…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.…”

mentioning

confidence: 99%

“…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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confidence: 99%

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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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