ONE- AND TWO-STAGE IMPLICIT INTERVAL METHODS OF RUNGE-KUTTA TYPE
Marciniak A. 1, Szyszka Barbara 2
1 Institute of Computing Science, 2Institute of Mathematics
Poznań Univeristy of Technology
Piotrowo 3a, 60-965 Poznań, Poland
DOI: 10.12921/cmst.1999.05.01.53-65
OAI: oai:lib.psnc.pl:500
Abstract:
The paper presents one- and two-stage implicit interval methods of Runge-Kutta type. It is shown that the exact solution of the initial value problem belongs to interval-solutions obtained by both kinds of these methods. Moreover, some approximations of the widths of interval-solutions are given.
References:
[ 1 ] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, J. Wiley & Sons, Chichester 1987.
[2] E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg 1987.
[3] S. A. Kalmykov, Ju. I. Šokin, Z. H. Juldašev, Methods of Interval Analysis [in Russian], Nauka,
Novosibirsk 1986.
[4] A. Krupowicz, Numerical Methods of Initial Value Problems of Ordinary Differential Equations [in Polish], PWN, Warsaw 1986.
[5] A. Marciniak, K. Gajda, A. Marlewski, B. Szyszka, The Concept of an Object-Oriented System for Solving the Initial Value Problem by Interval Methods of Runge-Kutta Type [in Polish], Pro Dialog 8 (1999), 39-82.
[6] A. Marciniak, A. Marlewski, Interval Representations of Non-Machine Numbers in Object Pascal [in Polish], Pro Dialog 7 (1998), 75-100.
[6] A. Marciniak, Interval Methods of Runge-Kutta Type in Floating-Point Interval Arithmetics [in Polish], Technical Report RB-027/99, Poznań University of Technology, Institute of Computing Science, Poznań 1999.
[7] Ju. I. Šokin, Interval Analysis [in Russian], Nauka, Novosibirsk 1981.
The paper presents one- and two-stage implicit interval methods of Runge-Kutta type. It is shown that the exact solution of the initial value problem belongs to interval-solutions obtained by both kinds of these methods. Moreover, some approximations of the widths of interval-solutions are given.
[ 1 ] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, J. Wiley & Sons, Chichester 1987.
[2] E. Hairer, S. P. Nørsett, G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg 1987.
[3] S. A. Kalmykov, Ju. I. Šokin, Z. H. Juldašev, Methods of Interval Analysis [in Russian], Nauka,
Novosibirsk 1986.
[4] A. Krupowicz, Numerical Methods of Initial Value Problems of Ordinary Differential Equations [in Polish], PWN, Warsaw 1986.
[5] A. Marciniak, K. Gajda, A. Marlewski, B. Szyszka, The Concept of an Object-Oriented System for Solving the Initial Value Problem by Interval Methods of Runge-Kutta Type [in Polish], Pro Dialog 8 (1999), 39-82.
[6] A. Marciniak, A. Marlewski, Interval Representations of Non-Machine Numbers in Object Pascal [in Polish], Pro Dialog 7 (1998), 75-100.
[6] A. Marciniak, Interval Methods of Runge-Kutta Type in Floating-Point Interval Arithmetics [in Polish], Technical Report RB-027/99, Poznań University of Technology, Institute of Computing Science, Poznań 1999.
[7] Ju. I. Šokin, Interval Analysis [in Russian], Nauka, Novosibirsk 1981.
