ON REPRESENTATIONS OF COEFFICIENTS IN IMPLICIT INTERVAL METHODS OF RUNGE-KUTTA TYPE
Marciniak A. 1,2, Szyszka Barbara 3
1 Poznań University of Technology, Institute of Computing Science
Piotrowo 3a, 60-965 Poznań, Poland
2Adam Mickiewicz University, Faculty of Mathematics and Computer Science
Umultowska 87, 61-614Poznań, Poland
3 Poznań University of Technology, Institute of Mathematics
Piotrowo 3a, 60-965 Poznań, Poland,
Received:
Rec. 12 November 2003
DOI: 10.12921/cmst.2004.10.01.57-71
OAI: oai:lib.psnc.pl:560
Abstract:
The paper presents one-, two- and three-stage implicit interval methods of Runge-Kutta type for solving the initial value problem. In our previous papers [1] and [2] it was shown that the exact solution belongs to the interval-solution obtained by both kinds of these methods. We continue the problem on the minimization of the widths of interval-solutions.
References:
[1] A. Marciniak and B. Szyszka, One- and Two-Stage Implicit Interval Methods of Runge-Kutta
Type, Computational Methods in Science and Technology 5, OWN, Poznań 1999.
[2] A. Marciniak, K. Gajda, and B. Szyszka, Three- and Four-Stage Implicit Interval Methodss of
Runge-Kutta Type, Computational Methods in Science and Technology 6, OWN, Poznań 2000.
[3] S. A. Kalmykov, Yu. I. Šokin, and Z. H. Yuldašew, Methodss of Interval Analysis [in Russian],
Nauka, Novosibirsk 1986.
[4] A. Marciniak., K. Gajda, A. Marlewski, and B. Szyszka, The Concept of Object-Oriented System for Solving the Initial Value Problem by Interval Methods of Runge-Kutta Type [in Polish], ProDialog 8, Nakom, Poznań 1999, 39-62.
[5] A. Marciniak, K. Gajda, and B. Szyszka, Interval Methods of Runge-Kutta Type in Floating-
Point Interval Arithmetics [in Polish], Technical Report RB-028/2000, Poznań University of
Technology, Institute of Computing Science, Poznań 2000.
[6] A. Krupowicz, Numerical Methods of Initial Value Problems of Ordinary Differential Equations [in Polish], PWN, Warszawa 1986.
[7] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and
General Linear Methods, J. Wiley&Sons, Chichester 1987.
[8] Yu. I. Šokin, Interval Analysis, [in Russian], Nauka, Novosibirsk 1981.
[9] J. Kudrewicz., Analizafunkcjonalna dla automatyków i elektroników, Państwowe Wydawnictwo Naukowe, Warszawa 1976.
[10] A. Marciniak, Finding the Integration Interval for Interval Methods of Runge-Kutta Type in
Floating-PointInterval Arithmetics, Pro Dialog 10, Nakom, Poznań 2000, 35-45.
[11] BorlandDelphi 6. Object Pascal. Language Guide, Borland, Scotts Valley 2001.
[12] A. Marciniak and A. Marlewski, Interval Representation of Non-Machine Numbers in Object
Pascal [in Polish], Pro Dialog 7, Nakom, Poznań 1998, 75-100.
[13] A. Marciniak and B. Szyszka, Interval Metods of Runge-Kutta Type in Floating-Point Interval
Arithmetics [in Polish], Technical Report RB-022/2001, Poznań University of Technology, Institute of Computing Science, Poznań 2001.
The paper presents one-, two- and three-stage implicit interval methods of Runge-Kutta type for solving the initial value problem. In our previous papers [1] and [2] it was shown that the exact solution belongs to the interval-solution obtained by both kinds of these methods. We continue the problem on the minimization of the widths of interval-solutions.
[1] A. Marciniak and B. Szyszka, One- and Two-Stage Implicit Interval Methods of Runge-Kutta
Type, Computational Methods in Science and Technology 5, OWN, Poznań 1999.
[2] A. Marciniak, K. Gajda, and B. Szyszka, Three- and Four-Stage Implicit Interval Methodss of
Runge-Kutta Type, Computational Methods in Science and Technology 6, OWN, Poznań 2000.
[3] S. A. Kalmykov, Yu. I. Šokin, and Z. H. Yuldašew, Methodss of Interval Analysis [in Russian],
Nauka, Novosibirsk 1986.
[4] A. Marciniak., K. Gajda, A. Marlewski, and B. Szyszka, The Concept of Object-Oriented System for Solving the Initial Value Problem by Interval Methods of Runge-Kutta Type [in Polish], ProDialog 8, Nakom, Poznań 1999, 39-62.
[5] A. Marciniak, K. Gajda, and B. Szyszka, Interval Methods of Runge-Kutta Type in Floating-
Point Interval Arithmetics [in Polish], Technical Report RB-028/2000, Poznań University of
Technology, Institute of Computing Science, Poznań 2000.
[6] A. Krupowicz, Numerical Methods of Initial Value Problems of Ordinary Differential Equations [in Polish], PWN, Warszawa 1986.
[7] J. C. Butcher, The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and
General Linear Methods, J. Wiley&Sons, Chichester 1987.
[8] Yu. I. Šokin, Interval Analysis, [in Russian], Nauka, Novosibirsk 1981.
[9] J. Kudrewicz., Analizafunkcjonalna dla automatyków i elektroników, Państwowe Wydawnictwo Naukowe, Warszawa 1976.
[10] A. Marciniak, Finding the Integration Interval for Interval Methods of Runge-Kutta Type in
Floating-PointInterval Arithmetics, Pro Dialog 10, Nakom, Poznań 2000, 35-45.
[11] BorlandDelphi 6. Object Pascal. Language Guide, Borland, Scotts Valley 2001.
[12] A. Marciniak and A. Marlewski, Interval Representation of Non-Machine Numbers in Object
Pascal [in Polish], Pro Dialog 7, Nakom, Poznań 1998, 75-100.
[13] A. Marciniak and B. Szyszka, Interval Metods of Runge-Kutta Type in Floating-Point Interval
Arithmetics [in Polish], Technical Report RB-022/2001, Poznań University of Technology, Institute of Computing Science, Poznań 2001.
