IMPLICIT INTERVAL MULTISTEP METHODS FOR SOLVING THE INITIAL VALUE PROBLEM
Jankowska Małgorzata 1, Marciniak A. 1,2
1Poznań University of Technology, Institute of Computing Science
Piotrowo 3a, 60-965 Poznań, Poland
2Adam Mickiewicz University, Faculty of Mathematics and Computer Science
Umultowska 87, 61-614 Poznań, Poland
Received:
Received 28 March, 2002
DOI: 10.12921/cmst.2002.08.01.17-30
OAI: oai:lib.psnc.pl:526
Abstract:
Implicit interval methods of Adams-Moulton type for solving the initial value problem are proposed. It is proved that the exact solution of the problem belongs to interval-solutions obtained by the considered methods. Furthermore, the widths of interval-solutions are estimated.
References:
[1] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, J. Wiley & Sons, Chichester 1987.
[2] Dormand, J. R., Numerical Methods for Differential Equations. A Computational Approach, CRC Press, Boca Raton 1996.
[3] Gajda, K., Marciniak, A., Szyszka, B., Three- and Four-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 6 (2000), 41-59.
[4] Jaulin, L., Kieffer, M., Didrit, O., Walter, É., Applied Interval Analysis, Springer-Verlag, London 2001.
[5] Hairer, E., Nørsett, S. P., Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg 1987.
[6] Kalmykov, S. A., Šokin, Ju. I., JuldaSev, Z. H., Methods of Interval Analysis [in Russian], Nauka, Novosibirsk 1986.
[7] Krupowicz, A., Numerical Methods of Initial Value Problems of Ordinary Differential Equations [in Polish], PWN, Warsaw 1986.
[8] Krückeberg, F., Ordinary Differential Equations, in: Topics in Interval Analysis, Hansen, E. (Ed.), Clarendron Press, Oxford 1969.
[9] Kudrewicz, J., Functional Analysis for Automatization and Electronics Specialists [in Polish], PWN, Warsaw 1976.
[10] Marciniak, A., Szyszka, B., One- and Two-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 5 (1999), 53-65.
[11] Marciniak, A., 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.
[12] Matwiejew, N. M., Methods for Integrating Ordinary Differential Equations [in Polish], PWN, Warsaw 1982.
[13] Moore, R. E., Interval Analysis, Prentice-Hall, Englewood Cliffs 1966.
[14] Rolewicz, S., Functional Analysis and Control Theory [in Polish], PWN, Warsaw 1974.
[15] Stetter, H. J., Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, Berlin 1973.
[16] Šokin, Ju. I., Interval Analysis [in Russian], Nauka, Novosibirsk 1981.
Implicit interval methods of Adams-Moulton type for solving the initial value problem are proposed. It is proved that the exact solution of the problem belongs to interval-solutions obtained by the considered methods. Furthermore, the widths of interval-solutions are estimated.
[1] Butcher, J. C., The Numerical Analysis of Ordinary Differential Equations. Runge-Kutta and General Linear Methods, J. Wiley & Sons, Chichester 1987.
[2] Dormand, J. R., Numerical Methods for Differential Equations. A Computational Approach, CRC Press, Boca Raton 1996.
[3] Gajda, K., Marciniak, A., Szyszka, B., Three- and Four-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 6 (2000), 41-59.
[4] Jaulin, L., Kieffer, M., Didrit, O., Walter, É., Applied Interval Analysis, Springer-Verlag, London 2001.
[5] Hairer, E., Nørsett, S. P., Wanner, G., Solving Ordinary Differential Equations I. Nonstiff Problems, Springer-Verlag, Berlin, Heidelberg 1987.
[6] Kalmykov, S. A., Šokin, Ju. I., JuldaSev, Z. H., Methods of Interval Analysis [in Russian], Nauka, Novosibirsk 1986.
[7] Krupowicz, A., Numerical Methods of Initial Value Problems of Ordinary Differential Equations [in Polish], PWN, Warsaw 1986.
[8] Krückeberg, F., Ordinary Differential Equations, in: Topics in Interval Analysis, Hansen, E. (Ed.), Clarendron Press, Oxford 1969.
[9] Kudrewicz, J., Functional Analysis for Automatization and Electronics Specialists [in Polish], PWN, Warsaw 1976.
[10] Marciniak, A., Szyszka, B., One- and Two-Stage Implicit Interval Methods of Runge-Kutta Type, Computational Methods in Science and Technology 5 (1999), 53-65.
[11] Marciniak, A., 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.
[12] Matwiejew, N. M., Methods for Integrating Ordinary Differential Equations [in Polish], PWN, Warsaw 1982.
[13] Moore, R. E., Interval Analysis, Prentice-Hall, Englewood Cliffs 1966.
[14] Rolewicz, S., Functional Analysis and Control Theory [in Polish], PWN, Warsaw 1974.
[15] Stetter, H. J., Analysis of Discretization Methods for Ordinary Differential Equations, Springer-Verlag, Berlin 1973.
[16] Šokin, Ju. I., Interval Analysis [in Russian], Nauka, Novosibirsk 1981.
