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The Preliminary Enclosing of the ODE Solutions on the Base of the Cauchy-Duhamel Identity

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Reliable Computing

Abstract

An iterative interval algorithm of a preliminary enclosing of the integral curve in the next step of an integration process is described. It is based on the well-known Cauchy-Duhamel identity. The conditions of the ending of the process are derived. The criterion of existence and uniqueness of the solution is discussed, too.

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References

  1. Bibikov, Yu. N.: General Course of Ordinary Differential Equations: Manual, LGU, Leningrad, 1981, p. 232 (in Russian).

  2. Dahlquist, G. A.: Special Stability Problem for Linear Multistep Methods, BIT 3 (1963), pp. 27-43.

    Google Scholar 

  3. Gavurin, M. K.: From the Experience of Numerical Intergration of Ordinary Differential Equations, Methods of Computing 1, LGU, Leningrad, 1963, pp. 45-51 (in Russian).

    Google Scholar 

  4. Menshikov, G. G.: Interval Analysis and Methods of Computing, Lecture Notes. Issues 1-9, NIIKH SPbGU, St.Petersburg, 1996-1999 (in Russian).

    Google Scholar 

  5. Menshikov, G.G. and Tomashevsky, A.V.:Width Estimate of Preliminary Localization of Integral Curves on Base of Cauchy-Duhamel Identity, in: Control Processes and Stability. Proceedings of XXXI Scientific Conference 1-7 April 2000, SPbGU, St.Petersburg, 2000, pp. 347-350.

  6. Moore, R. E.: Interval Analysis, Prentice-Hall, Englewood Cliffs, 1966.

    Google Scholar 

  7. Rakitskiy, Yu. V., Ustinov, S. M., and Chernorutskiy, I. G.: Numerical Methods of Stiff Systems Solving, Nauka, Moscow, 1979 (in Russian).

    Google Scholar 

  8. Scheu, G. and Adams, E.: Zur numerischen Konstruktion konvergenter Schrankenfolgen f¨ur Systeme nichtlinearer, gewöhnlicher Anfangswertaufgaben, in: Nickel, K. L. E. (ed.): Interval Mathematics, Lecture Notes in Computer Science 29, Springer, Berlin, 1975, pp. 279-287.

    Google Scholar 

  9. Shokin, Yu. I.: Interval Analysis, Nauka, Novosibirsk, 1981 (in Russian).

    Google Scholar 

  10. Lohner, R.: Einschließung der Losung gewohnlicher Anfangs-und Randwertaufgaben und Anwendungen, Dissertation, Karlsruhe Universität, 1988.

  11. Tsaritsyna, I.V.:On Numerical Integration of the Stiff Systems ofOrdinary Differential Equation, Methods of Computation 17, St. Petersburg University, 1995, pp. 195-203.

    Google Scholar 

  12. Tsaritsyna, I. V.: On Absolute Stability of Some Difference Methods for Systems of Ordinary Differential Equation, Methods of Computation 18, St. Petersburg University, 1999, pp. 220-229.

    Google Scholar 

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Authors and Affiliations

  1. Faculty of Applied Mathematics—Control Processes, Saint Petersburg State University, Bibliotechnaya Pl. 2, Stary Petershof, Saint Petersburg, 198904, Russia

    Gregory G. Menshikov

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  1. Gregory G. Menshikov
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Menshikov, G.G. The Preliminary Enclosing of the ODE Solutions on the Base of the Cauchy-Duhamel Identity. Reliable Computing 7, 485–495 (2001). https://doi.org/10.1023/A:1014702819312

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