Advertisement

Results in Mathematics

, Volume 42, Issue 3–4, pp 308–338 | Cite as

The index of linear differential algebraic equations with properly stated leading terms

  • Roswitha März
Article

Abstract

For linear differential-algebraic equations with properly stated leading terms and coefficients that are just continuous an index notion is introduced. The index criteria are given in terms of the original coefficients. The index is shown to be invariant under regular transformations of the unknown function, but also under refactorizations of the leading term. Inherent regular explicit ordinary differential equations are described in detail.

Keywords

Differential-algebraic equations regularity index 

Mathematics Subject Classification

34A09 34A30 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Ab et all]
    A. A. Abramov, K. Balla, V. I. Ul’yanova, L. F. Yukhno: O nelinejnoj samosopryazhennoj spektral’noj zadache dlya nekotorykh diffendzhial’no-algebraicheskikh uravnenij indeksa 1. Zhurnal Vych. Matern. i Matern. Fiz. 7(42) 2002.Google Scholar
  2. [BaMä]
    K. Balla und R. März: A unified approach to linear differential algebraic equations and their adjoint equations. Humboldt-Universität Berlin, Institut für Mathematik, Preprint 2000-18, to appear in Zeitschrift für Analysis und ihre Anwendungen 2, 2002.Google Scholar
  3. [Bo]
    Yu. Boyarintsev: Methods of solving singular systems of ordinary differential equations. John Wiley & Sons, 1992 (Russian original: 1988, Nauka, Siberian Division).Google Scholar
  4. [BrCaPe]
    K.E. Brenan, S.L. Campbell, L.R. Petzold: Numerical solution of initial-value problems in differential-algebraic equations. Elsevier Science Publ. Co, Inc. 1989.Google Scholar
  5. [Ca]
    S.L. Campbell: A general form for solvable linear time varying singular systems of differential equations. SIAM J. Math. Anal. 18(4) 1987, 1101–1115.MathSciNetMATHCrossRefGoogle Scholar
  6. [CaPe]
    S.L. Campbell, R.L. Petzold: Canonical forms and solvable singular systems of differential equations. SIAM J. Alg. Discr. Methods 4, 1983, 517–521.MathSciNetMATHCrossRefGoogle Scholar
  7. [CoCa]
    E. A. Coddington, R. Carlson: Linear ordinary differential equations. SIAM Philadelphia 1997.Google Scholar
  8. [Es et all]
    D. Estévez Schwarz, U. Feldmann, R. März, S. Sturtzel, C. Tischendorf: Finding beneficial DAE structures in circuit simulation. To appear in a special volume at Springer concerning mathematics for solving problems in industry and economy.Google Scholar
  9. [Gaj]
    I.V. Gajshun: Vvedenie v teoriyu linejnykh nestadzhionarnykh sistem. NAN Belarusi, Minsk 1999.Google Scholar
  10. [Ga]
    F.R. Gantmacher: Teoriya matrits. Moskva, Nauka 1966.Google Scholar
  11. [GePe]
    C.W. Gear and L.R. Petzold: ODE methods for the solution of differential-algebraic systems. SIAM J. Numer. Anal. 21(4) 1984, 716–728.MathSciNetMATHCrossRefGoogle Scholar
  12. [GrMä]
    E. Griepentrog, R. März: Basic properties of some differential-algebraic equations. Zeitschrift für Analysis und ihre Anwendungen 8, 1989, 25–40.MATHGoogle Scholar
  13. [GrMä1]
    E. Griepentrog, R. März: Differential-algebraic equations and their numerical tratment. Teubner, Leipzig, 1986.Google Scholar
  14. [Ha]
    B. Hansen: Linear time-varying differential-algebraic equations being tractable with the index k. Humboldt-Universität Berlin, Institut für Mathematik, Preprint 246, 1990.Google Scholar
  15. [HiMäTi]
    I. Higueras and R. März, C. Tischendorf: Numerically well formulated index-1DAEs. Humboldt-Universität Berlin, Institut für Mathematik, Preprint 2001–5.Google Scholar
  16. [KuMe]
    R. Kunkel, V. Mehrmann: Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comput. Appl. Math. 56, 1994, 225–251.MathSciNetMATHCrossRefGoogle Scholar
  17. [La]
    R. Lamour: Index determination for DAEs. Humboldt-Universität Berlin, Institut für Mathematik, Preprint 2001–19, 2001.Google Scholar
  18. [Mä]
    R. März: Differential algebraic systems anew. To appear in Applied Numerical Mathematics.Google Scholar
  19. [Mä1]
    R. März: Some results concerning index-3 differential-algebraic equations. J. Mathem. Analysis and Applications 140(1) 1989, 177–199.MATHCrossRefGoogle Scholar
  20. [Mä2]
    R. März: Numerical methods for differential-algebraic equations. Acta Numerica 1992, 141–198.Google Scholar
  21. [Mä3]
    R. März: Adjoint equations of differential-algebraic systems and optimal control problems. Proc. of the Institute of Mathematics, NAS of Belarus, Minsk, Vol. 7 2001.Google Scholar
  22. [RaRh]
    P.J. Rabier, W.C. Rheinboldt: Classical and generalized solutions of time-dependent linear differential-algebraic equations. Linear Algebra and its Applications 2145, 1996, 259–293.MathSciNetCrossRefGoogle Scholar
  23. [RaRh1]
    P.J. Rabier, W.C. Rheinboldt: Theoretical and Numerical Analysis of Differential-Algebraic Equations. Handbook of Numerical Analysis Vol. VIII, Elsevier Publ. Amsterdam 2002.Google Scholar
  24. [ReMaBa]
    G. Reißig, W.S. Martinson, P.J. Barton: Differential-algebraic equations of index 1 may have an arbitrarily high structural index. SIAM J. Scie. Comp. 21(6)2000, 1987–1990.MATHCrossRefGoogle Scholar
  25. [RiMä]
    R. Riaza, R. März: Singularities of linear time-varying DAEs. Humboldt-Universität Berlin, Institut für Mathematik, Preprint 2001–9, 2001.Google Scholar
  26. [Schu]
    I. Schumilina: Index-3 DAEs with properly stated leading term. Humboldt-Universität Berlin, Institut für Mathematik, Preprint 2001–20, 2001.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2002

Authors and Affiliations

  • Roswitha März
    • 1
  1. 1.Institut für Mathematik Unter den LindenHumboldt-Universität zu BerlinBerlinGermany

Personalised recommendations