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Indices and Solvability for General Systems of Differential Equations

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Computer Algebra in Scientific Computing CASC’99

Abstract

We consider general systems of ordinary and partial differential equations from a geometric point of view. This leads to simple interpretations of various index concepts introduced for differential algebraic equations. Especially, we obtain natural generalisations of these concepts to partial differential equations.

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References

  1. K.E. Brenan, S.L. Campbell, and L.R. Petzold. Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Classics in Applied Mathematics 14. SIAM, Philadelphia, 1996.

    Google Scholar 

  2. R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, and P.A. Griffiths. Exterior Differential Systems. Mathematical Sciences Research Institute Publications 18. Springer-Verlag, New York, 1991.

    Google Scholar 

  3. S.L. Campbell and C.W. Gear. The index of general nonlinear DAEs. Numer. Math., 72:173–196, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  4. S.L. Campbell and E. Griepentrog. Solvability of general differential algebraic equations. SIAM J. Sci. Comp., 16:257–270, 1995.

    MathSciNet  MATH  Google Scholar 

  5. S.L. Campbell and W. Marszalek. The index of an infinite dimensional implicit system. Math. Model. Syst., 1:1–25, 1996.

    Google Scholar 

  6. P.A.M. Dirac. Generalized Hamiltonian dynamics. Can. J. Math., 2:129–148, 1950.

    Article  MathSciNet  MATH  Google Scholar 

  7. E. Hairer, C. Lubich, and M. Roche. The Numerical Solution of Differential-Algebraic Equations by Runge-Kutta Methods. Lecture Notes in Mathematics 1409. Springer-Verlag, Berlin, 1989.

    Google Scholar 

  8. E. Hairer and G. Wanner. Solving Ordinary Differential Equations II. Springer Series in Computational Mathematics 14. Springer-Verlag, Berlin, 1996.

    Google Scholar 

  9. M. Janet. Sur les Systèmes d’quations aux Dérivées Partielles. J. Math. Pure Appl., 3:65–151, 1920.

    Google Scholar 

  10. B.N. Jiang, J. Wu, and L.A. Povelli. The origin of spurious solutions in computational electrodynamics. J. Comp. Phys., 125:104–123, 1996.

    Article  MATH  Google Scholar 

  11. F. John. Partial Differential Equations. Applied Mathematical Sciences 1. Springer-Verlag, New York, 1982.

    Google Scholar 

  12. P. Kunkel and V. Mehrmann. Canonical forms for linear differential-algebraic equations with variable coefficients. J. Comp. Appl. Math., 56:225–251, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Kunkel and V. Mehrmann. A new look at pencils of matrix valued functions. Lin. Alg. Appl., 212/213:215–248, 1994.

    Article  MathSciNet  Google Scholar 

  14. G. Le Vey. Some remarks on solvability and various indices for implicit differential equations. Num. Algo., 19:127–145, 1998.

    Article  MATH  Google Scholar 

  15. W. Lucht and K. Strehmel. Discretization based indices for semilinear partial differential algebraic equations. Technical Report 97–40, Fachbereich Mathematik und Informatik, Universität Halle-Wittenberg, 1997.

    Google Scholar 

  16. W. Lucht, K. Strehmel, and C. Eichler-Liebenow. Linear partial differential algebraic equations I: Indexes, consistent boundary/initial conditions. Technical Report 97–17, Fachbereich Mathematik und Informatik, Universität Halle- Wittenberg, 1997.

    Google Scholar 

  17. W. Lucht, K. Strehmel, and C. Eichler-Liebenow. Linear partial differential algebraic equations II: Numerical solution. Technical Report 97–18, Fachbereich Mathematik und Informatik, Universität Halle- Wittenberg, 1997.

    Google Scholar 

  18. J.F. Pommaret. Systems of Partial Differential Equations and Lie Pseudo-groups. Gordon &; Breach, London, 1978.

    Google Scholar 

  19. J.F. Pommaret. Partial Differential Equations and Group Theory. Kluwer, Dordrecht, 1994.

    MATH  Google Scholar 

  20. J.F. Pommaret and A. Haddak. Effective methods for systems of algebraic partial differential equations. In T. Mora and C. Traverso, editors, Proc. MEGA’90, pages 411–426. Birkhäuser, Boston, 1991.

    Google Scholar 

  21. P.J. Rabier and W.C. Rheinboldt. A geometric treatment of implicit differential algebraic equations. J. Diff. Eq., 109:110–146, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  22. S. Reich. On an existence and uniqueness theory for nonlinear differential- algebraic equations. Circ. Sys. Sig. Proc., 10:343–359, 1991.

    Article  MATH  Google Scholar 

  23. M. Renardy and R.C. Rogers. An Introduction to Partial Differential Equations. Texts in Applied Mathematics 13. Springer, New York, 1993.

    Google Scholar 

  24. J. Schü, W.M. Seiler, and J. Calmet. Algorithmic methods for Lie pseudogroups. In N. Ibragimov, M. Torrisi, and A. Valenti, editors, Proc. Modern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics, pages 337–344. Kluwer, Dordrecht, 1993.

    Chapter  Google Scholar 

  25. W.M. Seiler. Analysis and Application of the Formal Theory of Partial Differential Equations. PhD thesis, School of Physics and Materials, Lancaster University, 1994.

    Google Scholar 

  26. W.M. Seiler. On the arbitrariness of the general solution of an involutive partial differential equation. J. Math. Phys., 35:486–498, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  27. W.M. Seiler. Applying AXIOM to partial differential equations. Internal Report 95–17, Universität Karlsruhe, Fakultät für Informatik, 1995.

    Google Scholar 

  28. W.M. Seiler. Generalized tableaux and formally well-posed initial value problems. Preprint Lancaster University, 1995.

    Google Scholar 

  29. W.M. Seiler. Involution and symmetry reductions. Math. Comp. Model., 25:63–73, 1997.

    Article  MathSciNet  MATH  Google Scholar 

  30. W.M. Seiler. Numerical analysis of constrained Hamiltonian systems and the formal theory of differential equations. Math. Comp. Simul., 45:561–576, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  31. W.M. Seiler and R.W. Tucker. Involution and constrained dynamics I: The Dirac approach. J. Phys. A, 28:4431–4451, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  32. J.W. Sweeny. The D-Neumann problem. Acta Math., 120:223–251, 1968.

    Article  MathSciNet  Google Scholar 

  33. A. Szatkowski. Geometric characterization of singular differential algebraic equations. Int. J. Sys. Sci., 23:167–186, 1992.

    Article  MathSciNet  MATH  Google Scholar 

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Seiler, W.M. (1999). Indices and Solvability for General Systems of Differential Equations. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing CASC’99. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60218-4_30

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  • DOI: https://doi.org/10.1007/978-3-642-60218-4_30

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-66047-7

  • Online ISBN: 978-3-642-60218-4

  • eBook Packages: Springer Book Archive

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