Abstract
Differential-algebraic equations (DAEs) arise in many ways in many types of problems. In this expository paper we discuss a variety of situations where we have found mixed symbolic-numerical calculations to be essential. The paper is designed to both familiarize the reader with several fundamental DAE ideas and to present some applications. The situations discussed include the analysis of DAEs, the solution of DAEs, and applications which include DAEs. Both successes and challenges will be presented.
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K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations (SIAM, Philadelphia, PA, 1996).
S.L. Campbell, Numerical methods for unstructured higher index DAEs, Ann. Numer. Math. 1 (1994) 265-277.
S.L. Campbell, Linearization of DAEs along trajectories, Z. Angew. Math. Phys. (ZAMP) 46 (1995) 70-84.
S.L. Campbell and C.W. Gear, The index of general nonlinear DAEs, Numer. Math. 72 (1995) 173-196.
S.L. Campbell and E. Griepentrog, Solvability of general differential algebraic equations, SIAM J. Sci. Statist. Comput. 16 (1995) 257-270.
S.L. Campbell and R. Hollenbeck, Automatic differentiation and implicit differential equations, in: Computational Differentiation: Techniques, Applications, and Tools, eds. M. Berz, C. Bischof, G. Corliss and A. Griewank (SIAM, Philadelphia, PA, 1996) pp. 215-227.
S.L. Campbell, C.T. Kelley and K.D. Yeomans, Consistent initial conditions for unstructured higher index DAEs: A computational study, in: Proc. Computational Engineering in Systems Applications, Lille, France (1996) pp. 416-421.
S.L. Campbell and W. Marszalek, Mixed symbolic-numerical computations with general DAEs II: An applications case study, Numer. Algorithms (1998), this volume.
S.L. Campbell and E. Moore, Progress on a general numerical method for nonlinear higher index DAEs II, Circuits Systems Signal Process. 13 (1994) 123-138.
S.L. Campbell and E. Moore, Constraint preserving integrators for general nonlinear higher index DAEs, Numer. Math. 69 (1995) 383-399.
S.L. Campbell, E. Moore and Y. Zhong, Utilization of automatic differentiation in control algorithms, IEEE Trans. Automat. Control 39 (1994) 1047-1052.
S.L. Campbell and Y. Zhong, Jacobian reuse in explicit integrators for higher index DAEs, Appl. Numer. Math. 25 (1997) 391-412.
F. Delebecque and R. Nikoukhah, A mixed symbolic-numeric software environment and its application to control system engineering, in: Recent Advances in Computer Aided Control, eds. H. Hergert and M. Jamshidi (1992) pp. 221-245.
I. Duff and C.W. Gear, Computing the structural index, SIAM. J. Algebra Discrete Methods 7 (1986) 594-603.
A. Griewank, D. Juedes and J. Srinivasan, ADOL-C: A package for the automatic differentiation of algorithms written in C/C++, Preprint MCS-P180-1190, Mathematics and Computer Science Division, Argonne National Laboratory (1991).
E. Hairer, C. Lubich and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods (Springer, New York, 1989).
P. Kunkel and V. Mehrmann, Regular solutions of nonlinear differential-algebraic equations and their numerical determination, Preprint (1996).
P. Kunkel and V. Mehrmann, Local and global invariants of linear differential-algebraic equations and their relation, Electron. Trans. Numer. Anal. 4 (1996) 138-157 (electronic).
P. Kunkel and V. Mehrmann, A new class of discretization methods for the solution of linear differential-algebraic equations with variable coefficients, SIAM J. Numer. Anal. 33 (1996) 1941-1961.
P. Kunkel, V. Mehrmann, W. Rath and J. Weickert, A new software package for linear differential-algebraic equations, SIAM J. Sci. Comput. 18 (1997) 115-138.
R. März, Practical Lyapunov stability criteria for differential-algebraic equations, Preprint Nr. 91-28, Humboldt-Universität zu Berlin, Fachbereich Mathematik, Banach Center Publications (1994).
S. Mattsson and G. Söderlind, Index reduction in differential-algebraic equations using dummy derivatives, SIAM J. Sci. Statist. Comput. 14 (1993) 677-692.
C.C. Pantelides, The consistent initialization of differential-algebraic systems, SIAM J. Sci. Statist. Comput. 9 (1988) 213-231.
P.J. Rabier and W.C. Rheinboldt, A general existence and uniqueness theorem for implicit differential algebraic equations, Differential Integral Equations 4 (1991) 563-582.
P.J. Rabier and W.C. Rheinboldt, A geometric treatment of implicit differential-algebraic equations, J. Differential Equations 109 (1994) 110-146.
P.J. Rabier and W.C. Rheinboldt, Time-dependent linear DAEs with discontinuous inputs, Linear Algebra Appl. 247 (1996) 1-29.
P.J. Rabier and W.C. Rheinboldt, Discontinuous solutions of semilinear differential-algebraic equations. I. Distribution solutions, Nonlinear Anal. 27 (1996) 1241-1256.
P.J. Rabier and W.C. Rheinboldt, Discontinuous solutions of semilinear differential-algebraic equations. II. P-consistency, Nonlinear Anal. 27 (1996) 1257-1280.
P.J. Rabier and W.C. Rheinboldt, Classical and generalized solutions of time-dependent linear differential-algebraic equations, Linear Algebra Appl. 245 (1996) 259-293.
S. Reich, On the local qualitative behavior of differential-algebraic equations, Circuits Systems Signal Process. 14 (1995) 427-443.
C. Tischendorf, On the stability of solutions of autonomous index-1 tractable and quasi-linear index-2 tractable DAEs, Circuits Systems Signal Process. 13 (1994) 139-154.
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Campbell, S., Hollenbeck, R., Yeomans, K. et al. Mixed symbolic–numerical computations with general DAEs I: System properties. Numerical Algorithms 19, 73–83 (1998). https://doi.org/10.1023/A:1019154423096
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DOI: https://doi.org/10.1023/A:1019154423096