Differential-algebraic systems as differential equations on manifolds

Author:
Werner C. Rheinboldt

Journal:
Math. Comp. **43** (1984), 473-482

MSC:
Primary 58F99; Secondary 34A15, 65L99

DOI:
https://doi.org/10.1090/S0025-5718-1984-0758195-5

MathSciNet review:
758195

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Based on the theory of differential equations on manifolds, existence and uniqueness results are proved for a class of mixed systems of differential and algebraic equations as they occur in various applications. Both the autonomous and nonautonomous case are considered. Moreover, a class of algebraically incomplete systems is introduced for which existence and uniqueness results only hold on certain lower-dimensional manifolds. This class includes systems for which the application of ODE-solvers is known to lead to difficulties. Finally, some solution approach based on continuation techniques is outlined.

**[1]**V. I. Arnol′d,*Ordinary differential equations*, The M.I.T. Press, Cambridge, Mass.-London, 1973. Translated from the Russian and edited by Richard A. Silverman. MR**0361233****[2]**M. Bier, O. A. Palusinski, R. A. Mosher & D. A. Saville, "Electrophoresis: Mathematical modeling and computer simulation,"*Science*, v. 219, 1983, pp. 1281-1286.**[3]**James P. Fink and Werner C. Rheinboldt,*On the discretization error of parametrized nonlinear equations*, SIAM J. Numer. Anal.**20**(1983), no. 4, 732–746. MR**708454**, https://doi.org/10.1137/0720049**[4]**C. W. Gear, "Simultaneous numerical solution of differential-algebraic equations,"*IEEE Trans. Circuit Theory*, v. CT-18, 1971, pp. 89-95.**[5]**C. W. Gear & L. R. Petzold,*ODE Methods for the Solution of Differential/Algebraic Systems*, Tech. Report 82-1103, Dept. of Comp. Sci., Univ. of Illinois at Urbana-Champaign, 1982.**[6]**Serge Lang,*Introduction to differentiable manifolds*, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR**0155257****[7]**Linda Petzold,*Differential/algebraic equations are not ODEs*, SIAM J. Sci. Statist. Comput.**3**(1982), no. 3, 367–384. MR**667834**, https://doi.org/10.1137/0903023**[8]**Linda R. Petzold,*A description of DASSL: a differential/algebraic system solver*, Scientific computing (Montreal, Que., 1982) IMACS Trans. Sci. Comput., I, IMACS, New Brunswick, NJ, 1983, pp. 65–68. MR**751605****[9]**Werner C. Rheinboldt,*Solution fields of nonlinear equations and continuation methods*, SIAM J. Numer. Anal.**17**(1980), no. 2, 221–237. MR**567270**, https://doi.org/10.1137/0717020**[10]**Werner C. Rheinboldt and John V. Burkardt,*A locally parameterized continuation process*, ACM Trans. Math. Software**9**(1983), no. 2, 215–235. MR**715303**, https://doi.org/10.1145/357456.357460**[11]**G. Söderlind,*DASP3--A Program for the Numerical Integration of Partitioned Stiff ODE's and Differential Algebraic Systems*, Tech. Report TRITA-NA-8008, Dept. of Numer. Anal. and Comp. Science, The Royal Inst. of Technology, Stockholm, Sweden, 1980.

Retrieve articles in *Mathematics of Computation*
with MSC:
58F99,
34A15,
65L99

Retrieve articles in all journals with MSC: 58F99, 34A15, 65L99

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1984-0758195-5

Article copyright:
© Copyright 1984
American Mathematical Society