Abstract
Ordinary differential equations (ODEs), including high-order implicit equations, describe important problems in mechanical and chemical engineering. However, the use of self-validated methods providing rigorous enclosures of the solution has mostly been limited to explicit and weakly nonlinear problems, and no general-purpose algorithm for the validated integration of general ODE initial value problems has been developed. Since most integration techniques for Differential Algebraic Equations (DAEs) are based on transformation to implicit ODEs, the integration of DAE initial value problems has traditionally been restricted to few hand-picked problems from the relatively small class of low-index systems. The recently developed Taylor model method combines high-order differential algebraic descriptions of functional dependencies with intervals for verification. It has proven its power in several applications, including verified integration of ODEs under avoidance of the wrapping effect. Recognizing antiderivation (integration) as a natural operation on Taylor models yields methods that treat DEs within a fully differential algebraic context as implicit equations made of conventional functions and antiderivation. This method has the potential to be applied to high-index DAE problems and allows the computation of guaranteed enclosures of final conditions from large initial regions for large classes of initial value problems. In the framework of this method, a Taylor model represents the highest derivative of the solution function occurring in the DE and all lower derivatives are treated as antiderivatives of this Taylor model. Consequently, one obtains a set of implicit equations involving only the highest derivative. Utilizing methods of verified inversion of functional dependencies described by Taylor models allows the computation of a guaranteed enclosure of the solution in the form of a Taylor model. The performance of the method is illustrated by detailed examples.
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References
U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential–Algebraic Equations (SIAM, Philadelphia, PA, 1998).
M. Berz, Differential algebraic description of beam dynamics to very high orders, Particle Accelerators 24 (1989) 109.
M. Berz, Forward algorithms for high orders and many variables, in: Automatic Differentiation of Algorithms: Theory, Implementation and Application (SIAM, Philadelphia, PA, 1991).
M. Berz, Modern Map Methods in Particle Beam Physics (Academic Press, San Diego, 1999).
M. Berz, Constructive generation and verification of Lyapunov functions around fixed points of nonlinear dynamical systems, Internat. J. Comput. Res. (2001).
M. Berz, C. Bischof, G. Corliss and A. Griewank eds., Computational Differentiation: Techniques, Applications, and Tools (SIAM, Philadelphia, PA, 1996).
M. Berz and J. Hoefkens, Verified high-order inversion of functional dependencies and superconvergent interval Newton methods, Reliable Computing 7(5) (2001) 379–398.
M. Berz and G. Hoffstätter, Computation and application of Taylor polynomials with interval remainder bounds, Reliable Computing 4(1) (1998) 83–97.
M. Berz, G. Hoffstätter, W. Wan, K. Shamseddine and K. Makino, COSY INFINITY and its applications to nonlinear dynamics, in: [6], pp. 363–365.
M. Berz and K. Makino, Verified integration of ODEs and flows with differential algebraic methods on Taylor models, Reliable Computing 4(4) (1998) 361–369.
M. Berz and K. Makino, COSY INFINITY version 8.1 reference manual, Technical Report MSUCL-1195, National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824 (2001).
M. Berz, K. Makino and J. Hoefkens, Verified integration of dynamics in the Solar system, Nonlinear Anal. 47 (2001) 179–190.
K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential–Algebraic Equations (North-Holland, Amsterdam, 1989).
Y.F. Chang and G.F. Corliss, Solving ordinary differential equations using Taylor series, ACM Trans. Math. Software 8 (1982) 114–144.
Y.F. Chang and G.F. Corliss, ATOMFT: Solving ODEs and DAEs using Taylor series, Comput. Math. Appl. 28 (1994) 209–233.
B. Erdélyi, J. Hoefkens and M. Berz, Rigorous lower bounds for the domains of definition of extended generating functions, SIAM J. Appl. Dyn. Systems (2001) submitted.
H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, MA, 1980).
E. Griepentrog and R. März, Differential–Algebraic Equations and Their Numerical Treatment (Teubner, Stuttgart, 1986).
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential–Algebraic Problems (Springer, New York, 1991).
E.R. Hansen, Topics in Interval Analysis (Oxford Univ. Press, London, 1969).
J. Hoefkens, Rigorous numerical analysis with high-order Taylor models, Ph.D. thesis, Michigan State University, East Lansing, MI (2001); also MSUCL-1217.
J. Hoefkens and M. Berz, Verification of invertibility of complicated functions over large domains, Reliable Computing 8(1) (2002) 1–16.
J. Hoefkens, M. Berz and K. Makino, Efficient High-Order Methods for ODEs and DAEs (Springer, New York, 2001) pp. 343–350.
J. Hoefkens, M. Berz and K. Makino, Verified High-Order Integration of DAEs and Higher-Order ODEs (Kluwer Academic, Dordrecht, 2001) pp. 281–292.
R.B. Kearfott, Rigorous Global Search: Continuous Problems (Kluwer, Dordrecht, 1996).
E.R. Kolchin, Differential Algebraic Groups (Academic Press, New York, 1985).
R.J. Lohner, Enclosing the solutions of ordinary initial and boundary value problems, in: Computer Arithmetic: Scientific Computation and Programming Languages, eds. E.W. Kaucher, U.W. Kulisch and C. Ullrich, Wiley–Teubner Series in Computer Science (Teubner, Stuttgart, 1987) pp. 255–286.
R. Lohner, Einschließung der Lösung gewöhnlicher Anfangs-und Randwertaufgaben und Anwendungen, Ph.D. thesis, Universität Karlsruhe (1988).
K. Makino, Rigorous integration of maps and long-term stability, in: 1997 Particle Accelerator Conf., Vol. 2, 1997, pp. 1336–1340.
K. Makino, Rigorous analysis of nonlinear motion in particle accelerators, Ph.D. thesis, Michigan State University, East Lansing, MI, USA (1998); also MSUCL-1093.
K. Makino and M. Berz, Remainder differential algebras and their applications, in: [6], pp. 63–74.
K. Makino and M. Berz, Efficient control of the dependency problem based on Taylor model methods, Reliable Computing 5(1) (1999) 3–12.
K. Makino and M. Berz, Advances in verified integration of ODEs, in: SCAN'2000, 2000.
K. Makino and M. Berz, Global optimzation with Taylor models, Internat. J. Comput. Res. (2001).
K. Makino and M. Berz, Verified integration of transfer maps, Phys. Rev. ST-AB, submitted.
R.E. Moore, Interval Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1996).
R.E. Moore, Methods and Applications of Interval Analysis (SIAM, Philadelphia, PA, 1979).
N.S. Nedialkov, Computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation, Ph.D. thesis, University of Toronto (1999).
N.S. Nedialkov, K.R. Jackson and G.F. Corliss, Validated solutions of initial value problems for ordinary differential equations, Appl. Math. Comput. 105(1) (1999) 21–68.
C.C. Pantelides, The consistent initialization of differential–algebraic systems, SIAM J. Sci. Statist. Comput. 9(2) (1988) 213–231.
J.D. Pryce, A simple structural analysis method for DAEs, BIT 41(2) (2001) 364–394.
J.F. Ritt, Differential Equations from the Algebraic Viewpoint (Amer. Math. Soc., Washington, 1932).
J.F. Ritt, Integration in Finite Terms – Liouville's Theory of Elementary Methods (Columbia Univ. Press, New York, 1948).
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Hoefkens, J., Berz, M. & Makino, K. Computing Validated Solutions of Implicit Differential Equations. Advances in Computational Mathematics 19, 231–253 (2003). https://doi.org/10.1023/A:1022858921155
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DOI: https://doi.org/10.1023/A:1022858921155