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Gewöhnliche Differentialgleichungen

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Numerische Mathematik 2

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Zusammenfassung

Sehr viele Probleme aus den Anwendungsgebieten der Mathematik führen auf gewöhnliche Differentialgleichungen. Im einfachsten Fall ist dabei eine differenzierbare Funktion y = y(x) einer reellen Veränderlichen x gesucht, deren Ableitung y′(x) einer Gleichung der Form y′(x) = f (x, y(x)) oder kürzer

$$y' = f(x,y)$$

genügt; man spricht dann von einer gewöhnlichen Differentialgleichung. Im allg. besitzt (7.0.1) unendlich viele verschiedene Funktionen y als Lösungen. Durch zusätzliche Forderungen kann man einzelne Lösungen auszeichnen. So sucht man bei einem Anfangswertproblem eine Lösung y von (7.0.1), die für gegebenes x 0, y 0 eine Anfangsbedingung der Form

$$y({x_0}) = {y_0}$$

erfüllt.

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Literatur

  • Babuska, I., Prager, M., Vitâsek, E. (1966): Numerical Processes in Differential Equations. New York: Interscience.

    MATH  Google Scholar 

  • Bader, G., Deuflhard, P. (1983): A semi-implicit midpoint rule for stiff systems of ordinary systems of differential equations. Numer. Math. 41, 373–398.

    Article  MathSciNet  MATH  Google Scholar 

  • Bank, R. E., Bulirsch, R., Merten, K. (1990): Mathematical modelling and simulation of electrical circuits and semiconductor devices. ISNM. 93, Basel: Birkhäuser.

    Google Scholar 

  • Bock, H.G., Schlöder, J.P., Schulz, V.H. (1995): Numerik großer Differentiell-Algebraischer Gleichungen: Simulation and Optimierung, p. 35–80 in: Schuler,H. (ed.) Prozeßsimulation, Weinheim: VCH.

    Google Scholar 

  • Boyd, J. P. (1989): Chebyshev and Fourier Spectral Methods. Berlin-HeidelbergNew York: Springer.

    Book  Google Scholar 

  • Buchauer, O., Hiltmann, P, Kiehl, M. (1994): Sensitivity analysis of initial-value problems with application to shooting techniques. Numerische Mathematik 67, 151–159.

    Article  MathSciNet  MATH  Google Scholar 

  • Broyden, C.G. (1967): Quasi-Newton methods and their application to function minimization. Math. Comp. 21, 368–381.

    Article  MathSciNet  MATH  Google Scholar 

  • Bulirsch, R. (1971): Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen and Aufgaben der optimalen Steuerung. Report der CarlCranz-Gesellschaft.

    Google Scholar 

  • Bulirsch, R., Stoer, J. (1966): Numerical treatment of ordinary differential equations by extrapolation methods. Numer. Math. 8, 1–13.

    Article  MathSciNet  MATH  Google Scholar 

  • Butcher, J. C. (1964): On Runge-Kutta processes of high order. J. Auetral. Math. Soc. 4, 179–194.

    Article  MathSciNet  MATH  Google Scholar 

  • Byrne, D. G., Hindmarsh, A. C. (1987): Stiff o.d.e.-solvers: A review of current and coming attractions. J. Comp. Phys. 70, 1–62.

    Article  MathSciNet  MATH  Google Scholar 

  • Caracotsios, M., Stewart, W.E. (1985): Sensitivity analysis of initial value problems with mixed ODEs and algebraic equations. Computers and Chemical Engineering 9, 359–365.

    Article  Google Scholar 

  • Ciarlet, P. G., Lions, J. L., Eds. (1991): Handbook of Numerical Analysis. Vol. IL Finite Element Methods (Part 1). Amsterdam: North Holland.

    Google Scholar 

  • Ciarlet, P. G., Lions, J. L., Eds, Schultz, M. H., Varga, R. S. (1967): Numerical methods of high order accuracy for nonlinear boundary value problems. Numer. Math. 9, 394–430.

    Article  MathSciNet  Google Scholar 

  • Ciarlet, P. G., Lions, J. L., Eds, Wagschal, C. (1971): Multipoint Taylor formulas and applications to the finite element method. Numer. Math. 17, 84–100.

    MATH  Google Scholar 

  • Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A. (1987): Spectral Methods in Fluid Dynamics. Berlin-Heidelberg-New York: Springer.

    Google Scholar 

  • Clark, N. W. (1968): A study of some numerical methods for the integration of systems of first order ordinary differential equations. Report ANL-7428. Argonne National Laboratories.

    Google Scholar 

  • Coddington, E. A., Levinson, N. (1955): Theory of Ordinary Differential Equations. New York: McGraw-Hill.

    MATH  Google Scholar 

  • Collatz, L. (1960): The Numerical Treatment of Differential Equations. BerlinGöttingen-Heidelberg: Springer.

    Book  MATH  Google Scholar 

  • Funktionalanalysis and Numerische Mathematik. Berlin-HeidelbergNew York: Springer.

    Google Scholar 

  • Crane, P.J., Fox, P.A. (1969): A comparative study of computer programs for integrating differential equations. Num. Math. Computer Program library one–Basic routines for general use. Vol. 2, issue 2. New Jersey: Bell Telephone Laboratories Inc.

    Google Scholar 

  • Dahlquist, G. (1956): Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4, 33–53.

    MathSciNet  MATH  Google Scholar 

  • Dahlquist, G. (1959): Stability and error bounds in the numerical integration of ordinary differential equations. Trans. Roy. Inst. Tech. ( Stockholm ), No. 130.

    Google Scholar 

  • Dahlquist, G. (1963): A special stability problem for linear multistep methods. BIT 3, 27–43.

    Google Scholar 

  • Deuflhard, P. (1979): A stepsize control for continuation methods and its special application to multiple shooting techniques. Numer. Math. 33, 115–146.

    Article  MathSciNet  MATH  Google Scholar 

  • Deuflhard, P., Hairer, E., Zugck, J. (1987): One-step and extrapolation methods for differential-algebraic systems. Numer. Math. 51, 501–516.

    Article  MathSciNet  MATH  Google Scholar 

  • Diekhoff, H.-J., Lory, P., Oberle, H. J., Pesch, H.-J., Rentrop, P., Seydel, R. (1977): Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting. Numer. Math. 27, 449–469.

    Google Scholar 

  • Dormand, J. R., Prince, P. J. (1980): A family of embedded Runge-Kutta formulae. J. Comp. Appl. Math. 6, 19–26.

    Article  MathSciNet  MATH  Google Scholar 

  • Eich, E. (1992): Projizierende Mehrschrittverfahren zur numerischen Lösung von Bewegungsgleichungen technischer Mehrkörpersysteme mit Zwangsbedingungen und Unstetigkeiten. Fortschritt-Berichte VDI, Reihe 18, Nr. 109, VDI-Verlag, Düsseldorf.

    Google Scholar 

  • Engl, G., Kröner, A., Kronseder, T., von Stryk, O. (1999): Numerical Simulation and Optimal Control of Air Separation Plants, pp. 221–231 in: Bungartz, H.-J., Durst, F., Zenger, Chr.(eds.): High Performance Scientific and Engineering Computing, Lecture Notes in Computational Science and Engineering Vol. 8, New York: Springer.

    Google Scholar 

  • Enright, W. H., Hull, T. E., Lindberg, B. (1975): Comparing numerical methods for stiff systems of ordinary differential equations.BIT 15, 10–48.

    MATH  Google Scholar 

  • Fehlberg, E. (1964): New high-order Runge-Kutta formulas with stepsize control for systems of first-and second-order differential equations. Z. Angew. Math. Mech. 44, T17 — T29.

    Article  MathSciNet  MATH  Google Scholar 

  • Fehlberg, E. (1966): New high—order Runge—Kutta formulas with an arbitrary small truncation error. Z. Angew. Math. Mech. 46, 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  • Fehlberg, E. (1969): Klassische Runge—Kutta Formeln fiinfter und siebenter Ordnung mit Schrittweiten—Kontrolle. Computing 4, 93–106.

    Article  MathSciNet  MATH  Google Scholar 

  • Fehlberg, E. (1970): Klassische Runge-Kutta Formeln vierter und niedrigerer Ordnung mit Schrittweiten-Kontrolle und ihre Anwendung auf Wärmeleitungsprobleme. Computing 6, 61–71.

    Article  MathSciNet  MATH  Google Scholar 

  • Galin, S., Feehery, W.F., Barton, P.I. (1998): Parametric sensitivity functions for hybrid discrete/continuous systems. Preprint submitted to Applied Numerical Mathematics.

    Google Scholar 

  • Gantmacher, F. R. (1969): Matrizenrechnung II. Berlin: VEB Deutscher Verlag der Wissenschaften.

    Google Scholar 

  • Gear, C. W. (1971): Numerical initial value problems in ordinary differential equations. Englewood Cliffs, N.J.: Prentice-Hall.

    MATH  Google Scholar 

  • Gear, C. W. (1988): Differential algebraic equation index transformations. SIAM J. Sci. Statist. Comput. 9, 39–47.

    Article  MathSciNet  MATH  Google Scholar 

  • Gear, C. W., Petzold, L. R. (1984): ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal. 21, 716–728.

    Article  MathSciNet  MATH  Google Scholar 

  • Gill, P.E., Murray, W., Wright, H.M. (1995): Practical Optimization, 10th printing. London: Academic Press.

    Google Scholar 

  • Gottlieb, D. Orszag, S. A. (1977): Numerical Analysis of Spectral Methods: Theory and Applications Philadelphia: SIAM.

    Google Scholar 

  • Gragg, W. (1963): Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. Thesis, UCLA.

    Google Scholar 

  • Gragg, W. (1965): On extrapolation algorithms for ordinary initial value problems. J. SIAM Numer. Anal. Ser. B 2, 384–403.

    MathSciNet  Google Scholar 

  • Griepentrog, E., März, R. (1986): Differential-Algebraic Equations and Their Numerical Treatment. Leipzig: Teubner.

    MATH  Google Scholar 

  • Grigorieff, R.D. ( 1972, 1977): Numerik gewöhnlicher Differentialgleichungen 1, 2. Stuttgart: Teubner.

    Google Scholar 

  • Hairer, E., Lubich, Ch. (1984): Asymptotic expansions of the global error of fixed stepsize methods. Numer. Math. 45, 345–360.

    Article  MathSciNet  MATH  Google Scholar 

  • Hairer, E., Lubich, Ch, Roche, M. (1989): The numerical solution of differential-algebraic systems by Runge-Kutta methods. In: Lecture Notes in Mathematics 1409. BerlinHeidelberg-New York: Springer.

    Google Scholar 

  • Hairer, E., Lubich, Ch, Narsett, S. P., Wanner, G. (1993): Solving Ordinary Differential Equations I. Nonstiff Problems. 2nd ed., Berlin-Heidelberg-New York: Springer.

    MATH  Google Scholar 

  • Hairer, E., Lubich, Ch, Wanner, G. (1991): Solving Ordinary Differential Equations II. Stiff and Differential-Algebraic Problems. Berlin-Heidelberg-New York: Springer.

    Book  MATH  Google Scholar 

  • Heim, A., von Stryk, O. (1996): Documentation of PAREST — A multiple shooting code for optimization problems in differential-algebraic equations. Technical Report M9616, Fakultät für Mathematik, Technische Universität München.

    Google Scholar 

  • Henrici, P. (1962): Discrete Variable Methods in Ordinary Differential Equations. New York: John Wiley.

    MATH  Google Scholar 

  • Hestenes, M. R. (1966): Calculus of Variations and Optimal Control Theory. New York: John Wiley.

    MATH  Google Scholar 

  • Heun, K. (1900): Neue Methode zur approximativen Integration der Differentialgleichungen einer unabhängigen Variablen. Z. Math. Phys. 45, 23–38.

    MATH  Google Scholar 

  • Horneber, E. H. (1985): Simulation elektrischer Schaltungen auf dem Rechner. Fachberichte Simulation, Bd. 5. Berlin-Heidelberg-New York: Springer.

    Book  Google Scholar 

  • Hull, T. E., Enright, W. H., Fellen, B. M., Sedgwick, A. E. (1972): Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9, 603–637. [Errata, ibid. 11, 681 (1974).]

    Google Scholar 

  • Isaacson, E., Keller, H. B. (1966): Analysis of Numerical Methods. New York: John Wiley.

    MATH  Google Scholar 

  • Kaps, P. Rentrop, P. (1979): Generalized Runge—Kutta methods of order four with stepsize control for stiff ordinary differential equations. Numer. Math. 33 55–68.

    Google Scholar 

  • Keller, H. B. (1968): Numerical Methods for Two-Point Boundary-Value Problems. London: Blaisdell.

    MATH  Google Scholar 

  • Kiehl, M. (1999): Sensitivity analysis of ODEs and DAEs — theory and implementation guide. Optimization Methods and Software 10, 803–821.

    Article  MathSciNet  MATH  Google Scholar 

  • Krogh, F. T. (1974): Changing step size in the integration of differential equations using modified divided differences. In: Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations, 22–71. Lecture Notes in Mathematics 362. Berlin-Heidelberg-New York: Springer.

    Google Scholar 

  • Kutta, W. (1901): Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435–453.

    MATH  Google Scholar 

  • Lambert, J. D. (1973): Computational Methods in Ordinary Differential Equations. London-New York-Sidney-Toronto: John Wiley.

    MATH  Google Scholar 

  • Leis, J.R., Kramer, M.A. (1985): Sensitivity analysis of systems of differential and algebraic equations. Computers and Chemical Engineering 9, 93–96.

    Article  Google Scholar 

  • Morrison, K.R., Sargent, R.W.H. (1986): Optimization of multistage processes described by differential-algebraic equations, pp. 86–102 in: Lecture Notes in Mathematics 1230. New York: Springer.

    Google Scholar 

  • Na, T. Y., Tang, S. C. (1969): A method for the solution of conduction heat transfer with non-linear heat generation. Z. Angew. Math. Mech. 49, 45–52.

    Article  MATH  Google Scholar 

  • Oberle, H. J., Grimm, W. (1989): BNDSCO — A program for the numerical solution of optimal control problems. Internal Report No. 515–89/22, Institute for Flight Systems Dynamics, DLR, Oberpfaffenhofen, Germany.

    Google Scholar 

  • Oden, J. T., Reddy, J. N. (1976): An Introduction to the Mathematical Theory of Finite Elements. New York: John Wiley.

    MATH  Google Scholar 

  • Osborne, M. R. (1969): On shooting methods for boundary value problems. J. Math. Anal. Appl. 27, 417–433.

    Article  MathSciNet  MATH  Google Scholar 

  • Petzold, L. R. (1982): A description of DASSL — A differential algebraic system solver. IMACS Trans. Sci. Comp. 1, 65ff., Ed. H. Stepleman. Amsterdam: North Holland.

    Google Scholar 

  • Quarteroni, A., Valli, A. (1994): Numerical Approximation of Partial Differential Equations. Berlin-Heidelberg-New York: Springer.

    MATH  Google Scholar 

  • Rentrop, P. (1985): Partitioned Runge-Kutta methods with stiffness detection and stepsize control. Numer. Math. 47 545–564.

    Google Scholar 

  • Rentrop, P., Roche, M., Steinebach, G. (1989): The application of Rosenbrock-Wanner type methods with stepsize control in differential-algebraic equations. Numer. Math. 55 545–563.

    Google Scholar 

  • Rozenvasser, E. (1967): General sensitivity equations of discontinuous systems. Automation and Remote Control 28, 400–404

    Google Scholar 

  • Runge, C. (1895): Über die numerische Auflösung von Differentialgleichungen. Math. Ann. 46, 167–178.

    Article  MathSciNet  MATH  Google Scholar 

  • Schwarz, H. R. (1981): FORTRAN-Programme zur Methode der finiten Elemente. Stuttgart: Teubner.

    MATH  Google Scholar 

  • Schwarz, H. R., (1984): Methode der finiten Elemente. Stuttgart: Teubner.

    Book  Google Scholar 

  • Shampine, L. F., Gordon, M. K. (1975): Computer Solution of Ordinary Differential Equations. The Initial Value Problem. San Francisco: Freeman and Company., Watts, H. A., Davenport, S. M. (1976): Solving nonstiff ordinary differential equations — The state of the art. SIAM Review 18, 376–411.

    Google Scholar 

  • Shanks, E. B. (1966): Solution of differential equations by evaluation of functions. Math. Comp. 20, 21–38.

    Article  MathSciNet  MATH  Google Scholar 

  • Stetter, H. J. (1973): Analysis of Discretization Methods for Ordinary Differential Equations. Berlin-Heidelberg-New York: Springer.

    Book  MATH  Google Scholar 

  • Strang, G., Fix, G. J. (1973): An Analysis of the Finite Element Method. Englewood Cliffs, N.J.: Prentice-Hall.

    MATH  Google Scholar 

  • Troesch, B. A. (1960): Intrinsic difficulties in the numerical solution of a boundary value problem. Report NN-142, TRW, Inc. Redondo Beach, CA.

    Google Scholar 

  • Troesch, B. A. (1976): A simple approach to a sensitive two-point boundary value problem. J. Computational Phys. 21, 279–290.

    Article  MathSciNet  MATH  Google Scholar 

  • Velte, W. (1976): Direkte Methoden der Variationsrechnung. Stuttgart: Teubner. Wilkinson, J. H. (1982): Note on the practical significance of the Drazin Inverse. In: L.S. Campbell, Ed. Recent Applications of Generalized Inverses. Pitman Publ. 66, 82–89.

    Google Scholar 

  • Willoughby, R. A. (1974): Stiff Differential Systems. New York: Plenum Press. Zlâmal, M. (1968): On the finite element method. Numer. Math. 12, 394–409.

    Google Scholar 

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Authors and Affiliations

  1. Institut für Angewandte Mathematik und Statistik, Universität Würzburg, Am Hubland, 97074, Würzburg, Deutschland

    Prof. Dr. Josef Stoer

  2. Zentrum Mathematik, Technische Universität München, Arcisstr. 21, 80333, München, Deutschland

    Prof. Dr. Roland Bulirsch

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Stoer, J., Bulirsch, R. (2000). Gewöhnliche Differentialgleichungen. In: Numerische Mathematik 2. Springer-Lehrbuch. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-09025-1_2

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