# Ordinary Differential Equations

• J. Stoer
• R. Bulirsch
Part of the Texts in Applied Mathematics book series (TAM, volume 12)

## Abstract

Many problems in applied mathematics lead to ordinary differential equations. In the simplest case one seeks a differentiable function y = y(x) of one real variable x, whose derivative y’(x) is to satisfy an equation of the form y’(x) = f (x, y(x)), or more briefly,
$$y' = f\left( {x,y} \right);$$
(7.0.1)
one then speaks of an ordinary differential equation. In general there are infinitely many different functions y which are solutions of (7.0.1). Through additional requirements one can single out certain solutions from the set of all solutions. Thus, in an initial-value problem, one seeks a solution y of (7.0.1) which for given xo, y o satisfies an initial condition of the form
$$y\left( {{x_o}} \right) = {y_o}$$
(7.0.2)

## Keywords

Ordinary Differential Equation Extrapolation Method Multistep Method Midpoint Rule Linear Multistep Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References for Chapter 7

1. Babuska, I., Prager, M., Vitâsek, E.: Numerical Processes in Differential Equations. New York: Interscience (1966).
2. Bader, G., Deuflhard, P.: A semi-implicit midpoint rule for stiff systems of ordinary systems of differential equations. Numer. Math. 41, 373–398 (1983).
3. Bank, R. E., Bulirsch, R., Merten, K.: Mathematical modelling and simulation of electrical circuits and semiconductor devices. ISNM 93, Basel: Birkhäuser (1990).Google Scholar
4. Boyd, J. P.: Chebyshev and Fourier Spectral Methods. Berlin, Heidelberg, New York: Springer (1989).Google Scholar
5. Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen and Aufgaben der optimalen Steuerung. Report of the CarlCranz-Gesellschaft (1971).Google Scholar
6. Bulirsch, R., Stoer, J.: Numerical treatment of ordinary differential equations by extrapola tion methods. Numer. Math. 8, 1–13 (1966).
7. Butcher, J. C.: On Runge-Kutta processes of high order. J. Austral. Math. Soc. 4, 179–194 (1964).
8. Byrne, D. G., Hindmarsh, A. C.: Stiff o.d.e.-solvers: A review of current and coming attractions. J. Comp. Phys. 70, 1–62 (1987).
9. Ciarlet, P. G., Lions, J. L. (Eds.): Handbook of Numerical Analysis. Vol. II. Finite Element Methods (Part 1). Amsterdam: North Holland (1991).Google Scholar
10. Ciarlet, P. G., Schultz, M. H., Varga, R. S.: Numerical methods of high order accuracy for nonlinear boundary value problems. Numer. Math. 9, 394–430 (1967).
11. Ciarlet, P. G., Wagschal, C.: Multipoint Taylor formulas and applications to the finite element method. Numer. Math. 17, 84–100 (1971).
12. Canuto, C., Hussaini, M. Y., Quarteroni, A., Zang, T. A.: Spectral Methods in Fluid Dynamics. Berlin, Heidelberg, New York: Springer (1987).Google Scholar
13. Coddington, E. A., Levinson, N.: Theory of Ordinary Differential Equations. New York: McGraw-Hill (1955).
14. Collatz, L.: The Numerical Treatment of differential Equations. Berlin: Springer (1960).
15. Collatz, L.: Functional Analysis and Numerical Mathematics. New York: Academic Press (1966).Google Scholar
16. Dahlquist, G.: Convergence and stability in the numerical integration of ordinary differential equations. Math. Scand. 4, 33–53 (1956).
17. Dahlquist, G.: Stability and error bounds in the numerical integration of ordinary differen tial equations. Trans. Roy. Inst. Tech. (Stockholm), No. 130 (1959).Google Scholar
18. Dahlquist, G.: A special stability problem for linear multistep methods. BIT 3, 27–43 (1963).
19. Deuflhard, P.: A stepsize control for continuation methods and its special application to multiple shooting techniques. Numer. Math. 33, 115–146 (1979).
20. Deuflhard, P., Hairer, E., Zugck, J.: One-step and extrapolation methods for differential algebraic systems. Numer. Math. 51, 501–516 (1987).
21. Diekhoff, H.-J., Lory, P., Oberle, H. J., Pesch, H.-J., Rentrop, P., Seydel, R.: Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting. Numer. Math. 27, 449–469 (1977).
22. Dormand, J. R., Prince, P. J.: A family of embedded Runge-Kutta formulae. J. Comp. Appl. Math. 6, 19–26 (1980).
23. Enright, W. H., Hull, T. E., Lindberg, B.: Comparing numerical methods for stiff systems of ordinary differential equations. BIT 55, 10–48 (1975).
24. Fehlberg, E.: 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 (1964).
25. Fehlberg, E.: New high—order Runge—Kutta formulas with an arbitrary small truncation error. Z. Angew. Math. Mech. 46, 1–16 (1966).
26. Fehlberg, E.: Klassische Runge—Kutta Formeln fünfter and siebenter Ordnung mit Schrittweiten—Kontrolle. Computing 4, 93–106 (1969).
27. Gantmacher, R. R.: Matrizenrechnung II. Berlin: VEB Deutscher Verlag der Wissenschaften (1969).Google Scholar
28. Gear, C. W.: Numerical Initial Value Problems in Ordinary Differential Equations. Englewood Cliffs, N.J.: Prentice-Hall (1971).
29. Gear, C. W.: Differential algebraic equation index transformations. SIAM J. Sci. Statist. Comput. 9, 39–47 (1988).
30. Gear, C. W., Petzold, L. R.: ODE methods for the solution of differential/algebraic systems. SIAM J. Numer. Anal. 21, 716–728 (1984).
31. Gottlieb, D., Orszag, S. A.: Numerical Analysis of Spectral Methods: Theory and Applications. Philadelphia: SIAM (1977).Google Scholar
32. Gragg, W.: Repeated extrapolation to the limit in the numerical solution of ordinary differential equations. Thesis, UCLA (1963).Google Scholar
33. Gragg, W.: On extrapolation algorithms for ordinary initial value problems. J. SIAM Numer. Anal. Ser. B 2, 384–403 (1965).
34. Griepentrog, E., März, R.: Differential-Algebraic Equations and Their Numerical Treatment. Leipzig: Teubner (1986).
35. Grigorieff, R. D.: Numerik gewöhnlicher Differentialgleichungen 1, 2. Stuttgart: Teubner ( 1972, 1977 ).Google Scholar
36. Hairer, E., Lubich, Ch.: Asymptotic expansions of the global error of fixed step-size methods. Numer. Math. 45, 345–360 (1984).
37. Hairer, E., Lubich, Ch., Roche, M.: The numerical solution of differential-algebraic systemsGoogle Scholar
38. by Runge—Kutta methods. In: Lecture Notes in Mathematics 1409. Berlin, Heidelberg, New York: Springer (1989).Google Scholar
39. by Runge—Kutta methods, Nersett, S. P., Wanner, G.: Solving Ordinary Differential Equations I. Nonstiff Problems. Berlin, Heidelberg, New York: Springer (1987).Google Scholar
40. by Runge—Kutta methods, Wanner, G.: Solving Ordinary Differential Equations II. Stiff and Differential Algebraic Problems. Berlin, Heidelberg, New York: Springer (1991).Google Scholar
41. Henrici, P.: Discrete Variable Methods in Ordinary Differential Equations. New York: John Wiley (1962).
42. Hestenes, M. R.: Calculus of Variations and Optimal Control Theory. New York: John Wiley (1966).
43. Heun, K.: Neue Methode zur Approximativen Integration der Differentialgleichungen einer unabhängigen Variablen. Z. Math. Phys. 45, 23–38 (1900).
44. Horneber, E. H.: Simulation elektrischer Schaltungen auf dem Rechner. Fachberichte Simulation, Bd. 5. Berlin, Heidelberg, New York: Springer (1985).Google Scholar
45. Hull, E. E., Enright, W. H.,Fellen, B. M., Sedgwick, A. E.: Comparing numerical methods for ordinary differential equations. SIAM J. Numer. Anal. 9 603–637 (1972). [Errata, ibid. 11 681 (1974).]Google Scholar
46. Isaacson, E., Keller, H. B.: Analysis of Numerical Methods. New York: John Wiley (1966).
47. Kaps, P., Rentrop, P.: Generalized Runge—Kutta methods of order four with stepsize control for stiff ordinary differential equations. Numer. Math. 33, 55–68 (1979).
48. Keller, H. B.: Numerical Methods for Two-Point Boundary-Value Problems. London: Blaisdell (1968).
49. Krogh, F. T.: 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 (1974).Google Scholar
50. Kutta, W.: Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435–453 (1901).
51. Lambert, J. D.: Computational Methods in Ordinary Differential Equations. London, New York, Sydney, Toronto: John Wiley (1973).Google Scholar
52. Na, T. Y., Tang, S. C.: A method for the solution of conduction heat transfer with non-linear heat generation. Z. Angew. Math. Mech. 49, 45–52 (1969).
53. Oberle, H. J., Grimm, W.: 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 (1989).Google Scholar
54. Oden, J. T., Reddy, J. N.: An Introduction to the Mathematical Theory of Finite Elements. New York: John Wiley (1976).
55. Osborne, M. R.: On shooting methods for boundary value problems. J. Math. Anal. Appl. 27, 417–433 (1969).
56. Petzold, L. R.: A description of DASSL-A differential algebraic solver. IM ACS Trans. Sci. Comp. 1, 65ff., Ed. H. Stepleman. Amsterdam: North Holland (1982).Google Scholar
57. Rentrop, P.: Partitioned Runge–Kutta methods with stiffness detection and step-size control. Numer. Math. 47, 545–564 (1985).
58. Rentrop, P., Roche, M., Steinebach, G.: The application of Rosenbrock-Wanner type methods with stepsize control in differential-algebraic equations. Numer. Math. 55, 545–563 (1989).
59. Runge, C.: Über die numerische Auflösung von Differentialgleichungen. Math. Ann. 46, 167–178 (1895).
60. Schwarz, H. R.: Finite Element Methods. London, New York: Academic Press (1988).Google Scholar
61. Shampine, L. F., Gordon, M. K.: Computer Solution of Ordinary Differential Equations. The Initial Value Problem. San Francisco: Freeman and Company (1975).
62. Watts, H. A., Davenport, S. M.: Solving nonstiff ordinary differential equa tions—The state of the art. SIAM Review 18, 376–411 (1976).
63. Shanks, E. B.: Solution of differential equations by evaluation of functions. Math. Comp. 20, 21–38 (1966).
64. Stetter, H. J.: Analysis of Discretization Methods for Ordinary Differential Equations. Berlin, Heidelberg, New York: Springer (1973).Google Scholar
65. Strang, G., Fix, G. J.: An Analysis of the Finite Element Method. Englewood Cliffs, N.J.: Prentice-Hall (1973).
66. Troesch, B. A.: Intrinsic difficulties in the numerical solution of a boundary value problem. Report NN-142, TRW, Inc. Redondo Beach, CA (1960).Google Scholar
67. Troesch, B. A.: A simple approach to a sensitive two-point boundary value problem. J.Computational Phys. 21, 279–290 (1976).
68. Wilkinson, J. H.: Note on the practical significance of the Drazin inverse. In: L. S. Campbell, Ed. Recent Applications of Generalized Inverses. Pitman Publ. 66, 82–89 (1982).Google Scholar
69. Willoughby, R. A.: Stiff Differential Systems. New York: Plenum Press (1974). Zlâmal, M.: On the finite element method. Numer. Math. 12, 394–409 (1968).