One-step methods for ordinary differential equations with parameters.(English)Zbl 0701.65053

The paper is concerned with the problem of numerical determination of the parameters of nonlinear ordinary differential equations satisfying two- point boundary value conditions. Iterative methods based on the use of one-step integration procedures are proposed for two classes of problems. Sufficient conditions for the convergence of these methods are proved and error estimates are provided. Numerical examples illustrate the proposed methods.
Reviewer: A.Varga

MSC:

 65L10 Numerical solution of boundary value problems involving ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations 65L70 Error bounds for numerical methods for ordinary differential equations
Full Text:

References:

 [1] I. Babuška M. Práger E. Vitásek: Numerical processes in differential equations. Praha 1966. · Zbl 0156.16003 [2] R. Conti: Problemes lineaires pour les équations differentialles ordinaires. Math. Nachr. 23 (1961), 161-178. · Zbl 0107.28803 [3] J. W. Daniel R. E. Moore: Computation and theory in ordinary differential equations. San Francisco 1970. · Zbl 0207.08802 [4] A. Gasparini A. Mangini: Sul calcolo numerico delle soluzioni di un noto problema ai limiti per l’equazione $$y' = \lambda f(x,y)$$. Le Matematiche 22 (1965), 101-121. · Zbl 0137.33203 [5] P. Henrici: Discrete variable methods in ordinary differential equations. John Wiley, New York 1962. · Zbl 0112.34901 [6] T. Jankowski M. Kwapisz: On the existence and uniqueness of solutions of boundary-value problem for differential equations with parameter. Math. Nachr. 71 (1976), 237-247. · Zbl 0384.34046 [7] H. B. Keller: Numerical methods for two-point boundary-value problems. Blaisdell, London 1968. · Zbl 0172.19503 [8] A. V. Kibenko A. I. Perov: A two-point boundary value problem with parameter. (Russian), Azerbaidzan. Gos. Univ. Učen. Zap. Ser. Fiz.-Mat. i Him. Nauka 3 (1961), 21 - 30. · Zbl 0166.41101 [9] J. Lambert: Computational methods in ordinary differential equations. London 1973. · Zbl 0258.65069 [10] A. Pasquali: Un procedimento di calcolo connesso ad un noto problema ai limiti per l’equazione $$x'=f(t,x,\lambda)$$. Le Matematiche 23 (1968), 319-328. · Zbl 0182.22003 [11] Z. B. Seidov: A multipoint boundary value problem with a parameter for systems of differential equations in Banach space. (Russian). Sibirski Math. Z. 9 (1968), 223 - 228. [12] J. Stoer R. Bulirsch: Introduction to numerical analysis. New York, Heidelberg, Berlin 1980. · Zbl 0423.65002 [13] H. J. Stetter: Analysis of discretization methods for ordinary differential equations. New York, Heidelberg, Berlin 1973. · Zbl 0276.65001 [14] K. Zawischa: Über die Differentialgleichung $$y' = kf(x,y)$$ deren Lösungskurve durch zwei gegebene Punkte hindurchgehen soll. Monatsh. Math. Phys. 37 (1930), 103-124. · JFM 56.1048.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.