Abstract
Methods for the solution of nonlinear boundary-value problems for ordinary differential equations are discussed and classified as either finite-difference methods or initial-value methods. Within this framework, two algorithms, which are generated using the quasilinearization method, are presented and shown to be representative of these two methods. Consequently, both of the most widely used techniques for the solution of these problems can be formulated within the framework of the quasilinearization method. The computational properties of these algorithms are also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Kalaba, R.,On Nonlinear Differential Equations, the Maximum Operation, and Monotone Convergence, Journal of Mathematics and Mechanics, Vol. 8, No. 4, 1959.
Bellman, R., andKalaba, R.,Quasilinearization and Nonlinear Boundary-Value Problems, American Elsevier Publishing Company, New York, 1965.
Baird, C.,A Modified Quasilinearization Technique for the Solution of Boundary-Value Problems for Ordinary Differential Equations, Journal of Optimization Theory and Applications, Vol. 3, No. 4, 1969.
Henrici, P.,Discrete Variable Methods in Ordinary Differential Equations, John Wiley and Sons, New York, 1962.
Holt, J.,Numerical Solution of Nonlinear Two-Point Boundary-Value Problems Using Finite Difference Methods, Communications of the Association for Computing Machinery, Vol. 7, No. 6, 1964.
Wendroff, B.,Theoretical Numerical Analysis, Academic Press, New York, 1966.
Sylvester, R., andMeyer, F.,Two-Point Boundary Problems by Quasilinearization, SIAM Journal on Applied Mathematics, Vol. 13, No. 2, 1965.
Ortega, J., andRheinboldt, W.,On Discretization and Differentiation of Operators with Application to Newton's Method, SIAM Journal on Numerical Analysis, Vol. 3, No. 1, 1966.
Goodman, T., andLance, G.,The Numerical Solution of Two-Point Boundary-Value Problems, Mathematical Tables and Other Aids to Computation, Vol. 10, No. 54, 1956.
Miele, A.,Method of Particular Solutions for Linear, Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 2, No. 4, 1968.
Roberts, S., andShipman, J.,The Kantorovich Theorem and Two-Point Boundary-Value Problems, IBM Journal of Research and Development, Vol. 10, No. 5, 1966.
Kantorovich, L., andAkilov, G.,Functional Analysis and Normed Spaces, The Macmillan Company, New York, 1964.
McGill, R., andKenneth, P.,A Convergence Theorem on the Iterative Solution of Nonlinear Two-Point Boundary-Value Systems, Proceedings of the 14th International Astronautical Congress, Paris, 1963.
Antosiewicz, H. A.,Newton's Method and Boundary-Value Problems, Journal of Computer and System Sciences, Vol. 2, No. 2, 1968.
Bellman, R.,Successive Approximations and Computer Storage Problems in Ordinary Differential Equations, Communications of the Association for Computing Machinery, Vol. 4, No. 3, 1961.
Roberts, S., andShipman, J.,Continuation in the Shooting Methods for Two-Point Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 18, No. 1, 1967.
Roberts, S., Shipman, J., andRoth, C.,Continuation in Quasilinearization, Journal of Optimization Theory and Applications, Vol. 2, No. 3, 1968.
Author information
Authors and Affiliations
Additional information
Communicated by R. Kalaba
Rights and permissions
About this article
Cite this article
Baird, C.A. Quasilinearization and the methods of finite difference and initial values. J Optim Theory Appl 6, 320–330 (1970). https://doi.org/10.1007/BF00925380
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00925380