On some questions arising in the approximate solution of nonlinear differential equations
Authors:
R. Bellman and J. M. Richardson
Journal:
Quart. Appl. Math. 20 (1963), 333-339
MSC:
Primary 65.61
DOI:
https://doi.org/10.1090/qam/144472
MathSciNet review:
144472
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Abstract: A new approach to the approximate solution of nonlinear differential equations is explored. The basic idea is to rewrite the nonlinear equations in the form of an infinite sequence of coupled linear equations by application of the Carleman linearization process. The sequence is truncated at a finite stage by a linear closure approximation involving the minimization of the mean square error. Attention is given to the stability of the truncated sequence of linear equations with respect to propagation of error due to closure back to the earlier members of the sequence. The use of suitably defined orthogonal polynomials to simplify closure approximations is considered. The generalization of the general method to the multidimensional case is treated. Consideration is given to the concept of self-consistent closure methods in which the averaging of the squared closure error depends upon the approximate linear equations derived thereby. A specific example of the last is treated analytically in closed form and a numerical comparison is made with the exact solution.
- Robert Kalaba, On nonlinear differential equations, the maximum operation, and monotone convergence, J. Math. Mech. 8 (1959), 519–574. MR 0107731, DOI https://doi.org/10.1512/iumj.1959.8.58037
- Richard Bellman and John M. Richardson, Renormalization techniques and mean square averaging. I. Deterministic equations, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1191–1194. MR 126589, DOI https://doi.org/10.1073/pnas.47.8.1191
J. M. Richardson and R. Bellman, Perturbation techniques, Symposium on Nonlinear Oscillations, Kiev, U. S. S. R., to appear
T. Carleman, Ark. Mat. Astron. Fys. 22B, 1 (1932)
- R. Bellman, Teoriya ustoĭčivosti rešeniĭ differencial′nyh uraveniĭ, Izdat. Inostrannoĭ Lit., Moscow, 1954 (Russian). Translated by A. D. Myškis. MR 0075365
- Richard Bellman, Introduction to matrix analysis, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1960. MR 0122820
R. Kalaba, J. Math, and Mech. 8, 519 (1959)
R. Bellman and J. M. Richardson, Proc. Natl. Acad. Sci. U. S. 47, 1191 (1961)
J. M. Richardson and R. Bellman, Perturbation techniques, Symposium on Nonlinear Oscillations, Kiev, U. S. S. R., to appear
T. Carleman, Ark. Mat. Astron. Fys. 22B, 1 (1932)
R. Bellman, Stability Theory of Differential Equations (McGraw-Hill Book Company, Inc., New York, 1954)
R. Bellman, Introduction of Matrix Analysis (McGraw-Hill Book Company, Inc., New York, 1960)
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Article copyright:
© Copyright 1963
American Mathematical Society