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A method for integration of unstable systems of ordinary differential equation subject to two-point boundary conditions

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Abstract

Instability problems in systems of differential equations are discussed. A matrix technique is given for producing numerical solutions to a system of ordinary differential equations with boundary conditions specified at each end of the interval when the system contains dominant solutions which give rise to numerical instability in conventional integration methods. A method of “bringing up the initial conditions” is described, whereby the two-point nature of the problem is made use of to stabilize the system. Three numerical examples are included.

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References

  1. Frazer, R. A., Duncan, W. J., and Collar, A. R.,Elementary matrices and some applications to dynamics and differential equations, Cambridge 1938.

  2. Zurmühl, R.,Matrizen und ihre technischen Anwendungen, Berlin 1964.

  3. Falkenberg, J. C.,The General Analysis of Shells of Revolution as applied to Hyperbolic Cooling Towers under the action of Wind Forces and Dead Loads, University of Southampton 1966.

  4. Erugin, N. P.,Linear Systems of Ordinary Differential Equations with Periodic and Quasi-Periodic Coefficients, Academic Press, New York & London 1966.

    Google Scholar 

  5. Fox, L.,Numerical Solution of two-point boundary value problems, Oxford 1957.

  6. Midgley, J. E.,Calculation of subdominant solutions of linear differential equations, J. SIAM Numer. Anal. Vol. 3 No. 1, 1966.

  7. Roberts, S. M., and Shipman, J. S.,Continuation in Shooting methods for twopoint boundary value problems, Journ. Math. Anal. & Appl. 18, 1967.

  8. Goodman, T. R., and Lance, G. N.,Numerical Integration of Two-Point Boundary value problems, Math. Tables and other aids to comput. 10 (1956), p. 82–86.

    Google Scholar 

  9. Bellman, R. E., Kagiwada, H. H. and Kalaba, R. E.,Invariant embedding and the numerical integration of Boundary-Value problems for unstable linear systems of ordinary differential equations, Communications of the ACM February 1967.

  10. Conte, S. D.,The numerical solution of linear boundary value problems, SIAM rev. vol. 8 No. 3, July 1966.

  11. Godunov, S.,On the numerical solution of boundary value problems for systems of ordinary linear differential equations, Uspehi Mat. Nauk. 16 (1961), pp. 171–174.

    Google Scholar 

  12. Pipes, L. A.,A perturbation method for the solution of linear matrix differential equations, J. Franklin Institute, 283,5 May 1967.

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Authors and Affiliations

  1. Norwegian Building Research Institute, Blindern, Norway

    J. C. Falkenberg

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  1. J. C. Falkenberg
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Falkenberg, J.C. A method for integration of unstable systems of ordinary differential equation subject to two-point boundary conditions. BIT 8, 86–103 (1968). https://doi.org/10.1007/BF01939331

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  • DOI: https://doi.org/10.1007/BF01939331

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