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Galerkin solutions of stiff two-point boundary-value problems

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Abstract

Through the example of Conte (Ref. 6), the Galerkin procedure with a small number (N⩽6) of low-degree polynomial modes is illustrated as a computationally rapid and effective technique for solving extremely stiff linear two-point boundary-value problems. Numerical solutions are provided for eigenvalue spreads σ ranging from 20 through 106. They agree with the exact solution to at least 2N decimal places. The errors are insensitive to the eigenvalue spread. Comparisons are made with the continuation technique of Roberts and Shipman (Ref. 1), who did not succeed in solving this example for σ=√(36,000).

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References

  1. Roberts, S. M., andShipman, J. S.,Multipoint Solution of Two-Point Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 7, No. 4, 1971.

  2. Roberts, S. M., andShipman, J. S.,Continuation in Shooting Methods for Two-Point Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 18, No. 1, 1968.

  3. Roberts, S. M., andShipman, J. S.,Justification for the Continuation Method in Two-Point Boundary-Value Problems, Journal of Mathematical Analysis and Applications, Vol. 21, No. 1, 1968.

  4. Conte, S. D.,The Numerical Solution of Linear Boundary-Value Problems, SIAM Review, Vol. 8, No. 3, 1966.

  5. Kantorovich, L. V., andKrylov, V. I.,Approximate Methods of Higher Analysis, Interscience Publishers, New York, 1964.

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  6. Wilkinson, J. H.,The Algebraic Eigenvalue Problem, Oxford University Press, Oxford, England, 1965.

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Authors and Affiliations

  1. Carnegie-Mellon University, Pittsburgh, Pennsylvania

    C. P. Neuman (Associate Professor of Electrical Engineering)

  2. Department of Electrical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania

    F. G. Casasayas (Graduate Students)

Authors
  1. C. P. Neuman
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  2. F. G. Casasayas
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Additional information

Communicated by A. Miele

This work was supported in part by the National Science Foundation under Grant No. GJ-1075.

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Neuman, C.P., Casasayas, F.G. Galerkin solutions of stiff two-point boundary-value problems. J Optim Theory Appl 11, 203–212 (1973). https://doi.org/10.1007/BF00935884

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  • DOI: https://doi.org/10.1007/BF00935884

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