On the computation of solutions of boundary value problems on infinite intervals
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- by R. M. M. Mattheij PDF
- Math. Comp. 48 (1987), 533-549 Request permission
Abstract:
For solutions of linear boundary value problems defined on $[0,\infty )$ one has to study the stable or bounded solution manifold. A characterization of these manifolds is investigated here. A multiple shooting type algorithm is then developed to compute such solutions. This algorithm is fully adaptive and also covers problems where the ODE matrix does not tend to a limit (as is usually assumed), if the unstable manifold consists only of exponentially growing solutions. If the latter manifold also contains polynomially growing solutions, an extrapolation type approach is suggested. The theory is illustrated by a number of examples.References
- Alvin Bayliss, A double shooting scheme for certain unstable and singular boundary value problems, Math. Comp. 32 (1978), no. 141, 61–71. MR 464598, DOI 10.1090/S0025-5718-1978-0464598-6
- W. A. Coppel, Dichotomies in stability theory, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR 0481196 R. England, A Program for the Solution of Boundary Value Problems for Systems of Ordinary Differential Equations, Culham Laboratory Report, PDN 3/73, 1976.
- F. R. de Hoog and R. M. M. Mattheij, On dichotomy and well conditioning in BVP, SIAM J. Numer. Anal. 24 (1987), no. 1, 89–105. MR 874737, DOI 10.1137/0724008
- F. R. de Hoog and R. Weiss, An approximation theory for boundary value problems on infinite intervals, Computing 24 (1980), no. 2-3, 227–239 (English, with German summary). MR 620090, DOI 10.1007/BF02281727
- Frank R. de Hoog and Richard Weiss, On the boundary value problem for systems of ordinary differential equations with a singularity of the second kind, SIAM J. Math. Anal. 11 (1980), no. 1, 41–60. MR 556495, DOI 10.1137/0511003
- W. F. Langford, A shooting algorithm for the best least squares solution of two-point boundary value problems, SIAM J. Numer. Anal. 14 (1977), no. 3, 527–542. MR 436605, DOI 10.1137/0714032
- Marianela Lentini and Herbert B. Keller, Boundary value problems on semi-infinite intervals and their numerical solution, SIAM J. Numer. Anal. 17 (1980), no. 4, 577–604. MR 584732, DOI 10.1137/0717049
- Peter A. Markowich, A theory for the approximation of solutions of boundary value problems on infinite intervals, SIAM J. Math. Anal. 13 (1982), no. 3, 484–513. MR 653468, DOI 10.1137/0513033
- José Luis Massera and Juan Jorge Schäffer, Linear differential equations and function spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966. MR 0212324
- R. M. M. Mattheij, On approximating smooth solutions of linear singularly perturbed ODE, Numerical analysis of singular perturbation problems (Proc. Conf., Math. Inst., Catholic Univ., Nijmegen, 1978) Academic Press, London-New York, 1979, pp. 457–465. MR 556536
- R. M. M. Mattheij, Characterizations of dominant and dominated solution of linear recursions, Numer. Math. 35 (1980), no. 4, 421–442. MR 593837, DOI 10.1007/BF01399009
- R. M. M. Mattheij, Estimates for the errors in the solutions of linear boundary value problems, due to perturbations, Computing 27 (1981), no. 4, 299–318 (English, with German summary). MR 643401, DOI 10.1007/BF02277181
- R. M. M. Mattheij, Stable computation of solutions of unstable linear initial value recursions, BIT 22 (1982), no. 1, 79–93. MR 654744, DOI 10.1007/BF01934397
- R. M. M. Mattheij, The conditioning of linear boundary value problems, SIAM J. Numer. Anal. 19 (1982), no. 5, 963–978. MR 672571, DOI 10.1137/0719070
- R. M. M. Mattheij, Decoupling and stability of algorithms for boundary value problems, SIAM Rev. 27 (1985), no. 1, 1–44. MR 791753, DOI 10.1137/1027001 R. M. M. Mattheij & F. R. de Hoog, "On non-invertible boundary value problems," Proceedings of the Workshop on Numerical Boundary Value ODE’s (U. Ascher & R. D. Russell, eds.), Birkhäuser, Boston, 1985, pp. 55-75.
- R. M. M. Mattheij and G. W. M. Staarink, On optimal shooting intervals, Math. Comp. 42 (1984), no. 165, 25–40. MR 725983, DOI 10.1090/S0025-5718-1984-0725983-0
- R. M. M. Mattheij and G. W. M. Staarink, An efficient algorithm for solving general linear two-point BVP, SIAM J. Sci. Statist. Comput. 5 (1984), no. 4, 745–763. MR 765204, DOI 10.1137/0905053
- T. N. Robertson, The linear two-point boundary-value problem on an infinite interval, Math. Comp. 25 (1971), 475–481. MR 303742, DOI 10.1090/S0025-5718-1971-0303742-1
- Gustaf Söderlind and Robert M. M. Mattheij, Stability and asymptotic estimates in nonautonomous linear differential systems, SIAM J. Math. Anal. 16 (1985), no. 1, 69–92. MR 772869, DOI 10.1137/0516005
- G. W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev. 19 (1977), no. 4, 634–662. MR 461871, DOI 10.1137/1019104
Additional Information
- © Copyright 1987 American Mathematical Society
- Journal: Math. Comp. 48 (1987), 533-549
- MSC: Primary 65L10
- DOI: https://doi.org/10.1090/S0025-5718-1987-0878689-4
- MathSciNet review: 878689