Abstract
An iterative scheme, in which two-point boundary-value problems (TPBVP) are solved as multipoint boundary-value problems (MPBVP), which are independent TPBVPs in each iteration and on each subdomain, is derived for second-order ordinary differential equations. Several equations are solved for illustration. In particular, the algorithm is described in detail for the first boundary-value problem (FBVP) and second boundary-value problem (SBVP). A possible extension to higher-order BVPs is discussed briefly. The procedure may be used when the original TPBVP cannot be solved (does not converge) in a single long domain. It is suitable for implementation on computers with parallel processing. However, that issue is beyond the scope of this paper. The long domain is cut into a large number of subdomains and, based on assumed boundary conditions at the interface points, the resulting local BVPs are solved by any convenient conventional method. The local solutions are then patched by using simple matching formulas, which are derived below, rather than solving large systems of algebraic equations, as it is done in similar existing methods. Assuming that the local solutions are obtained by the most efficient methods, the overall convergence speed depends on the speed of matching. The proposed matching algorithm is based on a fixed-point iteration and has only a linear convergence rate. The rate can be made quadratic by applying standard accelerating schemes, which is beyond the scope of this article.
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References
Ascher, U., Mattheij, R., and Russel, R., Numerical Solution of Boundary-Value Problems for Ordinary Differential Equations, Prentice Hall, Englewood Cliffs, New Jersey, 1988.
Keller, H. B., Numerical Methods for Two-Point Boundary-Value Problems, Blaisdell Publishing Company, London, England, 1968.
Roberts, S., and Shipman, J., Two-Point Boundary-Value Problems: Shooting Methods, Elsevier, New York, New York, 1972.
Stoer, J., and Bulirsch, R., Introduction to Numerical Analysis, Springer, New York, New York, 1993.
Miele, A., and Wang, T., Parallel Computation of Two-Point Boundary-Value Problems via Particular Solutions, Journal of Optimization Theory and Applications, Vol. 79, pp. 5–29, 1993.
Anonymous, Overview of Research on the Parallel Computer SNI-KSR at Leibniz-Rechementrum Munchen, LRZ-Bericht 9401, 9.85-94, Munchen, Germany, 1994.
Bulirsch, R., Montrone, F., and Pesch, H. J., Abort Landing in the Presence of Windshear as a Minimax Optimal Control Problem, Part 2: Multiple Shooting and Homotopy, Journal of Optimization Theory and Applications, Vol. 70, pp. 223–254, 1991.
Kugelmann, B., and Pesch, H. J., New General Guidance Method in Constrained Optimal Control, Part 1: Numerical Method, Journal of Optimization Theory and Applications, Vol. 67, pp. 421–435, 1990.
Kiehl, M., A Vector Implementation of an ODE Code for Multipoint Boundary-Value Problems, Parallel Computing, Vol. 17, pp. 347–352, 1991.
Kugelmann, B., Performance of a Feedback Method with Respect to Changes in the Air Density during the Ascent of a Two-Stage-To-Orbit Vehicle, International Series of Numerical Mathematics, Vol. 115, pp. 329–338, 1994.
Kiehl, M., Mehlhorn, R., and Schumann, M., Parallel Multiple Shooting for Optimal Control Problems under NX, Optimization Methods and Software, Vol. 4, pp. 259–271, 1995.
Gasparo, M. G., and Macconi, M., Parallel Initial-Value Algorithms for Singularly Pertubed Boundary-Value Problems, Journal of Optimization Theory and Applications, Vol. 73, pp. 501–517, 1992.
Burden, R., and Faires, J., Numerical Analysis, 3rd Edition, Prindle, Weber, and Schmidt, Boston, Massachusetts, 1985.
Bellman, R., and Kalaba, R., Quasilinearization and Nonlinear Boundary-Value Problems, American Elsevier, New York, New York, 1965.
Bailey, P., Shampine, L., and Waltman, P., Nonlinear Two-Point Boundary-Value Problems, Academic Press, New York, New York, 1968.
Shampine, L. F., Numerical Solution of Ordinary Differential Equations, Chapman and Hall, New York, New York, 1994.
Sewell, G., IMSL Software for Differential Equations in One Space Variable, Technical Report 8202, IMSL, Houston, Texas, 1982.
Pereyra, V., PASVA3: An Adaptive Finite-Difference FORTRAN Program for First-Order Nonlinear Boundary-Value Problems, Lecture Notes in Computer Science, Springer, Berlin, Germany, Vol. 76, pp. 67–88, 1978.
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Pasic, H. Multipoint Boundary-Value Solution of Two-Point Boundary-Value Problems. Journal of Optimization Theory and Applications 100, 397–416 (1999). https://doi.org/10.1023/A:1021742521630
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DOI: https://doi.org/10.1023/A:1021742521630