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A comparison of several invariant imbedding algorithms for the solution of two-point boundary-value problems. (English) Zbl 0335.65031

65L10Boundary value problems for ODE (numerical methods)
34B05Linear boundary value problems for ODE
Full Text: DOI
[1] Stokes, G. G.: Mathematical and physical papers. 2 (1880) · Zbl 0171.24801
[2] Bellman, R. E.; Kalaba, R. E.: On the principle of invariant imbedding and propagation through inhomogeneous media. Proc. natl. Acad. sci. USA 42, 629-632 (1956) · Zbl 0071.41605
[3] Scott, M. R.: A bibliography on invariant imbedding and related topics. Sla-74-0284 (1974)
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[25] Matti, H.; Chow, C. K.; Stock, F. T.: Solution of ill-conditioned linear two-point boundary-value problems by the Riccati transformation. SIAM rev. 11, 616 (1969) · Zbl 0195.17202
[26] Tapley, B. D.; Williamson, W. E.: Comparison of linear and Riccati equations used to solve optimal control problems. Aiaa j. 10, 1154-1159 (1972) · Zbl 0244.49010
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[30] Casti, J. L.; Kalaba, R. E.; Scott, M. R.: A proposal for the calculation of characteristic functions for certain differential and integral operators via initial value procedures. J. math. Anal. appl. 41, 1-13 (1973) · Zbl 0251.65056
[31] Denman, E. D.; Jr., P. Nelson: Comparison of linear and Riccati equations used to solve optimal control problems. Aiaa j. 12, 575-576 (1974)
[32] M. R. Scott, A discussion of the stability of Riccati equations, to be published.
[33] Jr., P. Nelson; Giles, C. A.: A useful device for certain boundary-value problems. J. comput. Phys. 10, 374-377 (1972) · Zbl 0253.65051
[34] Bellman, R. E.; Kalaba, R. E.; Wing, G. M.: Invariant imbedding and mathematical physics--I: particle processes. J. math. Phys. 1, 280-308 (1960) · Zbl 0105.23202
[35] M. A. Golberg, Some functional relationships for two-point boundary-value problems--II: The inhomogeneous case, Appl. Math. Comput., to be published. · Zbl 0339.34014
[36] M. R. Scott and W. H. Vandevender, A new formulation of the addition formulas of invariant imbedding, to be published.
[37] Hull, T. E.; Emright, W. H.; Fellen, B. M.; Sedgwick, A. E.: Comparing numerical methods for ordinary differential equations. J. numer. Anal. 9, 603-637 (1972) · Zbl 0221.65115
[38] Krogh, F. T.: On testing a subroutine for the numerical integration of ordinary differential equations. J. a. C. m. 20, 545-562 (1973) · Zbl 0292.65039
[39] S. Davenport, L. F. Shampine, and H. A. Watts, Comparison of some codes for the initial value problem for ordinary differential equations, SIAM Review (to appear). · Zbl 0308.65042
[40] Partlett, B.; Wang, Y.: Can you write a decent Fortran subroutine without knowing the computer and the compiler which will process it. SIAM 1973 national meeting (18--21 June 1973)
[41] H. A. Watts and L. F. Shampine, rkf--a Runge--Kutta--Fehlberg fourth order integrator, Sandia Labs., Albuquerque, New Mexico.
[42] Shampine, L. F.; Gordon, M. K.: Computer solution of ordinary differential equations: the initial value problem. (1974) · Zbl 0347.65001
[43] M. R. Scott and H. A. Watts, stiff--a variable-order, variable step integrator for stiff equations, Sandia Lab. Albuquerque, N. M.
[44] Bailey, C. B.; Jones, R. E.: Brief instructions for using the sandia numerical mathematical subroutine library on the CDC 6600. (1974)
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[46] M. Lentini and V. Pereyra, A variable order, variable step, finite difference method for nonlinear multipoint boundary value problems, Math. Comput., to be published. · Zbl 0308.65054
[47] Greenspan, D.: Approximate solution of initial-value problems for ordinary differential equations by boundary-value techniques. J. math. Phys. sci. 1, 261-274 (1967) · Zbl 0208.41402
[48] Holt, J. F.: Numerical solution of nonlinear two-point boundary-value problems by finite difference methods. Commun. A. C. M. 7, 363-373 (1964) · Zbl 0123.11805
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[55] Keller, H. B.: Numerical methods for two-point boundary-value problems. (1968) · Zbl 0172.19503
[56] Roberts, S. M.; Shipman, J. S.: Two-point boundary-value problems: shooting methods. (1972) · Zbl 0239.65061
[57] Miele, A.; Iyer, R. R.: General technique for solving nonlinear, two-point boundary-value problems via the method of particular solutions. J. opt. Theory appl. 5, 382-399 (1970) · Zbl 0184.19905
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[59] Huddleston, R. E.: CDC 6600 routines for the interpolation of data and of data with derivatives. (1974)
[60] Jr., R. C. Allen; Scott, M. R.; Wing, G. M.: Solution of a certain class of nonlinear two-point boundary-value problems. J. comput. Phys. 4, 250-257 (1969) · Zbl 0202.16001
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[62] Wasserstorm, E.: Numerical solutions by the continuation method. SIAM rev. 15, 89-119 (1973)
[63] Roberts, S. M.; Shipman, J. S.: Solution of troesch’s two-point boundary value problem by a combination of techniques. J. comput. Phys. 10, 232-241 (1972) · Zbl 0247.65052
[64] Miele, A.; Aggarwal, A. K.; Tietze, J. L.: Solution of a two-point boundary-value problem with Jacobian matrix characterized by extremely large eigenvalues. J. comput. Phys. 15 (1974) · Zbl 0303.65075
[65] Jones, D. J.: Solution of troesch’s and other, two-point boundary-value problems by shooting techniques. J. comput. Phys. 12, 429-434 (1973) · Zbl 0264.65046
[66] M. R. Scott and H. A. Watts, A test example for nonlinear boundary-value codes, to be published.
[67] Jr., P. Nelson: A comparative study of invariant imbedding and superposition. Int. J. Comput. math. 3, 195-207 (1972) · Zbl 0263.65084
[68] J. B. Rivard and M. R. Scott, The two-phase, three-zone melting problem-- response to pulse heating, to be published. · Zbl 0353.76071
[69] Casti, J. L.: Reduction of dimensionality for systems of linear two-point boundary-value problems with constant coefficients. J. math. Anal. appl. 45, 522-531 (1974) · Zbl 0291.34012