A difference method for a singular boundary value problem of second order

Weinmüller, Ewa

doi:10.1090/S0025-5718-1984-0736446-0

A difference method for a singular boundary value problem of second order

Author: Ewa Weinmüller
Journal: Math. Comp. 42 (1984), 441-464
MSC: Primary 65L10; Secondary 39A12
DOI: https://doi.org/10.1090/S0025-5718-1984-0736446-0
MathSciNet review: 736446
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The standard three-point discretization applied to the numerical solution of linear boundary value problems for second order systems with a singularity at the origin is investigated. A number of numerical examples illustrating the theoretical results are presented.

[1] D. C. Brabston and H. B. Keller, A numerical method for singular two point boundary value problems, SIAM J. Numer. Anal. 14 (1977), no. 5, 779–791. MR 483475, https://doi.org/10.1137/0714054
[2] Frank R. de Hoog and Richard Weiss, Difference methods for boundary value problems with a singularity of the first kind, SIAM J. Numer. Anal. 13 (1976), no. 5, 775–813. MR 440931, https://doi.org/10.1137/0713063
[3] 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, https://doi.org/10.1137/0511003
[4] N. Dunford & J. T. Schwartz, Linear Operators, Part I, Interscience, New York, 1967.
[5] Pierre Jamet, On the convergence of finite-difference approximations to one-dimensional singular boundary-value problems, Numer. Math. 14 (1969/70), 355–378. MR 261799, https://doi.org/10.1007/BF02165591
[6] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
[7] Herbert B. Keller and Antoinette W. Wolfe, On the nonunique equilibrium states and buckling mechanism of spherical shells, J. Soc. Indust. Appl. Math. 13 (1965), 674–705. MR 183174
[8] H. B. Keller, Approximation methods for nonlinear problems with application to two-point boundary value problems, Math. Comp. 29 (1975), 464–474. MR 371058, https://doi.org/10.1090/S0025-5718-1975-0371058-7
[9] Yudell L. Luke, Mathematical functions and their approximations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0501762
[10] Frank Natterer, A generalized spline method for singular boundary value problems of ordinary differential equations, Linear Algebra Appl. 7 (1973), 189–216. MR 334530, https://doi.org/10.1016/0024-3795(73)90040-2
[11] Frank Natterer, Das Differenzenverfahren für singuläre Rand-Eigenwertaufgaben gewöhnlicher Differentialgleichungen, Numer. Math. 23 (1975), 387–409 (German, with English summary). MR 416042, https://doi.org/10.1007/BF01437038
[12] S. V. Parter, M. L. Stein & P. R. Stein, On the Multiplicity of Solutions of a Differential Equation Arising in Chemical Reactor Theory, Tech. Rep. 194, Dept. of Computer Sciences, Univ. of Wisconsin-Madison, 1973.
[13] Seymour V. Parter, A posteriori error estimates, Numerical solutions of boundary value problems for ordinary differential equations (Proc. Sympos., Univ. Maryland, Baltimore, Md., 1974), Academic Press, New York, 1975, pp. 277–292. MR 0405870
[14] P. Rentrop, Eine Taylorreihenmethode zur numerischen Lösung von Zwei-Punkt Randwertproblemen mit Anwendung auf singuläre Probleme der nichtlinearen Schalentheorie, TUM, Institut für Mathematik, München, 1977.
[15] R. D. Russell and L. F. Shampine, Numerical methods for singular boundary value problems, SIAM J. Numer. Anal. 12 (1975), 13–36. MR 400723, https://doi.org/10.1137/0712002
[16] E. Weinmüller, "On the boundary value problem for systems of ordinary differential equations with a singularity of the first kind," SIAM J. Math. Anal., v. 15, 1984. (To appear.)