On the numerical solution of singular boundary value problems of second order by a difference method

Author:
Ewa Weinmüller

Journal:
Math. Comp. **46** (1986), 93-117

MSC:
Primary 65L10; Secondary 34A50

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815834-X

MathSciNet review:
815834

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The standard three-point discretization applied to the numerical solution of nonlinear boundary value problems for second-order systems with a singularity of the first kind is investigated. The results are extended to the boundary value problems arising in practical problems from mechanics and chemistry. A number of numerical examples illustrating the theoretical results is presented.

**[1]**D. C. Brabston & H. B. Keller, "A numerical method for singular two point boundary value problems,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 779-791. MR**0483475 (58:3476)****[2]**F. R. de Hoog & R. Weiss, "Difference methods for boundary value problems with a singularity of the first kind,"*SIAM J. Numer. Anal.*, v. 13, 1976, pp. 775-813. MR**0440931 (55:13799)****[3]**P. Jamet, "On the convergence of finite difference approximations to one-dimensional singular boundary-value problems,"*Numer. Math.*, v. 14, 1970, pp. 355-378. MR**0261799 (41:6411)****[4]**T. Kato,*Perturbation Theory for Linear Operators*, Springer-Verlag, New York, 1966. MR**0203473 (34:3324)****[5]**H. B. Keller & A. W. Wolfe, "On the nonunique equilibrium states and buckling mechanism of spherical shells,"*J. Soc. Indust. Appl. Math.*, v. 13, 1965, pp. 674-705. MR**0183174 (32:656)****[6]**H. B. Keller, "Approximation methods for nonlinear problems with application to two-point boundary value problems,"*Math. Comp.*, v. 29, 1975, pp. 464-474. MR**0371058 (51:7279)****[7]**Y. L. Luke,*Mathematical Functions and Their Approximations*, Academic Press, New York, 1975. MR**0501762 (58:19039)****[8]**F. Natterer, "A generalized spline method for singular boundary value problems in ordinary differential equations,"*Linear Algebra Appl.*, v. 7, 1973, pp. 189-216. MR**0334530 (48:12849)****[9]**F. Natterer, "Das Differenzenverfahren für singuläre Rand-Eigenwertaufgaben gewöhnlicher Differentialgleichungen,"*Numer. Math.*, v. 23, 1975, pp. 387-409. MR**0416042 (54:4118)****[10]**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.**[11]**S. V. Parter,*A-Posteriori Error Estimates*, Tech. Rep. 214, Dept. of Computer Sciences, Univ. of Wisconsin-Madison, 1974. MR**0405870 (53:9662)****[12]**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.**[13]**R. D. Russell & L. F. Shampine, "Numerical methods for singular boundary value problems,"*SIAM J. Numer. Anal.*, v. 12, 1975, pp. 13-35. MR**0400723 (53:4553)****[14]**E. Weinmüller, "On the boundary value problem for systems of ordinary second order differential equations with a singularity of the first kind,"*SIAM J. Math. Anal.*, v. 15, 1984, pp. 287-307. MR**731868 (85m:34035)****[15]**E. Weinmüller, "A difference method for a singular boundary value problem of second order,"*Math. Comp.*, v. 42, 1984, pp. 441-464.

Retrieve articles in *Mathematics of Computation*
with MSC:
65L10,
34A50

Retrieve articles in all journals with MSC: 65L10, 34A50

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0815834-X

Article copyright:
© Copyright 1986
American Mathematical Society