Bifurcation in difference approximations to two-point boundary value problems

Author:
Richard Weiss

Journal:
Math. Comp. **29** (1975), 746-760

MSC:
Primary 65L10

DOI:
https://doi.org/10.1090/S0025-5718-1975-0383763-7

MathSciNet review:
0383763

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Abstract | References | Similar Articles | Additional Information

Abstract: Numerical methods for bifurcation problems of the form

() |

where and , are considered. Here

*y*is a scalar function, is a real scalar,

*L*is a linear differential operator and represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if is replaced by an appropriate difference scheme, then there exists a unique branch of nontrivial solutions of the discrete problem in a neighborhood of a branch of nontrivial solutions of bifurcating from the trivial solution and that the discrete branch converges to the continuous one. Error estimates are derived and an illustrative numerical example is included.

**[1]**K. E. ATKINSON, "The numerical solution of some bifurcation problems" (submitted).**[2]**M. G. CRANDALL & P. H. RABINOWITZ, "Bifurcation from simple eigenvalues,"*J. Functional Analysis*, v. 8, 1971, pp. 321-340. MR**44**#5836. MR**0288640 (44:5836)****[3]**N. DUNFORD & J. T. SCHWARTZ,*Linear Operators*. Vol. 1:*General Theory*, Pure and Appl. Math., vol. 7, Interscience, New York, 1958. MR**22**#8302.**[4]**H. B. KELLER, "Approximation methods for nonlinear problems" (submitted).**[5]**H. B. KELLER & W. F. LANGFORD, "Iterations, perturbations and multiplicities for nonlinear bifurcation problems,"*Arch. Rational Mech. Anal.*, v. 48, 1972, pp. 83-108. MR**0336478 (49:1252)****[6]**H.-O. KREISS, "Difference approximations for ordinary differential equations,"*Math. Comp.*, v. 26, 1972, pp. 605-631. MR**0373296 (51:9496)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1975-0383763-7

Keywords:
Ordinary differential equations,
boundary value problems,
bifurcation,
difference methods

Article copyright:
© Copyright 1975
American Mathematical Society