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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

Bifurcation in difference approximations to two-point boundary value problems


Author: Richard Weiss
Journal: Math. Comp. 29 (1975), 746-760
MSC: Primary 65L10
MathSciNet review: 0383763
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Abstract: Numerical methods for bifurcation problems of the form

$\displaystyle Ly = \lambda f(y),\quad By = 0,$ ($ \ast$)

where $ f(0) = 0$ and $ f'(0) \ne 0$, are considered. Here y is a scalar function, $ \lambda $ is a real scalar, L is a linear differential operator and $ By = 0$ represents some linear homogeneous two-point boundary conditions. Under certain assumptions, it is shown that if $ (\ast)$ 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 $ (\ast)$ 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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1975-0383763-7
PII: S 0025-5718(1975)0383763-7
Keywords: Ordinary differential equations, boundary value problems, bifurcation, difference methods
Article copyright: © Copyright 1975 American Mathematical Society