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Mathematics of Computation

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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: Numerical methods for bifurcation problems of the form \begin{equation}\tag {$\ast $} Ly = \lambda f(y),\quad By = 0,\end{equation} 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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Keywords: Ordinary differential equations, boundary value problems, bifurcation, difference methods
Article copyright: © Copyright 1975 American Mathematical Society