Bifurcation in difference approximations to two-point boundary value problems

Abstract

Numerical methods for bifurcation problems of the form ( ∗ ) L y = λ f ( y ) , B y = 0 , \begin{equation}\tag {$\ast $} Ly = \lambda f(y),\quad By = 0,\end{equation} where f ( 0 ) = 0 f(0) = 0 and f ′ ( 0 ) ≠ 0 f’(0) \ne 0 , are considered. Here y is a scalar function, λ \lambda is a real scalar, L is a linear differential operator and B y = 0 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.