A Posteriori Error Bounds for Two-Point Boundary Value Problems

Abstract

Consider a general two-point boundary value problem (TPBVP): \[ \begin{gathered} y'(t) = f(t,y), \hfill \\ B_1 y(a) + B_2 y(b) = w,\quad \hfill \\ \end{gathered} a \leqq t \leqq b, \] where $f:R^{n + 1} \to R^n ,f \in C^2 ,B_1 $ and $B_2 $ are $n \times n$ matrices and $w \in R^n $. It is shown how one can bound a posteriori the error made in the numerical solution of the TPBVP. The error bounds obtained are rigorous and include the truncation and the roundoff error. In addition, the computations establish the existence of solutions to the TPBVP. Numerical schemes are developed for the case where $f(t,y)$ is a polynomial in t and y. Examples are given of computational existence proofs for problems where analytical existence proofs are not known.