Numerical Solution of Boundary Value Problems in Differential-Algebraic Systems

Abstract

This paper extends the theory of shooting and finite-difference methods for linear boundary value problems (BVPs) in ordinary differential equations (ODEs) to BVPs in differential-algebraic equations (DAEs) of the form \[ \begin{gathered} E(t)\mathcal{Y}'(t) + F(t)\mathcal{Y}(t) = f(t),t \in [a,b], \hfill \\ B_a \mathcal{Y}(a) + B_b \mathcal{Y} (b) = \beta , \hfill \\ \end{gathered} \] where $E( \cdot )$, $F( \cdot )$, and $f( \cdot )$ are sufficiently smooth and the DAE initial value problem (IVP) is solvable. $E(t)$ may be singular on $[a,b]$ with variable rank, and the DAE may have an index that is larger than one. When $E(t)$ is nonsingular, the singular theory reduces to the standard theory for ODEs. The convergence results for backward differentiation formulas and Runge–Kutta methods for several classes of DAE IVPs are applied to obtain convergence of the corresponding shooting and finite-difference methods for these DAE boundary value problems. These methods can be implemented directly without having to (1) regularize the system to a lower index DAE or ODE or (2) convert the system to a particular canonical structure. Finally, some numerical experiments that illustrate these results are presented.