On a regularization of index 3 differential-algebraic equations
\textit{M. Hanke} [J. Math. Anal. Appl. 151, 236--253 (1990; Zbl 0722.34013)] introduced a regularisation for linear systems of differential algebraic equations (DAEs) with time-dependent coefficients \(A(t) x'(t) + B(t) x(t) = q(t)\) in case of tractability index 2. In the paper at hand, a corresponding regularisation is performed for such systems in case of tractability index 3, where linear two-point boundary conditions are considered. Based on an according matrix chain with smooth projectors, the authors arrange a modified system of DAEs depending on a parameter \(\varepsilon\). This system is proved to be transferable for sufficiently small parameters. Furthermore, assuming some properties, a solution of the modified system yields a solution of another larger system of DAEs, which represents a singular perturbation problem with respect to the parameter \(\varepsilon\). Vice versa, a solution of the singular perturbation problem forms a solution of the modified system of DAEs under analogue conditions. The authors prove that the norm of a projected difference between the solution of the original index-3 DAEs and the solution of the modified parameter-dependent system exhibits the magnitude \(\mathcal{O}(\sqrt{\varepsilon})\). Finally, two examples of initial value problems involving index-3 DAEs are discussed, where the exact solution is known explicitly. The corresponding modified systems are solved numerically via the implicit Euler scheme using several parameters \(\varepsilon\). The resulting differences verify that the predicted order of convergence is achieved or represents at least a lower bound.
scientific article; zbMATH DE number 5078424