Difference Methods for the Numerical Solution of Time-Varying Singular Systems of Differential Equations

摘要

This reprint introduces a class of difference methods for the numerical solution of differential equations of the form A(t)x'+B(t)x(t)=f(t) where A, B, and f are assumed sufficiently smooth in t in the interval I = (0,T) and A(t) is identically singular on I. These methods are straightforward extensions of the well-known Gear's backward difference methods (BDF's) and correspond to BDF's whenever A is constant. It is shown that the modified methods (MBDF's) work whenever the system can be transformed to a constant coefficient problem by a change of variable x = Ly, and also whenever a related system can be transformed into a certain canonical form. The author also investigates the relationship between the convergence of BDF's and the continuous regularization of the system by its pencil perturbation. In particular, he shows the existence of examples where the BDF's converge but the pencil perturbation is not a continuous regularization. (Author)