A stable numerical approach for implicit non-linear neutral delay differential equations
The paper considers implicit neutral functional differential equations or functional differential equations n Hale's form: \[ (y(t)-G(t,y(t-\tau(t))))' = F(t,y(t),y(t-\tau(t))), \quad t \geq t_0 \] under suitable conditions on \(F,G\) which ensure the existence of a unique solution subject to the specification of a suitable initial function. After preliminaries, the authors provide a reformulation of the equation as a delay differential algebraic equation and use the reformulated problem as the basis for an analysis of boundedness and stability of the zero solution under small perturbations in the initial function. Although the analysis is given for the nonlinear problem, careful attention is paid to the important linear case. Numerical schemes for solution of the problem are considered, based on a continuous Runge-Kutta method, and the authors give conditions under which the stability of the underlying problem is preserved in the numerical approximation. These results are backed up by the use of examples in which they are able to illustrate the stability properties of the Euler method and of the 2-stage Lobatto IIIC method.
implicit neutral functional differential equations
scientific article; zbMATH DE number 1960119