Asymptotic expansions for regularized state-dependent neutral delay equations
Nicola Guglielmi and Ernst Hairer
Abstract.
Singularly perturbed delay differential equations
arising from the regularization of state dependent neutral delay
equations are considered. Much insight into the solution of the
regularized problem and into its limit for epsilon to zeo is obtained by the
study of asymptotic expansions. Due to discontinuities in the derivative
of the solution of the neutral delay equation and the presence of
different time scales when crossing breaking points, new difficulties
have to be managed.
A new type of expansion (in powers ofthe square root of epsilon) turns out to be necessary
for the study of the transition from weak to classical solutions.
The techniques of this article can also be applied to the study of general
singularly perturbed delay equations.
Key Words.
Neutral delay differential equations, breaking points, termination, generalized solutions,
singularly perturbed delay differential equations, asymptotic expansions.