High Order Methods for State-Dependent Delay Differential Equations with Nonsmooth Solutions

Abstract

This work presents a theoretical basis for high order numerical methods to solve state-dependent delay differential equations of the form: \[\begin{gathered} \dot x(t) = f(t,x(t),x(\alpha (t,x(t))))\quad {\text{for }}t \in [a,b], \hfill \\ \alpha (t,x(t)) \leqq t, \hfill \\ x(t) = \phi (t)\quad {\text{for }}t \in [\bar a,a] \hfill \\ \end{gathered} \] where $\bar a = \min \alpha (t,x(t))$ for $t \in [a,b]$. The solutions to such equations typically have derivative jump discontinuities (jump points) which propagate from the initial jump point $t = a$. Thus, high order methods require an accurate determination of the location of jump discontinuities in lower order derivatives of the solution $x(t)$. The locations of these jump points are characterized as the zeros of certain nonlinear equations which themselves depend on $x(t)$. Because of this interdependence, the following nontrivial question arises: Can these unknown jump points be determined accurately enough to develop high order methods? This question is resolved affirmatively by exploiting special properties of delay differential equations.