State-dependent Delay Differential Equations Research Articles

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

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.

Characterization of jump discontinuities for state dependent delay differential equations

Cooke discussed the appropriateness of this model, and he presented some asymptotic theory. As early as 1960 Driver [8-111 pointed out that the two body problem of classical electrodynamics is properly modeled as an FDE with state dependent delays. Furthermore, he was (to the best of our knowledge) the first author to present an existence and uniqueness theory for such equations. The growth of interest in FDE has been stimulated by their wide-spread applications. Various forms of FDE provide the basis for mathematical models in numerous areas, including neural network theory, theory of learning, epidemiology, pharmacokinetics, and time lag control processes. For more detailed study of applications of FDE see [27]. In this paper we shall consider the scalar equation

Uniqueness of solutions of state dependent delay differential equations

Uniqueness of solutions of state dependent delay differential equations