An efficient unified approach for the numerical solution of delay differential equations

Abstract

In this paper we propose a new framework for designing a delay differential equation (DDE) solver which works with any supplied initial value problem (IVP) solver that is based on a standard step-by-step approach, such as Runge-Kutta or linear multi-step methods, and can provide dense output. This is done by treating a general DDE as a special example of a discontinuous IVP. Using this interpretation we develop an efficient technique to solve the resulting discontinuous IVP. We also give a more clear process for the numerical techniques used when solving the implicit equations that arise on a time step, such as when the underlying IVP solver is implicit or the delay vanishes.The new modular design for the resulting simulator we introduce, helps to accelerate the utilization of advances in the different components of an effective numerical method. Such components include the underlying discrete formula, the interpolant for dense output, the strategy for handling discontinuities and the iteration scheme for solving any implicit equations that arise.

Accurate First-Order Sensitivity Analysis for Delay Differential Equations

“…We have developed and implemented an approach that determines accurate and reliable approximations to the first-order sensitivities of the solution of a system of DDEs. The approach can be applied to any numerical DDE method with discontinuity location capability (such as those discussed in [16] or [12]). It is shown that (as the specified tolerance, TOL, goes to zero) the max error in the sensitivities will be bounded by a small multiple of TOL.…”

“…In [16] we showed that DDEs can be considered as a special subclass of discontinuous IVPs. Here we briefly review this correspondence.…”

“…We use a simple system of DDEs to avoid notational complications. The transformation for general DDEs is discussed in [16].…”

Angular dependence of the acoustic signal of a microbubble cloud

“…We exploited the fact that the delay times of the delayed terms are always greater than zero, which is valid, because the physical distance between bubbles is always positive. This allows the DDE system to be formulated as a series of initial value problems (Zivaripiran and Enright, 2009). Essentially, we created a feedback loop, where the pressure emitted by the bubbles are fed back into the pressure input terms of the bubble system every time step smaller than the smallest delay time.…”