Computing breaking points in implicit delay differential equations

Guglielmi, Nicola; Hairer, Ernst

doi:10.1007/s10444-007-9044-5

Cited by 65 publications

(69 citation statements)

References 15 publications

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“…We include results for a new version [10] of RADAR5 (Guglielmi and Hairer [9]), and DDE SOLVER (Thompson and Shampine [20]). We have set all absolute tolerances and the relative tolerance to TOL with TOL=10 −6 ,10 −9 for all solvers.…”

Section: Resultsmentioning

confidence: 99%

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

ZivariPiran

Enright

2009

Numer Algor

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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.

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Section: Resultsmentioning

confidence: 99%

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

ZivariPiran

Enright

2009

Numer Algor

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“…This is the general form of a differential-algebraic delay equation. Codes that are written for such systems (like RADAR5 [GH01,GH08]) can therefore be applied to neutral state-dependent delay equations.…”

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confidence: 99%

Asymptotic Expansions for Regularized State-Dependent Neutral Delay Equations

Guglielmi¹,

Hairer²

2012

SIAM J. Math. Anal.

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Abstract. Singularly perturbed delay differential equations arising from the regularization of state dependent neutral delay equations are considered. Asymptotic expansions of their solutions are constructed and their limit for ε → 0 + is studied. 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 two-dimensional dynamical system is presented which characterizes whether classical or weak solutions are approximated by the regularized problem. A new type of expansion (in powers of √ ε) 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.

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“…In addition, Baker and Willé [23], investigated the propagation of discontinuities in the solutions of scalar and systems of DVIDEs. Guglielmi and Hairer have discussed how to accurately compute these breaking points in [24]. The use of arbitrary meshes will in general result in a reduction in the order of accuracy due to the presence of these discontinuities.…”

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confidence: 99%

Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays

Shakourifar¹,

Enright²

2011

SIAM J. Sci. Comput.

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Abstract.Volterra integro-differential equations with time-dependent delay arguments (DVIDEs) can provide us with realistic models of many real-world phenomena. Delayed LoktaVolterra predator-prey systems arise in Ecology and are well-known examples of DVIDEs first introduced by Volterra in 1928. We investigate the numerical solution of systems of DVIDEs using an adaptive stepsize selection strategy. We will present a generic variable stepsize approach for solving systems of neutral DVIDEs based on an explicit continuous Runge-Kutta method using defect error control and study the convergence of the resulting numerical method for various kind of delay arguments. We will show that the global error of the numerical solution can be effectively and reliably controlled by monitoring the size of the defect of the approximate solution and adjusting the stepsize on each step of the integration. Numerical results will be presented to demonstrate the effectiveness of this approach.

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Computing breaking points in implicit delay differential equations

Cited by 65 publications

References 15 publications

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

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

Asymptotic Expansions for Regularized State-Dependent Neutral Delay Equations

Reliable Approximate Solution of Systems of Volterra Integro-Differential Equations with Time-Dependent Delays

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